Question

If $a\left|\right|b$ $m\angle 2=63$and $m\angle 9=105$, find the measure of each missing angle.

Answer 1

Step-1: Evaluating for $\angle 1$.

Given, $m\angle 2=63°;m\angle 9=105°,$ and line $a$ is parallel to $b$.

$m\angle 1+m\angle 2=m\angle 9(\mathrm{alternate}\mathrm{exterior}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 1+63°=105°\left(\mathrm{substitution}\right)$

Subtract $63$ from each side:

$$\begin{gathered} m\angle 1=105°-63° \\ m\angle 1=42° \end{gathered}$$

Step-2: Evaluating for $\angle 3$.

$m\angle 3+m\angle 2+m\angle 1=180°(\mathrm{angles}\mathrm{in}\mathrm{a}\mathrm{straight}\mathrm{line})$

$m\angle 3+63°+42°=180°\text{(substitution)}$

$$\begin{gathered} m\angle 3+105°=180° \\ m\angle 3=180°-105° \\ m\angle 3=75° \end{gathered}$$

Step-3: Evaluating for $\angle 4$:

$m\angle 4=m\angle 1(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 4=42°$

Step-4: Evaluating for $\angle 5$:

$m\angle 5=m\angle 2(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 5=63°$

Step-5: Evaluating for $\angle 6$.

$m\angle 6=m\angle 3(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 6=75°$

Step-6: Evaluating for $\angle 7$.

$m\angle 7=m\angle 9(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 7=105°$

Step 7. Evaluating for $\angle 8$.

$m\angle 8=m\angle 9(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 8=75°$

Step-8: Evaluating for $\angle 10$.

$m\angle 8=m\angle 10(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 10=75°$

Step-9: Evaluating for $\angle 11$.

$m\angle 11=m\angle 4(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 11=42°$

Step-10: Evaluating for $\angle 12$.

$m\angle 12=180°-m\angle 11(\mathrm{linear}\mathrm{pair}\mathrm{theorem})$

$m\angle 12=180°-42°\left(\mathrm{substitution}\right)$

$m\angle 12=138°$

Step-11: Evaluating for $\angle 13$.

$m\angle 13=m\angle 11(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 13=42°\left(\mathrm{substitution}\right)$

Step-12: Evaluating for $\angle 14$

$m\angle 14=m\angle 12(\mathrm{vertical}\mathrm{angles}\mathrm{are}\mathrm{congruent})$

$m\angle 14=138°$

Hence,

$$\begin{gathered} m\angle 1=42° \\ m\angle 3=75° \\ m\angle 4=42° \\ m\angle 5=63° \\ m\angle 6=75° \\ m\angle 7=105° \\ m\angle 8=m\angle 10=75° \\ m\angle 11=m\angle 13=42° \\ m\angle 12=m\angle 14=138° \end{gathered}$$