The purpose of this activity is for students to use 180-degree rotations to create a diagram of 3 congruent triangles. As students create the diagram, they prepare informal arguments for congruent angles in the figure. Using these informal arguments, they find the sum of the interior angle measures of the original triangle.
A similar diagram will be used to generalize the sum of interior angles of a triangle in a later activity. 

Some students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the first rotation, and allow them to do the second rotation on their own.

The goal of this discussion is for students to explain how rigid transformations connect with the sum of interior angles of a triangle.
Here are some questions for discussion:

• "How can you tell that line is parallel to line ?" (They were made using a 180-degree rotation, they are both on grid lines)
• "How can you tell that the 3 angles around point make a straight angle?" (They all line up along a grid line, so they form a straight line.)
• If there were no grid lines, could we know that the three angles around point A would make a straight angle? (Since we used a 180-degree rotation of the same line segment but around 2 different points, and have to be parallel to each other. Since they share point , they have to be on the same line.)
• How does this tell us about the sum of the angles for triangle ? (A straight angle is 180 degrees, and the three angles around point that make a straight angle are congruent to the three angles in triangle , so they must also total to 180 degrees.)

Students will have more time to make the connection about parallel lines and a similar figure without grid lines, so it is okay if they cannot explain why they create a straight angle without using grid lines at this time.

In this activity, students will continue to explore the interior angles of a triangle. The purpose of this activity is to provide a complete argument, not depending on the grid, of why the sum of the three angles in a triangle is .

Students may use the structure of parallel lines cut with a transversal to identify alternate interior angles on the figure or use a 180-degree rotation about the midpoint of triangle side lengths to identify corresponding parts of the figure (MP7). In either case, this argument works for any triangle in general, since no angle measures are given or needed. 

The goal of this activity is for students to explain that the sum of the interior angles of any triangle must be 180 degrees. 

Display both of these images for all to see: 

Ask students how they are similar and how they are different.

Similarities students may notice:

• Triangle is in both images.
• Triangle is between 2 parallel lines.
• The 3 angles around point are a straight angle.
• The sum of the angles inside triangle is 180 degrees.

Differences students may notice:

• There are grid lines on the first image but not the second.
• We can measure the height of the triangles in the first image.
• The first image has 3 triangles but the second only has 1 triangle.
• The first image has parallel segments but the second has parallel lines.

Tell students that the first image helps us show that the sum of the angles in that particular triangle is 180 degrees, but we can create a diagram like the second one for any triangle. That means we can know that the sum of the angles in any triangle is 180 degrees.

This activity is optional because it provides additional opportunity for students to identify corresponding parts of figures using rigid transformations. This image will also be helpful in a later unit as students investigate the Pythagorean theorem. 

In this activity, students use the structure of rigid transformations to find angle measures (MP7). Some strategies that students may use include:

• Rigid transformations take one triangle to a congruent triangle.
• Congruent triangles have congruent corresponding angles.
• The measures of the angles that make up a straight line add up to 180 degrees.
• Parallel lines cut by a transversal have congruent alternate interior angles.

Display the image of the 4 triangles for all to see. Invite students to share how they calculated one of the other unknown angles in the image, adding to the image until all the unknown angles are filled in.

 If no student points it out, highlight that angles , , , and are all right angles.