91Ó°ÊÓ

• $a$is transversal to parallel lines$c$and$d$
• $b$is transversal to parallel lines $c$and $d$
• $c$is transversal to parallel lines$a$and$b$
• $d$is transversal to parallel lines$a$and$b$

Corresponding angle to $∠2$is $∠4$and $∠10$

Corresponding angle to $∠4$is$∠12$

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

Thus,$∠2â‰\dots ∠4,∠2â‰\dots ∠10,\text{and}∠4â‰\dots ∠12$

Vertically opposite angle to $∠2$is$∠5$

Vertically opposite angle to $∠4$is $∠7$

Vertically opposite angle to $∠10$is$∠13$

Vertically opposite angle to $∠12$is $∠15$

Theorem 2-3: Vertically opposite angles are congruent, so $∠2â‰\dots ∠5,∠4â‰\dots ∠7,∠10â‰\dots ∠13,\text{and}∠12â‰\dots ∠15$

If $a=b$and $b=c$then $a=c$

$\begin{array}{c}∠2â‰\dots ∠4,∠4â‰\dots ∠12⇒∠2â‰\dots ∠12\\ ∠2â‰\dots ∠4,∠4â‰\dots ∠7⇒∠2â‰\dots ∠7\\ ∠2â‰\dots ∠10,∠10â‰\dots ∠13⇒∠2â‰\dots ∠13\\ ∠2â‰\dots ∠12,∠12â‰\dots ∠15⇒∠2â‰\dots ∠15\end{array}$

Thus, angles congruent to $∠2$is$∠4,∠5,∠7,∠10,∠12,∠13,\text{and}∠15$.