Unit 2, Lesson 3
Congruent Triangles, Part 1
Geometry
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If triangle is congruent to triangle . . .
Noah and Priya were playing Invisible Triangles. Priya told Noah all the sides and angles that are congruent.
Triangles A B C and D E F. Segments A B and D E each have 1 tick mark, B C and E F each have 2 tick marks, A C and D F each have 3 tick marks. Angles A C B and D F E each have 1 tick mark, A B C and D E F each have 2 tick marks, B A C and E D F each have 3 tick marks.
Here are the steps Noah had to tell Priya to do before all 3 vertices coincided:
Now Noah and Priya are working on explaining why their steps worked, and they need some help. Answer their questions to help them fill in the missing parts of their proof.
First, we translate triangle by the directed line segment from to . Point will coincide with because we defined our transformation that way. Then, rotate the image, triangle , by the angle so that rays and line up. Point will coincide with point because . Finally, reflect the image, triangle , across . Ray and ray will line up because . Point and point will coincide because . All 3 points in the triangles coincide, so this is a sequence of transformations that takes triangle to triangle .
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.