In this activity, students practice transforming a figure on the coordinate plane. Students may choose to use tracing paper and perform these transformations as if there were no grid. Other students may notice the structure of gridlines and look for patterns in the coordinates. During the Activity Synthesis, students are reminded that rigid transformations produce congruent figures. This helps prepare students for the next activity, in which they reason that given two congruent figures, there must be a sequence of transformations carrying one figure to the other.

Making dynamic geometry software and tracing paper available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite students to share strategies such as “Reflecting across the -axis makes the -values negative and keeps the -values the same.” If students do not notice patterns like this one, there is no need to mention them. In a subsequent lesson, students will investigate the effect of transformations on coordinates.

Ask students what they notice about the three figures. (The figures are trapezoids. The figures have two right angles. All three figures are congruent.) Ask students how they know the figures are congruent. (They are congruent by definition of rigid transformations.)

In this activity students calculate side lengths and angle measures of triangles on the coordinate plane. In the process they demonstrate the two triangles are congruent. During the synthesis they discuss the minimum requirements for a proof of triangle congruence, since calculating all side lengths and angle measures goes above and beyond what is necessary. Finally, students specify a sequence of rigid transformations taking one triangle to the other.

If students are stuck on finding the measures of the angles, suggest they look at their reference chart for concepts from a prior unit that can help.

Invite a student to share their solution for calculating the measure of angle . Ask the class why they can use trigonometric ratios to find angle measures. (Triangle is a right triangle.)

Invite students to share how they determined that the triangles were congruent. Here are some questions for the discussion:

• "Did we have to show that every pair of corresponding side lengths and angles are congruent in order to determine that the triangles are congruent?" (No. It is possible to use other strategies, like congruence theorems or rigid transformations.)
• "How did calculating the side lengths help show that the triangles are congruent?" (Three side lengths means that we can use the Side-Angle-Side Triangle Congruence Theorem.)
• "Is there a different Triangle Congruence Theorem you could have used with this figure?" (Since the triangles are right triangles, we could have use the Side-Angle-Side Triangle Congruence Theorem.)
• "How does finding a sequence of rigid motions help us determine whether shapes are congruent?" (If we didn't have side lengths or angle measures, we could still use a sequence of rigid motions to show that two figures are congruent.