# Angle-Side-Angle Triangle Congruence

ID: jabuk-bitoh
Illustrative Mathematics, CC BY 4.0
Subject: Geometry
Standards: HSG-CO.B.8HSG-CO.B.7HSG-CO.C.10HSG-CO.A.5HSG-CO.C.9HSG-CO.C.11

12 questions

# Angle-Side-Angle Triangle Congruence

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##### Notice and Wonder: Assertion

1) What do you notice? What do you wonder?

Assertion: Through two distinct points passes a unique line. Two lines are said to be ﻿$\textit{distinct}$﻿ if there is at least one point that belongs to one but not the other. Otherwise, we say the lines are the same. Lines that have no point in common are said to be ﻿$\textit{parallel}$﻿.

Therefore, we can conclude: given two distinct lines, either they are parallel, or they have exactly one point in common.

##### Proving the Angle-Side-Angle Triangle Congruence Theorem

2) Two triangles have 2 pairs of corresponding angles congruent, and the corresponding sides between those angles are congruent. Sketch 2 triangles that fit this description.

Label the triangles ﻿$WXY$﻿ and ﻿$DEF$﻿, so that angle ﻿$W$﻿ is congruent to angle ﻿$D$﻿, angle ﻿$X$﻿ is congruent to angle ﻿$E$﻿, and side ﻿$WX$﻿ is congruent to side ﻿$DE$﻿.

3) Use a sequence of rigid motions to take triangle ﻿$WXY$﻿ onto triangle ﻿$DEF$﻿. For each step, explain how you know that one or more vertices will line up.

##### Find the Missing Angle Measures

Lines ﻿$\ell$﻿ and ﻿$m$﻿ are parallel. ﻿$a$﻿ = 42. Find the indicated values.

Given: ﻿$\ell \parallel m$﻿

4) ﻿$b =$﻿

5) ﻿$c =$﻿

6) ﻿$d =$﻿

7) ﻿$e =$﻿

8) ﻿$f =$﻿

9) ﻿$g =$﻿

10) ﻿$h =$﻿

##### What Do We Know For Sure About Parallelograms?

Quadrilateral ﻿$ABCD$﻿ is a ﻿$\textbf{parallelogram}$﻿. By definition, that means that segment ﻿$AB$﻿ is parallel to segment ﻿$CD$﻿, and segment ﻿$BC$﻿ is parallel to segment ﻿$AD$﻿.

11) Sketch parallelogram ﻿$ABCD$﻿ and then draw an auxiliary line to show how ﻿$ABCD$﻿ can be decomposed into 2 triangles.

12) Prove that the 2 triangles you created are congruent, and explain why that shows one pair of opposite sides of a parallelogram must be congruent.