options: Define trigonometric ratios and solve problems involving right triangles.

Solve real-world and mathematical problems involving area, surface area, and volume.

Prove geometric theorems.

Analyze, compare, create, and compose shapes.

Use coordinates to prove simple geometric theorems algebraically.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Make geometric constructions.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Understand similarity in terms of similarity transformations.

Understand and apply theorems about circles.

Classify two-dimensional figures into categories based on their properties.

Prove theorems involving similarity.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Find arc lengths and areas of sectors of circles.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Understand and apply the Pythagorean Theorem.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Explain volume formulas and use them to solve problems.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand congruence in terms of rigid motions.

Apply trigonometry to general triangles.

Apply geometric concepts in modeling situations.

Translate between the geometric description and the equation for a conic section.

Reason with shapes and their attributes.

Experiment with transformations in the plane.
problem_activity: Optional activity
This is a matching activity where each student receives a card showing a triangle and works to form a group of three. Each card has a triangle with the measure of only one of its angles given. Students use what they know about transformations and estimates of angle measures to find partners with triangles congruent to theirs. Each unique triangle’s three interior angles are then displayed for all to see. Students notice that the sum of the measures of the angles in each triangle is 180 degrees.
During this activity, students can use MP7, thinking about applying rigid motions to their triangle to see if it might match up with another student's triangle. Or they may identify that their triangle is acute, right, or obtuse and use this as a criterion when they search for a congruent copy (also MP7).
You will need the Find All Three blackline master for this activity.
Launch
Provide access to geometry toolkits. Distribute one card to each student, making sure that all three cards have been distributed for each triangle. (If the number of students in your class is not a multiple of three, it’s okay for one or two students to take ownership over two cards showing congruent triangles.) Explain that there are two other students who have a triangle congruent to theirs that has been re-oriented in the coordinate plane through combinations of translations, rotations, and reflections. Instruct students to look at the triangle on their card and estimate the measures of the other two angles. With these estimates and their triangle in mind, students look for the two triangles congruent to theirs with one of the missing angles labeled.
Prepare and display a table for all to see with columns angle 1, angle 2, angle 3 and one row for each group of three students. It should look something like this:
| student groups | angle 1 | angle 2 | angle 3 |
|----------------|---------|---------|---------|
|                |         |         |         |
|                |         |         |         |
Once the three partners are together, they complete one row in the posted table for their triangle’s angle measures. Whole-class discussion to follow.
Students might ask if they can use tracing paper to find congruent triangles. Ask how they would use it and listen for understanding of transformations to check for congruence. Respond that this seems to be a good idea.
Representation: Provide Access for Perception.
Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory
Student Facing
Your teacher will give you a card with a picture of a triangle.
The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.
Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
Enter the three angle measures for your triangle on the table your teacher has posted.
Student Response
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Activity Synthesis
Begin the discussion by asking students:
"What were your thoughts as you set about to find your partners?"
"How did you know that you found a correct partner?"
Expect students to discuss estimates for angle measures and their experience of how different transformations influence the position and appearance of a polygon.
Next look at the table of triangle angles and ask students:
"Is there anything you notice about the combinations of the three angle measures?"
"Is there something in common for each row?"
Guide students to notice that the sum in each row is the same, 180 degrees. Ask whether they think this is always be true for any triangle. Share with students that in the next activity, they will work towards considering whether this result is true for all triangles.


correct_option_index: 15