Chapter Test
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∠ 1 and ∠ 2 are corresponding angles in △ RWZ and △ TYV.
See solution.
To show that ∠ 1 and ∠ 2 are congruent, we should prove congruence between △ RWZ and △ TYV where ∠ 1 and ∠ 2 would be corresponding angles.
△ VSX≅ △ ZSX by the SSS Congruence Theorem. Also, since ∠ VXW and ∠ ZXY are vertical angles, we know they are congruent by the Vertical Angles Congruence Theorem. Now we have enough information to prove that two more triangles are congruent. △ VXW≅ △ XZY by the SAS Congruence Theorem. Let's mark some congruent corresponding parts that we will need to prove congruence in △ RWZ and △ TYV.
Notice that VX+XY=VY and WX+XZ=WZ. Because we know that VX≅ XY≅ WX≅ XZ, it can be proven that WZ≅ WZ. By separating the triangles, it becomes more clear which theorem we should use.
Two angles and the included side in △ RWZ are congruent with two angles and the included side in △ TYV. This means we can prove that these triangles are congruent by using the ASA Congruence Theorem. Since ∠ 1 and ∠ 2 are corresponding angles in these triangles, we can finally say that they are congruent. ∠ 1 ≅ ∠2