G.2.7.4 What Do We Know For Sure About Parallelograms?
The angle-side-angle theorem could be proven in a similar method to what we did for the side-angle-side theorem in the last class. Instead of spending time proving, today we are going to use the angle-side-angle theorem to prove things about parallelograms.
The angle-side-angle theorem says that two triangles are congruent when two pairs of corresponding angles and the included corresponding sides are congruent.
A parallelogram is a quadrilateral with two pairs of opposite parallel sides.
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)
Sketch parallelogram ABCD and then draw an auxiliary line to show how ABCD can be decomposed into 2 triangles.
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.