Question

If $c\parallel d$, name all angels that must be congruent to $\angle 1$.

Short answer

Angles congruent to$\angle 1$is$\mathbf{\angle }\mathbf{6}\mathbf{,}\mathbf{}\mathbf{\angle }\mathbf{9}\mathbf{,}$and$\mathbf{\angle }\mathbf{14}$.

Step-by-step solution

Step 1. Observations from figure

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

Step 2. Find corresponding angles

Corresponding angle to $\angle 1$is $\angle 9$

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

Thus,$\angle 1\cong \angle 9$

Step 3. Find vertically opposite angles

Vertically opposite angle to $\angle 1$is$\angle 6$

Vertically opposite angle to $\angle 9$is $\angle 14$

Theorem 2-3: Vertically opposite angles are congruent, so $\angle 1\cong \angle 6$and$\angle 9\cong \angle 14$.

Step 4. Use transitive property

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

Substitute $\angle 1$for $a$,$\angle 6$ for $b$, and $\angle 14$for $c$

Since, $\angle 1\cong \angle 6$and $\angle 6\cong \angle 14$implies$\angle 1\cong \angle 14$

Thus, angles congruent to $\angle 1$is$\angle 6,\angle 9,$and$\angle 14$