Unit 2, Lesson 3
Congruent Triangles, Part 1
Geometry
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If 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
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
We know that rays
If necessary, reflect triangle
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.