Unit 1, Lesson 11
Are Those Triangles Congruent?
Integrated Math 2
11.1
Warm-up
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Geometry toolkits (HS)Activity Narrative
The purpose of this activity is to determine what prior knowledge students have about triangle congruence. Students will need to recall the Side-Side-Side Triangle Congruence Theorem and identify corresponding parts of congruent triangles.
Launch
Tell students that their reference charts may be helpful for this activity. Ask students to use colored pencils to highlight corresponding sides.
Activity
NoneTriangles and are congruent.
- Which triangle congruence theorem can you use to prove the triangles are congruent?
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Name the 3 sets of corresponding sides.
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Name the 3 sets of corresponding angles.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to share what they remember about triangle congruence theorems and practice using previously learned language for triangles, congruence, and corresponding parts.
- “What are the three triangle congruence theorems? What does each of them mean?” (Angle-Side-Angle, Side-Angle-Side, and Side-Side-Side; Given a certain set of three corresponding congruent pairs in two triangles, we can be sure those two triangles are congruent.)
- “What does ‘corresponding parts’ mean for two figures?” (Corresponding parts are those that coincide after transformations.)
- “What is true about corresponding parts for congruent figures?” (Corresponding parts of congruent figures are congruent.)
MLR8 Discussion Supports. Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Advances: Speaking
Lesson Synthesis
Display parallelogram , without any congruent parts marked.
Ask students, “Suppose we want to prove triangle is congruent to triangle by the Side-Side-Side Triangle Congruence Theorem. What information would we need to write this proof?” As students share, annotate the image and ask clarifying questions, such as:
- “How do you know these two parts are congruent?” (Sides and are congruent and sides and are congruent because opposite sides of a parallelogram are congruent. Sides and are congruent because they are the same line segment.)
- “Is there another way to prove triangle is congruent to triangle ?” (Yes. Since is parallel to and is parallel to , there are two pairs of alternate interior angles that are congruent. Using any pair of congruent sides, we can prove triangle is congruent to triangle by the Angle-Side-Angle Triangle Congruence Theorem.)
Student Lesson Summary
To know whether two figures are congruent, we can show that there is a translation, rotation, or reflection, or a sequence of these rigid transformations, that takes the first figure exactly to the second figure. Alternatively, we could show that every pair of corresponding sides and angles are congruent. In a previous grade we learned some theorems that can help simplify this process for triangles.
The Side-Angle-Side Triangle Congruence Theorem allows us to prove two triangles are congruent if two pairs of corresponding sides and the pair of corresponding angles between those sides are congruent.
The Angle-Side-Angle Triangle Congruence Theorem allows us to prove two triangles are congruent if two pairs of corresponding angles and a pair of corresponding sides are congruent. Because of the Triangle Angle Sum Theorem, given the measures of two angles in a triangle, we can always find the measure of the third angle, so this theorem works even if the congruent side is not between the two given angles.
The Side-Side-Side Triangle Congruence Theorem allows us to prove two triangles are congruent if all 3 pairs of corresponding sides are congruent.
To prove that two triangles are congruent, look at the diagram and given information and think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then check to see if all the information matches the Angle-Side-Angle, Side-Angle-Side, or Side-Side-Side Triangle Congruence Theorem.