Q14. This figure is like the one that... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 4: Q14. (page 150)

This figure is like the one that Euclid used to prove that the base angles of an isosceles triangle are congruent (our Theorem 4-1). Write a paragraph proof following the key steps shown below. Given: $\overset{炉}{AB} 鈮� \overset{炉}{AC}; \overset{炉}{AB}$ and $\overset{炉}{AC}$ are extended so $\overset{炉}{BD} 鈮� \overset{炉}{CE}$ .
Prove: $鈭� ABC 鈮� 鈭� ACB$
Key steps of proof:
1. $螖 DAC 鈮� 螖 EAB$
2. $螖 DBC 鈮� 螖 EBC$
3. $螖 DBC 鈮� 螖 EBC$
4. $鈭� ABC 鈮� 鈭� ACB$

Short Answer

Expert verified

$螖 DAC 鈮� 螖 EAB$ by SAS postulate then by corresponding parts of congruent triangles $\overset{炉}{CD} 鈮� \overset{炉}{BE}$ . $螖 DBC 鈮� 螖 ECB$ by SSS postulate. As $\overset{炉}{AC} 鈮� \overset{炉}{AB}$ then by isosceles triangle theorem, $鈭� ABC 鈮� 鈭� ACB$ .

Step by step solution

Step 1. Description of step.

Consider $螖 D A C$ and $螖 E A B$ . From the given figure it can be observed that $\overset{炉}{A D} 鈮� \overset{炉}{A E}$ , $鈭� A 鈮� 鈭� A$ and $\overset{炉}{A C} 鈮� \overset{炉}{A B}$ then by SAS postulate, $螖 D A C 鈮� 螖 E A B$ .

Step 2. Description of step.

As $螖 D A C 鈮� 螖 E A B$ then by corresponding parts of congruent triangles $\overset{炉}{C D} 鈮� \overset{炉}{B E}$ .

Step 3. Description of step.

Consider $螖 D B C$ and $螖 E B C$ . From the given figure it can be observed that $\overset{炉}{D B} 鈮� \overset{炉}{C E}$ , $\overset{炉}{B C} 鈮� \overset{炉}{B C}$ and $\overset{炉}{C D} 鈮� \overset{炉}{B E}$ then by SSS postulate, $螖 D B C 鈮� 螖 E B C$ .

Step 4. Description of step.

As $\overset{炉}{A C} 鈮� \overset{炉}{A B}$ then by isosceles triangle theorem, $鈭� A B C 鈮� 鈭� A C B$ .

Step 5. Write a paragraph proof.

$螖 D A C 鈮� 螖 E A B$ by SAS postulate then by corresponding parts of congruent triangles $\overset{炉}{C D} 鈮� \overset{炉}{B E}$ . $螖 D B C 鈮� 螖 E C B$ by SSS postulate. As $\overset{炉}{A C} 鈮� \overset{炉}{A B}$ then by isosceles triangle theorem, $鈭� A B C 鈮� 鈭� A C B$ .