Question

two parallel lines are cut by a transversal as shown below. suppose m∠4 = 146°. find m∠6 and m∠7.

Explanation:

Step1: Identify angle - relationship for ∠4 and ∠6

∠4 and ∠6 are alternate - interior angles. When two parallel lines are cut by a transversal, alternate - interior angles are congruent.
$m\angle6=m\angle4$

Step2: Find the measure of ∠6

Since $m\angle4 = 146^{\circ}$, then $m\angle6=146^{\circ}$

Step3: Identify angle - relationship for ∠6 and ∠7

∠6 and ∠7 are supplementary angles (linear pair). The sum of the measures of two angles in a linear pair is $180^{\circ}$. So $m\angle6 + m\angle7=180^{\circ}$

Step4: Find the measure of ∠7

We know $m\angle6 = 146^{\circ}$, then $m\angle7=180^{\circ}-m\angle6$. Substitute $m\angle6 = 146^{\circ}$ into the equation: $m\angle7=180 - 146=34^{\circ}$

Answer:

$m\angle6 = 146^{\circ}$
$m\angle7 = 34^{\circ}$