Elementary Geometry for College Students (7th Edition) Clone

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

Chapter 3 - Section 3.2 - Corresponding Parts of Congruent Triangles - Exercises - Page 152: 34

Answer

- Prove that $\triangle DAB\cong\triangle CAE$ by method ASA to deduce $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$ - Then show that $\overline{DE}\cong\overline{CB}$ - Then we can prove that $\triangle DEC\cong\triangle CBD$ by method SAS.

Work Step by Step

- Prove that $\triangle DAB\cong\triangle CAE$ by method ASA to deduce $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$ - Then show that $\overline{DE}\cong\overline{CB}$ - Then we can prove that $\triangle DEC\cong\triangle CBD$ by method SAS. * Prove that $\triangle DAB\cong\triangle CAE$ 1) $\overline{DB}\bot\overline{BC}$ and $\overline{CE}\bot\overline{ED}$ (Given) 2) $\angle DBA$ and $\angle CEA$ are right $\angle$s (if 2 lines are perpendicular with each other, then the angles that they make up are right angles) 3) $\angle DBA\cong\angle CEA$ (2 corresponding right angles are congruent) 4) $\overline{AB}\cong\overline{AE}$ (Given) 5) $\angle DAB\cong\angle CAE$ (2 vertical angles are congruent) So now 2 angles and the included side of $\triangle DAB$ are congruent with 2 corresponding angles and the included side of $\triangle CAE$. 5) $\triangle DAB\cong\triangle CAE$ (ASA) 6) $\overline{DB}\cong\overline{CE}$ and $\overline{DA}\cong\overline{CA}$ (CPCTC) * Prove that $\triangle DCB\cong\triangle CDE$ We see that $\overline{DA}\cong\overline{CA}$ (proved above) and $\overline{AE}\cong\overline{AB}$ (given) So $\overline{DA}+\overline{AE}\cong\overline{CA}+\overline{AB}$ That means $\overline{DE}\cong\overline{CB}$ 7) $\overline{DE}\cong\overline{CB}$ (proved above) 8) $\overline{CE}\cong\overline{DB}$ (proved in 6) 9) $\angle CED\cong\angle DBC$ (proved in 3) So now 2 sides and the included angle of $\triangle DEC$ are congruent with 2 corresponding sides and the included angle of $\triangle CBD$. 10) $\triangle DEC\cong\triangle CBD$ (SAS)
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