Question

two parallel lines are crossed by a transversal. if ( mangle6 = 123.5^{circ} ), then ( mangle1 ) is ( 56.5^{circ} ). ( 136.5^{circ} ). ( 67.5^{circ} ). ( 123.5^{circ} ).

Explanation:

Step1: Find the measure of $\angle 3$

Since $\angle 6$ and $\angle 3$ are alternate interior angles, $m\angle 3=m\angle 6 = 123.5^{\circ}$ (Alternate Interior Angles Theorem).

Step2: Use the linear - pair relationship

$\angle 1$ and $\angle 3$ form a linear pair. So, $m\angle 1+m\angle 3 = 180^{\circ}$ (Linear Pair Postulate).
Substitute $m\angle 3 = 123.5^{\circ}$ into the equation: $m\angle 1+123.5^{\circ}=180^{\circ}$.
Solve for $m\angle 1$: $m\angle 1=180^{\circ}-123.5^{\circ}=56.5^{\circ}$.

Answer:

$56.5^{\circ}$