Unit 2, Lesson 3
Congruent Triangles, Part 1
Integrated Math 1
3.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
This Warm-up both reviews previous work on corresponding parts of congruent figures and previews the subsequent activity in which students will use rigid transformations to take one figure onto another.
Students start by drawing conclusions based on a congruence statement and then are given a specific figure to reason about. This gives the opportunity to talk about how, often in geometry, the static picture is just one example of many different possible images that match the given information.
Launch
Arrange students in groups of 2. First, display the congruence statement, “Triangle is congruent to triangle ” for all to see. Tell students their job is to think of at least one thing they know must be true, one thing that could possibly be true, and one thing that definitely can’t be true about each statement or image. Give students 1 minute of quiet think time and then 1 minute to discuss the things on their lists with their partner.
Then display the image for all to see. Give students an additional 1 minute of quiet think time to add to or change their lists based on what they see.
Activity Synthesis
Ask students to share what they wrote down before they saw the image. Next, invite a student to mark congruent sides and angles of the image for all to see.
If students made claims like segment can’t possibly be congruent to segment , ask for a counterexample. If you have access to the digital version of the curriculum, drag points on the applet to demonstrate the case in which triangle is isosceles with segment congruent to segment .
Lesson Synthesis
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
If all pairs of corresponding sides and all pairs of corresponding angles are congruent, then the triangles must be congruent. (Theorem)
, so .
Invite students to share the big ideas of a transformation proof. (One does a transformation and then uses what they know about a side length or angle to prove that corresponding parts coincide.)
Create a display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. There is a blackline master with the final version which will be built over several lessons. After this lesson, it should include:
Transformations:
- Translate from to .
- Rotate using as the center by angle .
- Rotate using as the center so that coincides with .
- Reflect across .
- Reflect across the perpendicular bisector of .
Justifications:
- We know the image of is congruent to because rigid motions preserve measure.
- Points and coincide after translating because we defined our translation that way!
- Since points and are the same distance along the same ray from , they have to be in the same place.
- Rays and coincide after rotating because we defined our rotation that way!
- The image of must be on ray since both and are on the same side of and make the same angle with it at .
Student Lesson Summary
If all corresponding parts of two triangles are congruent, then one triangle can be taken exactly onto the other triangle using a sequence of translations, rotations, and reflections. The congruence of corresponding parts justifies that the vertices of the triangles will line up exactly.
One of the most common ways to line up the vertices is through a translation to get one pair of vertices to line up followed by a rotation to get a second pair of vertices to line up, and if needed, a reflection to get the third pair of vertices to line up. There are multiple ways to justify why the vertices must line up if the triangles are congruent, but here is one way to do it:
First, translate triangle by the directed line segment from to . Points and coincide after translating because we defined our translation that way! Then, rotate the image of triangle using as the center, so that rays and line up.
We know that rays and line up because that’s how we defined the rotation. The distance is the same as the distance , because translations and rotations don’t change distances. Since points and are the same distance along the same ray from , they have to be in the same place.
If necessary, reflect triangle across so that the image of is on the same side of as . We know angle is congruent to angle because translations, rotations, and reflections don’t change angles.
Triangle A B C, blue, on left. Triangle D E F on the right. Triangle A prime B prime C double prime, after a transformation, overlapping Triangle D E F. Point D equals Point A prime, Point E equals Point B prime, Point F equals Point C double prime.
must be on ray since both and are on the same side of and make the same angle with it at . We know the distance is the same as the distance , so that means is the same distance from as is from (because translations and rotations preserve distance). Since and are the same distance along the same ray from , they have to be in the same place.