Chapter 5.2, Problem 20PSA

To determine

To Prove: $\overline{QT}\parallel \overline{RS}$ .

Explanation of Solution

Given:

  

  $\begin{array}{l}\angle 1\text{comp}\angle 2\\ \angle 3\text{comp}\angle 2\end{array}$

Concept Used:

When parallel lines is intersected by a transversal corresponding angles are equal. Sum of complementary angles is right angle.

Proof:

Here, we have

  $\begin{array}{l}\angle 1\text{comp}\angle 2\\ \Rightarrow \angle 1\text{+}\angle 2={90}^{\circ }\\ \angle 3\text{comp}\angle 2\\ \Rightarrow \angle 3\text{+}\angle 2={90}^{\circ }\\ \Rightarrow \angle 1\text{+}\angle 2=\angle 3\text{+}\angle 2\\ \Rightarrow \angle 1=\angle 3\\ \text{Since, we know that when parallel lines is intersected by a transversal corresponding angles are equal}\text{.}\\ \Rightarrow \overline{QT}\parallel \overline{RS}\text{with transversal }\overline{QR}\end{array}$

As we extend lines TA and GO, we get two alternate exterior angles $\angle FAT\text{and}\angle HOG$ such that