Angle-Side-Angle Triangle Congruence

ID: jabuk-bitoh
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-CO.B.8HSG-CO.B.7HSG-CO.C.10HSG-CO.A.5HSG-CO.C.9HSG-CO.C.11
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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 distinct\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 parallel\textit{parallel}.

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

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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 WXYWXY and DEFDEF, so that angle WW is congruent to angle DD, angle XX is congruent to angle EE, and side WXWX is congruent to side DEDE.

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

Find the Missing Angle Measures

Lines \ell and mm are parallel. aa = 42. Find the indicated values.

Given: m\ell \parallel m

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4) b=b = 

5) c=c = 

6) d=d = 

7) e=e = 

8) f=f = 

9) g=g = 

10) h=h = 

What Do We Know For Sure About Parallelograms?

Quadrilateral ABCDABCD is a parallelogram\textbf{parallelogram}. By definition, that means that segment ABAB is parallel to segment CDCD, and segment BCBC is parallel to segment ADAD.

11) Sketch parallelogram ABCDABCD and then draw an auxiliary line to show how ABCDABCD 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.