Question

Assume that $a\parallel b$and $c\parallel d$, name all angels congruent to $\angle 2$.

Short answer

Angles congruent to $\angle 2$is$\mathbf{\angle }\mathbf{4},\mathbf{\angle }\mathbf{5},\mathbf{\angle }\mathbf{7},\mathbf{\angle }\mathbf{10},\mathbf{\angle }\mathbf{12},\mathbf{\angle }\mathbf{13},$and$\mathbf{\angle }\mathbf{15}$.

Step-by-step solution

Step 1. Observations from figure

• $a$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$b$

Step 2. Find corresponding angles

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

Corresponding angle to $\angle 4$is$\angle 12$

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

Thus,$\angle 2\cong \angle 4,\angle 2\cong \angle 10,\text{and}\angle 4\cong \angle 12$

Step 3. Find vertically opposite angles

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

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

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

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

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

Step 4. Use transitive property

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

$\begin{array}{c}\angle 2\cong \angle 4,\angle 4\cong \angle 12\Rightarrow \angle 2\cong \angle 12\\ \angle 2\cong \angle 4,\angle 4\cong \angle 7\Rightarrow \angle 2\cong \angle 7\\ \angle 2\cong \angle 10,\angle 10\cong \angle 13\Rightarrow \angle 2\cong \angle 13\\ \angle 2\cong \angle 12,\angle 12\cong \angle 15\Rightarrow \angle 2\cong \angle 15\end{array}$

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