SOL G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and d) using deductive reasoning, including the law of syllogism. Hints and Notes Conditional: If p, then q. p → q Inverse: Insert “nots” If not p, then not q. ~p → ~q Converse: Change If q, then p. q→ p Contrapositive: Change order and insert “nots” ~q → ~p Symbols: ~ not and or → if then, implies ↔ if and only if therefore Biconditional: p if and only if q. p↔q Law of Syllogism: If a→b and b→c, then a→c. Law of Detachment: If a→b is true. a is true. b is true. PRACTICE G.1 Which is the converse of the sentence, “If Sam leaves, then I will stay.”? A If I stay, then Sam will B If Sam does not then I will not stay C If Sam leaves then I will not stay. D If I do not stay, then Sam will not According to the Venn Diagram, is true? A Some football players play offense and defense B All football players play defense C No football players play offense and defense D All football players play offense or defense Let a represent “x is an even number.” Let b represent “x is a multiple of 4.” When , of the following is true? A B C D Which statement is the inverse of “If the waves are small, I do not go surfing”? A If the waves are not small, I do not go surfing. B If I do not go surfing, the are small. C If I go surfing, the waves are not small. D If the waves are not small, I go surfing.
HINTS and NOTES Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Ex: AB + BC > AC AC + BC > AB AB + AC > BC To determine whether a triangle can exist: Add the 2 smallest and that must be greater than the third side To determine the range of a third side: The 3rd side must be greater than the 2nd side – 1st side OR less than the 1st side + 2nd side & NOTES Comparing sides and angles of a triangle: If one side of a triangle is longer than another side, then the opposite the longer side is larger PRACTICE G.5 On the shores of a river, surveyors marked locations A, B, and C. The measure of , and the measure of . Which lists the distances between these locations in order, greatest to least? A A to B, B to C, A to C B B to C, A to B, A to C C A to B, A to C,B to C D A to C, A to B, B to C Which of the following could be lengths of the sides of ? A B C D In the drawing, the measure of and the measure of . Which is the shortest side of the triangle? A B C D Sara is building a triangular pen for her pet rabbit. If two of the sides measure 8 feet and 15 feet, the length of the third side could be
and Drag. These questions give you the choices for your answer or answers. You must click on each correct answer and drag it to the appropriate box. You must get all of them correct to get the answer correct. For our purposes, just write the correct answers in the boxes. Determine the reason to justify each statement. You may choose from the Definitions, Theorems, and Postulates listed in the 2-column Given AAS Transitive Property of Congruence SAS Reflexive Property of Congruence ASA Symmetric Property of Congruence SSA If 2 sides of a triangle are congruent, then the angles opposite them are congruent Substitution Property of Equality Definition of Angle Bisector Definition of Congruency Definition of Isosceles Triangle Reflexive Property of Equality A Kite has 2 congruent sides Symmetric Property of Equality bisects . is isosceles with base . Prove: Statements Reasons 1. bisects . 1. Given 2. 2. 3. 3. 4. 4. 5. 5. Determine if the pair of triangles is similar. If similar, highlight the similarity theorem to the box next to the triangle If not, then highlight “Not Similar”. Not Similar AA Similarity SAS Similarity SSS Similarity Not Similar AA Similarity SAS Similarity SSS Similarity Not Similar AA Similarity SAS Similarity SSS Similarity Not Similar AA Similarity SAS Similarity SSS Similarity
HINTS and NOTES Sum of the exterior angles of any polygon is always = 360o. Remember: Number of sides = number of interior angles Equilateral – means all sides are equal Equiangular - all angles are congruent. Regular – means all sides are equal in length and all angles are equal in measure Practice G.10 Some of the angle measures are given for one of the fish shaped polygons in this tessellation. What is the value of x? A B C D In the drawing, a regular polygon is partially covered by the trapezoid. How many sides does the covered polygon have? A 8 B 6 C 5 D 4 The figure has angle measures as shown. What is the measure of ?
In the diagram below of circle O, chords and are drawn such that. Identify one pair of inscribed angles that are congruent to each other by highlighting the angles and give their measure. SOL G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. HINTS and NOTES Circle- locus of points equidistant from a given point (center) Standard Form Equation of Circle- Coordinates of the center of the circle are (h, k) and r is the length of the radius. Write the standard equation of a circle with its center at the origin and radius of 7. A B C D Write the standard equation of a circle with center (4, -4) and radius 4. A B C D The standard equation of a circle with center (-4, 3) and radius 7 is ___. A B C D