Unit 3, Lesson 10
Other Conditions for Triangle Similarity (Optional)
Geometry
10.1
Warm-up
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
The purpose of this activity is to remind students of strategies they can use to show that triangles are congruent. Students must use the structure of the images to find additional angles or sides that can be shown to be congruent (MP7). Because the triangles have multiple ways in which they could be proven to be congruent, comparing ideas brings out structure.
Launch
Display one problem at a time. Give students quiet think time for each problem, and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Supports accessibility for: Memory; Organization
Activity
NoneIs there enough information to determine if the pairs of triangles are congruent? If so, what theorem(s) would you use? If not, what additional piece of information could you use?
- Triangle and triangle
- Triangle and triangle
- Triangle and triangle
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. After each response, ask students if anyone found another way to show congruence or has another explanation for why the triangles are not congruent.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example:
- "First, I _____ because _____."
- "I noticed _____, so I _____."
Some students may benefit from the opportunity to rehearse with a partner what they will say before they share with the whole class.
Advances: Speaking, RepresentingLesson Synthesis
Ask students, “Which of these pairs of triangles can we now prove are similar using one of our shortcuts?” (The pair of triangles in A are similar by the Angle-Angle Triangle Similarity Theorem. The pair of triangles in B are similar by the Side-Angle-Side Triangle Similarity Theorem. There’s not enough information to decide if the triangles in C or D are similar. The triangles in E are congruent by the Side-Side-Side Triangle Congruence Theorem, and all congruent triangles are similar.)
A
B
C
D
E
Student Lesson Summary
Besides the Angle-Angle Triangle Similarity Theorem, what other criteria are sufficient to prove triangles similar?
When two sides of one triangle are proportional to two corresponding sides of a second triangle, using the same scale factor, , and the pair of angles between these sides are congruent, then the triangles are similar by the Side-Angle-Side Triangle Similarity Theorem.
For example, angles and are vertical angles and so they are congruent, and there are two pairs of corresponding sides with the same scale factor.
Two triangles, E D F and B D C. Angles E D F and B D C are vertical angles. Point E prime plotted on segment D E. Point F prime plotted on segment F D. Segment E prime F prime drawn. Segment D C labeled k h. Segment D B labeled k j. Segment D E prime labeled k j. Segment D F prime labeled k h.
Dilate triangle using center and a scale factor of . Because , is now congruent to , and is congruent to . The dilation did not change the size of the angles. Therefore, triangle is congruent to triangle by the Side-Angle-Side Triangle Congruence Theorem. This means that there is a sequence of rigid motions that takes triangle to triangle . That means that triangle is similar to triangle because there is a dilation and a sequence of rigid motions that takes one to the other. There wasn’t anything special about these two triangles. Therefore, any pair of triangles with two pairs of sides whose lengths are in the same proportion and with the angle between them congruent must be similar.
We can also show that if all three pairs of corresponding sides are proportional and use the same scale factor, , this is sufficient to prove that the triangles are similar. We call this the Side-Side-Side Triangle Similarity Theorem.