Question

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

Short answer

Angles congruent to$\angle 1$is$\mathbf{\angle }\mathbf{3},\mathbf{}\mathbf{\angle }\mathbf{6},$and$\mathbf{\angle }\mathbf{8}$.

Step-by-step solution

Step 1. Observations from figure

$c$is transversal to parallel$a$ lines$b$.

Step 2. Find corresponding angles

Corresponding angle to $\angle 1$is $\angle 3$

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

Thus,$\angle 1\cong \angle 3$

Step 3. Find vertically opposite angles

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

Vertically opposite angle to $\angle 3$is $\angle 8$

Theorem 2-3: Vertically opposite angles are congruent, so$\angle 1\cong \angle 6$ and$\angle 3\cong \angle 8$.

Step 4. Use transitive property

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

Substitute $\angle 1$for $a$,$\angle 3$for $b$, and $\angle 8$for $c$

Since, $\angle 1\cong \angle 3$and $\angle 3\cong \angle 8$implies $\angle 1\cong \angle 8$

Thus, angles congruent to $\angle 1$is $\angle 3,\angle 6,\text{and}\angle 8$