Angles on a Straight Line Add to 180 Angles On One Side of A Straight Line Angles on one side of a straight line will always add to 180 degrees. If a line is split into 2 and you know one angle you can always find the other one. 30° + 150° = 180° Example: If we know one angle is 45° what is angle "a" ? Angle a is 180° − 45° = 135° This method can be used for several angles on one side of a straight line. Example: What is angle "b" ? Angle b is 180° less the sum of the other angles. Sum of known angles = 45° + 39° + 24° Sum of known angles = 108° Angle b = 180° − 108° Straight Angles A straight angle is 180 degrees This is a straight angle A straight angle changes the direction to point the opposite way. Sometimes people say "You did a complete 180 on that!" ... meaning you completely changed your mind, idea or direction.

All the angles below are straight angles: Angles Angles Angle - The union of two line segments or two rays that have a common endpoint. Vertex - The point where two rays or two line segments meet. Ray - Part of a line that starts at one endpoint and continues in one direction. The arrows on the sides of the angle tell you that you can extend the sides (These are called rays). The space between the two rays (the two sides) is what we call an angle. The point where the two rays meet is called the vertex. An angle separates a plane into three parts: the interior of the angle, the exterior of the angle, and the angle itself. The exterior of the angle includes all the points in the plane not on the angle or in its interior. Neither the interior nor the exterior of an angle contains points on the angle. What letter represents the vertex of the angle shown to the right? Enter your answer here and press tab External Links to Extend Basic Understanding Top of Page Angles in Your World Carpenters use many different tools to make sure that all the pieces of wood they are working with fit together. One tool they use is a miter box. A miter box guides the saw so that the correct angle is cut.

You can see a corner that was cut with a miter box probably in your own home or at your school. For instance, look at the top of a wooden door frame (the wood around the opening of the door). A carpenter used a miter box to cut the wood so that the pieces fit together to form the corner of the frame. Building picture frames is another area where using a miter box to cut angles is useful. Frame shop personnel also cut mats at angles to accent pictures and paintings. Top of Page Naming an Angle An angle can be named by its vertex, or it can be named using a point from each side and the vertex. So the angle below can be called angle B, angle ABC or angle CBA. We can use the symbol to mean "angle." So, instead of saying angle B, we could say B. What are three different ways to name the angle to the right? Use the less than sign, <, to represent an angle symbol when you input your answer. Enter each answer in the left column followed by pressing tab External Links to Extend Basic Understanding Classifying Angles at Math.com : Read about and practice classifying and naming angles. Angle Bisector - A ray dividing an angle into two equal parts.

A ray (BD) is an angle bisector of ABC if and only if D is in the interior of the angle and ABD is congruent to Check for Understanding Ray MA is an angle bisector of <TMH. If the measure of <TMA is 26 degrees, then state the measure of: a) <AMH, and Enter each answer in the left column followed by pressing tab a) Harcourt's Animated Math Glossary : Read another definition of an angle bisector. Related Investigations on InterMath to Challenge Teachers Relationships : Explore various relationships in a given construction involving a parallelogram. Angle bisector : Explore relationships of parallelograms and angle bisectors. Inscribed quadrilateral : Do the angle bisectors of the opposite sides of an inscribed quadrilateral meet at right angles? Angle Bisector : Given a triangle and an angle bisector, determine proportions comparing the lengths of the various segments that form the triangle. Moving Walls : Determine the ideal position to stand when you are surrounded. Degree of an Angle - The unit for measuring angles. Protractor - A measuring device that can be used to approximate the measure of an angle.

An angle is classified by its measure (the number of degrees or amount space between the two rays forming the angle). Various classifications of angles are in the following table. Acute Angle An acute angle is an angle whose measure is between 0 and 90 . As you can see an acute angle is bigger than 0 and less than 90 A right angle forms a square corner and measures exactly 90 . An obtuse angle is greater than 90 but less than A straight angle is formed by a straight line and measures exactly 180 Reflex Angle A reflex angle measures more than 180 , but less than 360 See also Definitions Page: Triangles Types of Angles in Your World Buildings and architectural drawings often display good examples of right angles. There are a lot of right angles in the drawing of the Sears Tower, a skyscaper in Chicago, shown at right. In the floor plan of an apartment shown below, locate a right angle, obtuse angle, and an acute angle. Why do you think there are fewer acute angles in the plan than right or obtuse angles? Some buildings don't contain very many right angles, as you can see in the model for a Concert Hall below.

The architect, Frank Gehry, is known for using unusual curved shapes in his designs. Check for Understanding Identify the types of angles in the diagram to the right: a) <CIE, c) <BIC (region from B to C in counterclockwise direction) d) <CIA, and Choose each answer from the menus. a) Angles at Hamilton's "Math to Build On" : Read more about parts of angles and naming them. Related Investigations on InterMath to Challenge Teachers Slopes and Angles of Elevation : Determine the relationship between the slope of a line and the angle the line makes with a horizontal line. Tessellation Restrictions : Determine which polygons can tessellate. Sum of Angles in a Polygon : Explore the relationship between the number of sides of a polygon and the sum of its interior angles. Twice Reflected over Intersecting Lines : Find an equation which relates the angles formed by intersecting lines and points reflected over these lines. Angles in a Circle : Investigate how the location of an angle's vertex affects its relationship to the measure(s) of the arc(s) it intercepts.

Inscribed Angle - An angle is an inscribed angle if and only if its vertex lies on a circle and its sides contain chords of the circle. Examples of Inscribed Angles Non-Examples of Inscribed Angles This angle is not inscribed because although the angle's vertex lies on a circle, one of its sides does not contain a chord of the circle. This angle is not inscribed because the angle's vertex does not lies on a circle. See also Definitions Page: Circle Check for Understanding Which of the following angles is an inscribed angle? Enter your answer to the left and press tab < External Links to Extend Basic Understanding Circles at Math.com : Scroll down the page to read another definition of an inscribed angle. Related Investigations on InterMath to Challenge Teachers Angles in a Circle : Investigate how the location of an angle's vertex affects its relationship to the measure(s) of the arc(s) it intercepts. Top of Page Location, Location, Location Interior Angle - An interior angle of a polygon is formed by two adjacent sides. Also, when a third line (a transversal) crosses two other lines, the angles formed in between the other two lines are called interior angles.

(see alternate interior angles) Exterior Angle - An exterior angle of a polygon is formed when you extend a side of a polygon. Also, when a third line (a transversal) crosses two other lines, the angles formed outside the other two lines are called exterior angles. (see alternate exterior angles) An angle is an exterior angle of a polygon if and only if it forms a linear pair with one of the angles of the polygon. An interior angle of a polygon is formed by two consecutive sides of a polygon. You will sometimes see an interior angle called a vertex angle. See also Definitions Page: Polygon Check for Understanding Use three letters to identify: a) an interior angle of the quadrilateral, and b) an exterior angle of the quadrilateral. Use the less than sign, <, to represent an angle symbol when you input your answer. Enter each answer in the left column followed by pressing tab a) External Links to Extend Basic Understanding Exterior Angles Conjectures : Read more about the exterior angles of polygons. Related Investigations on InterMath to Challenge Teachers Sum of Angles in a Polygon : Explore the relationship between the number of sides of a polygon and the sum of its interior angles.

Sum of Exterior Angles : Determine the sum of the exterior angles in any polygon. Interior angles : Explore the sum of interior angles of a quadrilateral. Outside Looking In : Identify the sum of the exterior angles in a triangle. Remotely Interior : Compare exterior angles with remote interior angles. Top of Page Pairs of Angles Adjacent angles are angles in the same plane that have a common vertex and a common side but no common interior points. For example, angle HKF and angle FKI are adjacent angles. Can you find some more pairs of adjacent angles in the diagram? How many adjacent angles are in the picture at the left? Vertical angles (or opposite angles) are two nonadjacent angles formed by intersecting lines. Their sides form two lines (EF & GH); the vertical angles are the ones opposite each other (EKG & HKF). Notice that the angles have only one point in common, the vertex (K). How many pairs of vertical angles are in this diagram? What is the relationship between the angles opposite each other? Congruent angles are the same size; they have exactly the same measure. Perhaps you noticed that all vertical angles are also congruent angles. Complementary angles are two angles whose sum is 90 .

Notice that at all times angle BAT + angle CAT = 90 .The diagram to the left is the typical representation of complementary angle but the angles do not have to be adjacent as shown here to be complementary. Supplementary angles are two angles whose sum in 180 . Notice that at all times angle ABG + angle CBG = 180 . If they are the angles are adjacent (as in this diagram) they may be called a linear pair because they form a straight line. In the figure at the left, angles 1, 2, 3, and 4 are exterior angles. Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. Angles 1 and 4 are alternate exterior angles. Angles 2 and 3 are also alternate exterior angles. Line t (red) is called a transversal, a line crossing two or more lines. When the transversal cuts through 2 parallel lines, the alternate exterior angles formed have a special relationship. Using GSP, what do you notice about this situation?

In the figure at the left, angles 1,2,3, and 4 are interior angles. Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. Angles 1 and 4 are alternate interior angles. Angles 2 and 3 are also alternate interior angles. Line t (red) is called a transversal, a line crossing two or more lines. When the transversal cuts through 2 parallel lines, the alternate interior angles formed have a special relationship. Using GSP, what do you notice in this situation? In the figure at the left, angles 1 and 3 are corresponding angles. Angles 2 and 4 are also corresponding angles. Corresponding angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are in the same relative position. For example, both angles 1 and 3 are in the upper left position at each intersection. Line t (red) is called a transversal, a line crossing two or more lines.

When the transversal cuts through 2 parallel lines, the alternate interior angles formed have a special relationship. Using GSP, what do you notice in this situation? Pairs of Angles in Your World The railing on a set of stairs provides a good example of corresponding angles. Can you spot pairs of corresponding angles in the picture below? The use of alternate interior angles allows an ironing board to stay parallel to the floor or a clothes rack to stay parallel to a wall. The Math Dude : Why Does a Triangle Have 180 Degrees? :: Quick and Dirty Tips ™ Cheryl Butler Why Does a Triangle Have 180 Degrees? How many degrees do the three angles of a triangle contain? They've got 180 of 'em, right? Why 180 and not some other number? And do all triangles really contain 180 degrees? Keep on reading to find out! By May 21, 2016 Episode #216 You know how the angles of a triangle always add up to 1800? Why is that?

After all, 1800 is the angle that stretches from one side of a straight line to another—so it’s kind of weird that that’s the number of degrees in the angles of a triangle. What in the world does a triangle have to do with a single straight line? As it turns out, quite a lot. And triangles also have a lot to do with rectangles, pentagons, hexagons, and the whole family of multi-sided shapes known as polygons. We’ll see exactly what I mean by this over the next few weeks. But for today, we’re going to start by figuring out exactly why it is that the angles of a triangle always add up to 1800. Or so you thought…because we're also going to see that sometimes they don't.; Interior and Exterior Angles Before we get too far into our story about triangles and the total number of degrees in their three angles, there's one little bit of geometric vocabulary that we should talk about. And that is the difference between an interior and an exterior angle. The easiest way to describe the difference between these two things is with an example. Since today's theme is the triangle, let's talk about the interior and exterior angles of a triangle.

In short, the interior angles are all the angles within the bounds of the triangle. In other words, they're the kind of angles we've been talking about all along. The exterior angles of a triangle are all the angles between one side of the triangle and the line you get by extending a neighboring side outside the bounds of the triangle. If you think about it, you'll see that when you add any of the interior angles of a triangle to its neighboring exterior angle, you always get 1800—a straight line. Why Do Triangles Have 180 Degrees? Which brings us to the main question for today: Why is it that the interior angles of a triangle always add up to 1800? As it turns out, you can figure this out by thinking about the interior and exterior angles of a triangle. To see what I mean, either grab your imagination or a sheet of paper because it’s time for a little mathematical arts-and-crafts drawing project. Start by drawing a right triangle with one horizontal leg, one vertical leg, and with the hypotenuse extending from the top left to the bottom right.

Now make a copy of this triangle, rotate it around 1800, and nestle it up hypotenuse-to-hypotenuse with the original (just as we did when figuring out how to find the area of a triangle ). Finally, make yet another copy of the original triangle and shift it to the right so that it’s sitting right next to the newly-formed rectangle. With me so far? If so, your picture should look like this: What’s the point of this picture? Take a look at the interior angle at the bottom right of the original triangle (the one labeled “A”). Now take a look at the two angles that make up the exterior angle for that corner of the triangle (the ones labeled “B” and “C”). As we know, if we add up the interior and exterior angles of one corner of a triangle, we always get 1800. And our little drawing shows that the exterior angle in question is equal to the sum of the other two angles in the triangle. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the angle in the bottom right corner to make a 1800 angle.

For the sake of simplicity, we’ve made our drawing using a right triangle. But it turns out that you can make an exactly analogous drawing using any triangle you fancy, and you’ll always end up reaching the same conclusion. Try making a few drawings starting with different triangles of your choosing to see this for yourself. As an example, here’s another one that I’ve made: The inevitable conclusion of this game is that the interior angles of a triangle must always add up to 1800. Our lovely and elegant little drawing proves that this must be so. Can Triangles Have More Than 180 Degrees? Or does it? Might there be some limitation to our drawing that is blinding us to some other more exotic possibility? Here’s something for you to think about or try. Procure an uninflated balloon, lay it on a flat surface, and draw as close to as perfect of a triangle on it as you can. If you have a protractor handy, it’d be great to measure and add up the triangle’s interior angles and check that they’re pretty close to 1800. Now blow up the balloon and take a look at your triangle. What happened to it?

If you have that protractor, try once again to sum up its interior angles. What happened to this sum? Do you still get 1800? What does this all mean when it comes to the question of whether or not the interior angles of a triangle always add up to 1800 as we seem to have found? Unfortunately, we’re all out of time for today. But be sure to check back next time as we start exploring the strange and wonderful world known as non-Euclidean geometry. Wrap Up Please be sure to check out my book, The Math Dude’s Quick and Dirty Guide to Algebra . And remember to become a fan of The Math Dude on Facebook , where you’ll find lots of great math posted throughout the week. If you’re on Twitter , please follow me there, too. High School Trigonometry/Angles in Triangles - Wikibooks, open books for an open world High School Trigonometry/Angles in Triangles From Wikibooks, open books for an open world 9 Vocabulary The word trigonometry derives from two Greek words meaning triangle and measure.

As you will learn throughout this chapter, trigonometry involves the measurement of angles, both in triangles, and in rotation (e.g, like the hands of a clock.) Given the important of angles in the study of trigonometry, in this lesson we will review some important aspects of triangles and their angles. We’ll begin by categorizing different kinds of triangles. Categorize triangles by their sides and angles. Determine the measures of angles in triangles using the triangle angle sum. Determine whether or not triangles are similar. Solve problems using similar triangles. Triangles and Their Interior Angles[ edit ] Formally, a triangle is defined as a 3−sided polygon. This means that a triangle has 3 sides, all of which are (straight) line segments. We can categorize triangles either by their sides, or by their angles. The table below summarizes the different types of triangles. Table 1.6: Types of triangles Name A triangle with three equal sides and three congruent angles. This type of triangle is acute Isosceles A triangle with two equal sides and two equal angles. An equilateral triangle is also isosceles. Scalene A triangle with no pairs of equal sides.

Right A triangle with one 90° angle. It is not possible for a triangle to have more than one 90° angle (see below). Acute A triangle in which all three angles measure less than 90°. Obtuse A triangle in which one angle is greater than 90°. It is not possible for a triangle to have more than one obtuse angle (see below). In the following example, we will categorize specific triangles. Example 1 Determine which category best describes the triangle: a. A triangle with side lengths 3, 7, and 8. b. A triangle with side lengths 5, 5, and 5. c. A triangle with side lengths 3, 4, and 5. Solution: a. This is a scalene triangle. b. This is an equilateral, or equiangular triangle. It is also acute. c. This is a scalene triangle, but it is also a right triangle. While there are different types of triangles, all triangles have one thing in common: the sum of the interior angles in a triangle is always 180°.

You can see why this true if you remember that a straight line forms a "straight angle", which measures 180°. Now consider the diagram below, which shows the triangle ABC, and a line drawn through vertex B, parallel to side AC. Below the figure is a proof of the triangle angle sum. If we consider sides AB and CB as transversals between the parallel lines, then we can see that angle A and angle 1 are alternate interior angles. Similarly, angle C and angle 2 are alternate interior angles. Therefore angle A and angle 1 are congruent, and angle C and angle 2 are congruent. Now note that angles 1, 2, and B form a straight line. Therefore the sum of the three angles is 180°. We can complete the proof using substitution: m {\displaystyle m\angle 1+m\angle B+m\angle 2=180} m 180 {\displaystyle m\angle A+m\angle B+m\angle C=180} We can use this result to determine the measure of the angles of a triangle. In particular, if we know the measures of two angles, we can always find the third.

Example 3 Find the measures of the missing angles. a. A triangle has two angles that measures 30° and 50°. b. A right triangle has an angle that measures 30°. c. An isosceles triangle has an angle that measures 50°. Solution: The triangle is a right triangle, which means that one angle measures 90°. So we have: 180 − 90 − 30 = 60 c. 50° and 80°, or 65° and 65° There are two possibilities. First, if a second angle measures 50°, then the third angle measures 80° as 180 − 50 − 50 = 80. In the second case, the 50° angle is not one of the congruent angles. In this case, the sum of the other two angles is 180 − 50 = 130. Therefore the two angles each measure 65°. Notice that information about the angles of a triangle does not tell us the lengths of the sides.

For example, two triangles could have the same three angles, but the triangles are not congruent. That is, the corresponding sides and the corresponding angles do not have the same measures. However, these two triangles will be similar. Next we define similarity and discuss the criteria for triangles to be similar. Similar Triangles[ edit ] Consider the situation in which two triangles have three pairs of congruent angles. These triangles are similar. This means that the corresponding angles are congruent, and the corresponding sides are proportional.

In the triangles shown above, we have the following: Three pairs of congruent angles: ∠ ∠ F {\displaystyle \angle A\cong \angle D,\angle B\cong \angle E{\text{and }}\angle C\cong \angle F} The ratios of sides within one triangle are equal to the ratios of sides within the second triangle: A F ¯ {\displaystyle {\tfrac {\overline {AB}}{\overline {BC}}}={\tfrac {\overline {DE}}{\overline {EF}}},{\tfrac {\overline {AB}}{\overline {AC}}}={\tfrac {\overline {DE}}{\overline {DF}}},{\text{and }}{\tfrac {\overline {AC}}{\overline {BC}}}={\tfrac {\overline {DF}}{\overline {EF}}}} The ratios of corresponding sides are equal: A F ¯ {\displaystyle {\tfrac {\overline {AB}}{\overline {DE}}}={\tfrac {\overline {BC}}{\overline {EF}}},{\tfrac {\overline {AB}}{\overline {DE}}}={\tfrac {\overline {

AC}}{\overline {DF}}},{\text{and }}{\tfrac {\overline {BC}}{\overline {EF}}}={\tfrac {\overline {AC}}{\overline {DF}}}} Example 4 In the triangles shown above, A {\displaystyle {\overline {AB}}=8,{\overline {BC}}=7,{\overline {AC}}=5,{\text{and }}{\overline {DE}}=4} .

What are the lengths of sides DF and EF? Solution: {\displaystyle {\tfrac {\overline {AB}}{\overline {DE}}}={\tfrac {\overline {BC}}{\overline {EF}}}} , we have = 3.5 {\displaystyle {\tfrac {8}{4}}={\tfrac {7}{EF}}\Rightarrow 2={\tfrac {7}{EF}}\Rightarrow 2EF=7\Rightarrow EF={\tfrac {7}{2}}=3.5} . {\displaystyle {\tfrac {\overline {AB}}{\overline {DE}}}={\tfrac {\overline {AC}}{\overline {DF}}}} , we have = 2.5 {\displaystyle {\tfrac {8}{4}}={\tfrac {5}{EF}}\Rightarrow 2={\tfrac {5}{EF}}\Rightarrow 2EF=5\Rightarrow EF={\tfrac {5}{2}}=2.5} . Recall that these triangles are considered to be similar because they have three pairs of congruent angles. This is just one of three ways to determine that two triangles are similar.

The table below summarizes criteria for determining if two triangles are similar. Table 1.7: Criteria for similar triangles Criteria Two triangles are similar if they have have three pairs of congruent angles SSS Two triangles are similar if all three pairs of corresponding sides are in the same proportion SAS Two triangles are similar if two pair of corresponding sides are in the same proportion, and the included angles are congruent. A special case of SSS is "HL", or "hypotenuse leg". This is the case of two right triangles being similar. This case is examined in example 5 below. Example 5 Determine if the triangles are similar. Solution: The triangles are similar. Recall that for every right triangle, we can use the Pythagorean Theorem to find the length of a missing side. In ABC we have: (AC)2 + 82 = Similarly, in triangle DEF we have: (DF)2 + 42 = 3 Therefore the sides of the triangles are proportional, with a ratio of 2:1.

Because we will always be able to use the Pythagorean Theorem in this way, two right triangles will be similar if the hypotenuse and one leg of one triangle are in proportion with the hypotenuse and one leg of the second triangle. This is the HL criteria. Applications of Similar Triangles[ edit ] Similar triangles can be used to solve problems in which lengths or distances are proportional. The following example will show you how to solve such problems. Example 6 Use similar triangles to solve the problem: A tree casts a shadow that is 24 feet long. A person who is 5 feet tall is standing in front of the tree, and his shadow is 8 feet long. Approximately how tall is the tree? Solution: The picture shows us similar right triangles: the person and his shadow are the legs of one triangle, and the tree and its shadow form the legs of the larger triangle. The triangles are similar because of their angles: they both have a right angle, and they share one angle. Therefore the third angles are also congruent, and the triangles are similar. The ratio of the triangles' lengths is 3:1.

If we let h represent the height of the tree, we have: h Review Questions[ edit ] Triangle ABC is an isosceles triangle. If side AB is 5 inches long, and side BC is 7 inches long, how long is side AC? Can a right triangle be an obtuse triangle? Explain. A triangle has one angle that measures 48° and a second angle that measures 28°. What is the measure of the third angle in the triangle? Claim: the two non-right angles in any right triangle are complements. (a) Explain why this claim is true. (b) Use this claim to find the measure of the third angle in the triangle below. In triangle DOG, the measure of angle O is twice the measure of angle D, and the measure of angle G is three times the measure of angle D. What are the measures of the three angles? Triangles ABC and DEF shown below are similar. What is the length of D Angles and Lines Sundials, Gnomons, Eclipses, and Transits Astronomical events dictate the day, lunar month, and year. When man wanted to subdivide a day, the sundial proved useful.

Here you will find geometry activities with a sundial theme. There are many variations on the sundial theme. However, one vocabulary word from this is gnomon, the part of the sundial which casts the shadow. For a horizontal sundial, what is the appropriate angle this makes with the horizon? A visit here might be useful. Another study of angles involves eclipses and transits . The student is assumed to be familiar with such events, especially, the pair of venus transits in June 2004 and 2012. These are rare enough that no one alive has seen one. Pairs of Angles Angles often come in pairs which might have special names depending on their relative size or location. Adjacent angles are angles which share a side or ray. A pair of complementary angles sum to 90°. A pair of supplementary angles sum to 180°. A linear pair of angles are both supplementary and adjacent. Vertical angles share a vertex and are formed by extending each ray though the vertex. The letter X provides some good examples. The angle opening upward and the angle opening downward are vertical angles as are the pair opening left and right. Vertical angles have an easily proved property given below.

You can prove this property by noting that these two angles are supplementary to the same angle. Vertical angles are equal. Angles separate their plane into three regions: the angle; the region inside or interior to the angle; and the region outside or exterior to the angle. For angles of measure between 0° and 180°, the interior is the convex set. A ray which extends from the vertex of an angle through the interior such that it divides the angle into two congruent (same shape and measure) angles is called an angle bisector. Bisect means to cut into two equal pieces. Angles in Relative Positions Angles are also formed when a transversal intersects a pair of lines, which are not necessarily parallel. A prototypical example would be the not equal symbol: . Angles which are in the same relative position but on the other line are called corresponding angles (angles 1 and 5 in the figure below right). The word alternate is often used to indicate things on opposite sides, like some leaves on a stem. In this context it means on opposite sides of the transversal. The word interior refers to the region between the lines, whereas, the word exterior refers to the regions outside the lines.

We refer to those angles between the lines as interior angles (angles 3, 4, 5 and 6) and those outside the lines as exterior angles (angles 1, 2, 7 and 8). (Note also below the use of interior / exterior angles as applied to polygons.) Thus when we have a transversal cutting two lines (not necessarily parallel) we have alternate interior angles (angles 4 and 6 below right) and alternate exterior angles (angles 1 and 7 below right). If the pair of lines are parallel, we have the following important properties. Two lines cut by a transversal are parallel if and only if: Alternate interior angles are equal in measure; or Alternate exterior angles are equal in measure; or Corresponding angles are equal in measure. The angle between two sides of a polygon is an interior angle, whereas the angle formed by one side and extending the other side of an angle in a polygon is an exterior angle. They form a linear pair. Examples of interior angles would be those labelled x and 60º in the figure left. The angle labelled (2x+4)º is an exterior angle.

We can also differentiate between the interior and exterior of a polygon by noting that your right side is toward the interior as you travel clockwise around a closed figure or on your left side as you travel counter-clockwise or anti-clockwise as they say on the other side of the "pond." Angles from Trigonometry There are also angles which are especially important in trigonometry. First, in trigonometry the angle vertex is placed at the origin of a coordinate system with one ray coincident with the x-axis. Angles are then measured as positive if measured counter-clockwise from the x-axis or as negative if measured clockwise from the x-axis. This is called standard position. Our geometry differentiates between rotation and angle. Thus in trigonometry angles can be more than 180° or less than 0°, whereas in geometry only rotations can be. The reference angle is then the acute (or right) angle between the ray and the x-axis. It is always between 0° and 90° inclusive. An angle in standard position is a quadrantal angle whenever its terminal side lies on an axis. Quadrantal angles are thus integer multiples of 90°.

Coterminal angles are angles which share the same sides, such as 120° and -240° or 90° and 450°. Coterminal angles differ by an integral multiple of 360° or 2 radians. Angles inside circles are either central angles if their vertex is the center of the circle, or inscribed angles if their vertex is on the circle. (We assume each side includes a point on the circle other than the vertex.) A central angle intercepts a portion of the circumference of the circle termed an arc. An arc is a minor arc if its central angle is less than 180°. An arc is a major arc if its central angle is more than 180°. If an arc is exactly 180° it is termed a semicircle. Minor arcs are specified using two points (letters) under a curved line, whereas semicircles and major arcs are specified using three points. There is the following relationship between a central angle and its corresponding inscribed angle. An inscribed angle is half the measure of the central angle intercepting the same arc.

If you know the measure of a central angle, you can also calculate its arc length as the proportion between this measure over 360° equal to this unknown arc length over the circumference. Assumptions: Valid and Invalid In Geometry, we never take anything for granted, but build knowledge by proving everything step by step. This tradition dates back to the ancient Greek Pythagorean school and was codified many years later by Euclid in a 13 volume set of books called Elements. Proofs often go awry because steps are not done in a proper order or steps are omitted. Another common mistake is to make improper assumptions or not properly support assumptions. Please become familiar with the following lists! The following items can be assumed from figures: collinearity and betweenness of points on lines; intersection of lines at a given point; and points interior to, on, or exterior to an angle, n-gon, or circle. The following items cannot be assumed from figures: collinearity of points not on lines; whether or not lines are parallel or perpendicular, angles are acute or obtuse; angle measure, and segment length, especially congruence of such. Always remember: Geometric diagrams are not drawn to scale.

Although we presented many axioms about the real numbers previously in Numbers Lesson 13, we will review and summarized them again here since they apply to our geometry. Postulates of Equalities, Inequality, and Operations: Example: We presented before ( Numbers Lesson 12 ) the definition of slope and various forms of linear equations, especially y = mx + b, where m is the slope and b is the y-intercept. If two lines are parallel, then their slopes are equal. If two lines are perpendicular, then the product of their slopes is -1. Numbers whose product are -1 are termed negative reciprocals. If the slope of one line is -x and the slope of another is 1/x, then -x  1/x = -1. Perpendicular lines, of course, always form 90° angles. Other terms for perpendicular are: orthogonal, normal, and, of course, right. The transitivity property may be used to show two lines parallel to a third line are parallel to each other. This is often termed the Transitivity of Parallelism Theorem. If two lines are perpendicular to the same line, then the two lines are parallel to each other, if they are coplanar.

If a line is perpendicular to a line that is parallel to another, then the first line is perpendicular to the other two, again if they are all coplanar. The Euclidean or Plane Geometry we are studying will make this basic assumption. Proof Mathematical proofs are generally needed for the following three reasons. 1) People sometimes disagree because something obvious to one may not be so obvious to another. 2) Unexpected results can be verified. 3) Some statements resist proof for a long time. It is thus possible that it cannot be proved or just isn't true. Proof arguments can be written in many styles, just like there are many different styles of writing: essay, lists of concepts, etc. The two most common forms of proof are two column and paragraph. Examples of each have occurred already: compare the paragraph proof used in Numbers Lesson 3 to show that the primes form an infinite set with the two column proof used in Numbers Lesson 12 to derive the quadratic formula. Traditionally (since 1900) Geometry has used the two column proof with the left column containing the statement and the right column containing the corresponding justification. It is something you must become familiar with. Proofs can also be direct or indirect as described in Numbers Lesson 5 .

They can also be based on inductive or deductive reasoning as well. Rules for Constructing, Drawing, and Sketching The ancient Greeks established some simple rules for geometric construction which lead to some interesting mathematical development over the next several thousand years as discussed previously in Numbers Lesson 14. Thus the word construct means you are to follow these rules which restrict your methods considerable. Specifically, you may use only an unmarked straight-edge and a compass. These three rules are as follows. Point rule: A point must either be given or be the intersection of figures that have already been constructed. These might be called legal points. Straightedge rule: A straightedge can be used to draw the line through legal points A and B. Compass rule:A compass can be used to draw an arc containing a second legal point B. Generally this means that a compass can be lifted keeping the same radius. At other times you may be asked to draw a geometric figure. In such cases, you may also use a ruler and a protractor. Finally, there are times you may be asked to sketch a geometric figure. In such cases you are not required to use a straight-edge to ensure your lines are straight or a protractor to ensure your angles are properly measured.

The following basic construction algorithms must be mastered. More will be added later . Perpendicular bisector of a given segment. Perpendicular bisector of a segment allowing access on one side only. Regular (equilateral) triangle. Mazes and Labyrinths The following link has a maze and a free downloadable Applet to draw perfect mazes: . Perfect mazes are not only solvable, but well-connected and contain no closed loops. Mazes and labyrinths have a long history, probably evolving from defense works against an enemy. Later they evolved into magical devices against evil spirits. Labyrinths often contain blind alleys and are technically different from mazes. In Greek mythology, the Minotaur was confined in the famous labyrinth of Crete. Mazes of hedges or maize (corn) are still common in such gardens as the Versailles, Hampton Court in London, or out in Iowa. Perhaps next year we can take a local field trip to: Wick's Apple House Maize Maze or Barbott Farms. A maze having only one entrance and one exit can be solved by placing one hand against either wall and keeping it there as one traverses the maze.

This will usually not be the shortest solution, but the exit can always be reached in this manner. For this to work in a labyrinth, there must be no closed loops. If it has no closed loops it is simply connected otherwise it is multiply connected. Mazes are also used to test learning behavior particularly in mice. Below/on the next page is a maze from your homework. For homework 3.7. Straight Angle definition - Math Open Reference Math Open Reference Straight angleAlso Flat Angle Definition: An angle whose measure is exactly 180° - a straight line Try this Adjust the angle below by dragging an orange dot and see how the angle ∠ABC behaves. Note that it is a Straight Angle only when it is exactly 180° Straight (or Flat) angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. The measure of a flat angle is thus always 180° Types of angle Altogether, there are six types of angle as listed below. Click on an image for a full description of that type and a corresponding interactive applet.

Angles: Types and Labeling - Concept - Geometry Video by Brightstorm Angles: Types and Labeling Univ. of Wisconsin J.D. Univ. of Wisconsin Law school Brian was a geometry teacher through the Teach for America program and started the geometry program at his school Thank you for watching the video. To unlock all 5,300 videos, start your free trial. Angles: Types and Labeling - Concept Brian McCall Univ. of Wisconsin J.D. Univ. of Wisconsin Law school Brian was a geometry teacher through the Teach for America program and started the geometry program at his school Share angle acute right obtuse straight vertex side An Angle is something that we use throughout Geometry, we talk about it all the way to the very end when we're talking about Trigonometry. Well an angle is formed by two rays that share common end point. It's measured in degrees and is between 0 and 180 degrees, if it's over 180 then you're going to subtract that number. So let's say you had 220 degrees you're going to subtract 180 from that so it's actually a 40 degree angle.

If we look at an example where we have angle a, b, c, there's two ways that you could label this. You can write this as angle abc or since there are no other adjacent angles that is an adjacent angle would be something like this where it would share that vertex that common end point. Since there are no other adjacent angles you could also just label this based on the vertex which is b. Now there's something very specific about the way that I wrote angle abc, whenever you write the angle its vertex must be the middle letter. But what is the vertex? The vertex is this point that is the common end point of your rays. So I'm going to label this as the vertex, so the rays form what are called the sides. So bc, ray bc is one side of this angle and ray ba is another side. So again you can label an angle two different ways, one using three letters that make up the two sides and the vertex making sure that your vertex is the middle letter or if there aren't any other adjacent angles you can just label it based on its vertex.

There are four key types of angles first one is a acute, so if I drew this angle and I said that's angle x if it is acute that means that is less than 90 degrees but also greater than 0 degrees. So it has to be somewhere in between them, it cannot be exactly 90 degrees it cannot be exactly 0 degrees. A right angle if this is x is equal to exactly 90 degrees a right angle. So we're going going to label all of our right angles in Geometry using these two segments which will tell you the student that this is a 90 degree angle. The third type is an obtuse angle. So here if we measured x, x is going to be less than 180 degrees but more than 90 degrees. Because if this angle was able to be exactly 90 degrees it would be a right angle if it was less than 90 it would be acute. The last one which is key to a lot of proofs is a straight angle. If you have a straight angle it is the equal to exactly 180 degrees. Which means if we had the rotation about point x that was form the full rotation around any given point is 360 degrees.

So 2x=360 degrees. So keep this in mind and remember obtuse is going to be in between 180 and 90, right is going to be exactly 90, acute is going to be between 90 and 0. And that there are two different ways of labeling your angles. And the way that will always work is using three letters. Lines, rays, and angles - a free geometry lesson with exercises Reviews Lines, Rays, and Angles This fourth grade geometry lesson teaches the definitions for a line, ray, angle, acute angle, right angle, and obtuse angle. We also study how the size of the angle is ONLY determined by how much it has "opened" as compared to the whole circle. The lesson contains many varied exercises for students. A When two points are connected with a straight line, we get a line segment. We call this line segment AB or line segment AB (note the bar on top). The sides of a triangle are line segments. A line has no beginning point or end point. Imagine it continuing indefinitely in both directions. We can illustrate that by little arrows on both ends. We can name a line using two points on it.

This is line EF or line (note the arrowheads). Or, we can name a line using a lowercase letter: this is line s. A ray starts out at a point and continues off to infinity. We can show that by drawing an arrow at one end of the ray. Think of the sun's rays: they start at the sun and go on indefinitely. We can name a ray using its starting point and one other point that is on the ray: this is ray QP or ray (note the one arrowhead). Or, we can name a ray using a lowercase letter: this is ray r. What is an angle? Many people think that an angle is some kind of slanted line. But in geometry an angle is made up of two rays that have the same beginning point. That point is called the vertex and the two rays are called the sides of the angle. To name an angle, we use three points, listing the vertex in the middle. This is angle DEF or ∠DEF. We can use the symbol ∠ for angle. 1. Write if each figure is a line, ray, line segment, or an angle, and name it. 2. a. Find the angle formed by the rays DE and DF.

How do we name it? b. Find the angle formed by the rays CA and CE. How do we name it? c. What is BD? (a line, a line segment, or a ray)? 3. a. Draw two points, D and E. Then draw line DE. b. Draw point Q not on the line. c. Draw rays DQ and EQ. d. Find angles EDQ and DEQ in your drawing. Imagine that the two sides of the angle start side by side, and then open up to a certain point. When the two sides “open up”, they draw an imaginary arc of a circle. (You can illustrate this with two pencils as the two sides of an angle. Keep one pencil stationary while you rotate the other.) If the angle opens up to a full circle, we say the angle is 360 degrees (360°). This angle is half of the full circle, so it measures 180°. It is called the straight angle. Your two pencils (rays) are lying down flat or straight on the floor. This is one-fourth of the full circle, so it is 90°. This is called the right angle.

Table and book In each of these pictures the angle is opened more and more and keeps getting bigger. The arc of the circle is larger. These angles are acute angles, which means they are less than a right angle (less than 90°). Think of acute angles as sharp angles. If someone stabbed you with the vertex of an acute angle, it would feel sharp. The angle is opened even more now. It is an obtuse angle: an angle that is more than a right angle, yet less than a straight angle. Think of obtuse angles as dull angles. Here's another way of thinking about angles. Think of a sun rising in the morning in the horizon, gradually getting higher, and traveling through the sky along an arc of a circle. How big is the angle? It does not matter how long the sides of the angle are. Remember, they are rays, and rays go on indefinitely. But when we draw them on paper, we have to draw them as ending somewhere. The sides of the angle might even seem to have different lengths. That doesn't matter either. The size of the angle is ONLY determined by how much it has “opened” as compared to the whole circle.

Think how big an arc of a circle the sides have drawn, as compared to a whole circle. Which of these two angles is bigger? Look at how much the angle has opened? How big a part of a circle have the sides drawn? The second angle (on the right) is bigger. Illustrative Mathematics Task Letters can be thought of as geometric figures. How many line segments are needed to make the letter A? How many angles are there? Are they acute, obtuse, or right angles? Are any of the line segments perpendicular? Are any of the line segments parallel? We can build all of these letters from line segments and arcs of circles. Build all of the capital letters with the smallest number of "pieces," where each piece is either a line segment or an arc of a circle. Which letters have perpendicular line segments? Which letters have parallel line segments? Which letters have no line segments? Do any letters contain both parallel and perpendicular lines? What makes the lower case letters "i" and "j" different than all of the capital letters? IM Commentary The purpose of this task is for students to analyze the geometry of letters. Letters provide a good opportunity for students to broaden their understanding of what constitutes a 2-dimensional geometric figure.

Most students recognize polygons, circles, ellipses, and simple closed curves with some kind of symmetry as "shapes," but a 2-D geometric figure in general is simply a specified collection of points in the plane. So, for example, you can have a geometric figure that is not connected (like the lower case letters "i" and "j"), or you can have a geometric figure that is composed of a simple closed curve with a line segment sticking out of it (like the letter "P"). The benefit of working with letters as opposed to made-up figures with these characteristics is that students have already accepted them as objects, and so it seems reasonable to study them from a geometrical perspective. This task has students composing and decomposing figures, which is an important way of looking at geometric figures and a skill that students start working on in kindergarten and continue building throughout elementary school and beyond. Students should have access to a plastic or metal tracing ruler with different sized circles, a straight edge, and some colored pencils or markers. Students are often confused about whether two line segments are parallel or perpendicular. For example, parallel lines are lines that never meet. Does this mean that these line segments are parallel?

Even though these segments do not meet, the answer is "no." The trick is to realize that every line segment is contained in an (infinite) line. So two line segments are parallel if and only if the lines that contain them are parallel. Likewise, two line segments are perpendicular if and only if the lines that contain them are perpendicular. This is what allows us to see that the letter F, for example, is composed of three line segments, two of which are parallel to each other and both of which are perpendicular to the third. Note that whenever two lines segments meet at an endpoint, two angles are implicitly defined. Often students only focus on the angle that is less than 180 degrees, but for example, the letter L can be thought of as defining both an angle that is 90 degrees and one that is 270 degrees. It would be good to address this directly in a whole-class discussion. Solution The letter A is composed of 3 line segments which meet in three places and form 5 angles less than 180 degrees. Three of these angles are acute, and two are obtuse.

Note that students might also count angles that are greater than 180 degrees, so it is important for students to explicitly identify the angles they see. For example, a student might see an angle greater than 180 degrees at the top of the letter A. Alternatively, the a student might see the letter A as being composed of 5 line segments and see two straight-angles on the sides of the letter A where the horizontal segment meets them. Of course, this isn't the smallest number of pieces needed to build the letter A, but it is a correct decomposition and analysis of it. None of the line segments are perpendicular or parallel. The hardest ones are the ones composed of both segments and curves.

B can be made from 4 segments and two semi-circles, for example: These letters have perpendicular line segments: B, D, E, F, H, L, P, R, T For example, Here is an analysis of the letter B which is composed of 4 line segments (three of which are parallel to each other and are all perpendicular to the fourth) and two arcs of a circle: These letters have parallel line segments: B, D, E, F, H, M, N, R, U, Z These letters have no line segments: C, O, S. Depending on the font (or your handwriting), Q might or might not contain a line segment. These letters have both parallel and perpendicular line segments: B, D, E, F, H, R The lower case letters "i" and "j" are different than all of the capital letters because they are disconnected. That is, they are made out of pieces that don't touch.

Question: How many degrees are there in the angle on either side of a straight line?
Answer: 180