Elementary Geometry for College Students (7th Edition)

Published by Cengage
ISBN 10: 978-1-337-61408-5
ISBN 13: 978-1-33761-408-5

Chapter 3 - Section 3.1 - Congruent Triangles - Exercises - Page 142: 5


1) $\angle A$ is the common angle. $\overline{AB}$ is the common side. 2) If $\overline{BC}\cong\overline{BD}$, we cannot conclude that $\triangle ABC$ and $\triangle ABD$ are congruent. 3) SSA is not a reason to prove 2 triangles congruent.

Work Step by Step

1) A common angle is the angle that both triangles have. In the image, we see that both triangles $\triangle ABC$ and $\triangle ABD$ share the angle $\angle A$. So. $\angle A$ is the common angle. Similary, a common side is the side that both triangles have. Here, both triangles share the side $\overline{AB}$. Therefore, $\overline{AB}$ is the common side. 2) There are 4 ways to prove that 2 triangles are congruent: SSS, SAS, ASA and AAS (which is actually ASA). Here, since both triangles $\triangle ABC$ and $\triangle ABD$ have common angle $\angle A$ and common side $\overline{AB}$, we need either of the following to get congruence: - $\overline{AC}\cong\overline{AD}$ to have SAS - $\angle ABC\cong\angle ABD$ to have ASA - $\angle ACB\cong\angle ADB$ to have AAS. Instead, having $\overline{BC}\cong\overline{BD}$ means we only have SSA, which is not enough to prove 2 triangles are congruent. 3) As said above, SSA is not a reason to prove triangles congruent. To prove that, we can refer to Figure 3.9 in the textbook to show an example where 2 triangles have SSA but are not congruent.
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