Chapter 5.3, Problem 20PSB

To determine

To Prove: $\overleftrightarrow{YZ}\text{ bisects }\angle AYB$ .

Explanation of Solution

Given:

  

  $\overline{CY}\cong \overline{AY},\overline{YZ}\parallel \overline{CA}$

Concept Used:

When two parallel lines are intersected by a transversal:

• Alternate interior angles are equal
• Corresponding angles are equal

Proof:

Here, we have

  $\overline{CY}\cong \overline{AY},\overline{YZ}\parallel \overline{CA}$

Since when two parallel lines are intersected by a transversal alternate interior angles are equal.

  $\begin{array}{l}\because \overline{YZ}\parallel \overline{CA}\\ \Rightarrow \angle CAY=\angle AYZ\left(\text{Alternate interior angles}\right)\left(1\right)\\ \Rightarrow \angle YCA=\angle BYZ\left(\text{Corresponding angles}\right)\\ \because \overline{CY}\cong \overline{AY}\\ \Rightarrow \angle YCA=\angle YAC\left(\text{Angles opposite to equal sides are equal}\right)\left(2\right)\\ \text{From }\left(\text{1}\right)\text{and }\left(\text{2}\right)\\ \Rightarrow \angle AYZ=\angle BYZ\\ \Rightarrow \overleftrightarrow{YZ}\text{ bisects }\angle AYB\end{array}$