Question

6.) given: line m || line n, m∠2 = 5x - 14, and m∠3 = 3x + 10. prove: m∠1 = 134°

Explanation:

Step1: Use corresponding - angles property

Since line m || line n, ∠2 and ∠3 are corresponding angles, so $m\angle2=m\angle3$.
$5x - 14=3x + 10$

Step2: Solve the equation for x

Subtract 3x from both sides: $5x-3x - 14=3x-3x + 10$, which simplifies to $2x-14 = 10$.
Add 14 to both sides: $2x-14 + 14=10 + 14$, so $2x=24$.
Divide both sides by 2: $x = 12$.

Step3: Find the measure of ∠3

Substitute x = 12 into the expression for $m\angle3$: $m\angle3=3x + 10=3\times12+10=36 + 10=46^{\circ}$.

Step4: Use the linear - pair property

∠1 and ∠3 form a linear - pair. So $m\angle1+m\angle3 = 180^{\circ}$.
Substitute $m\angle3 = 46^{\circ}$ into the equation: $m\angle1+46^{\circ}=180^{\circ}$.
Subtract 46° from both sides: $m\angle1=180^{\circ}-46^{\circ}=134^{\circ}$.

Answer:

$m\angle1 = 134^{\circ}$