Question

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

Explanation:

Step1: Find the measure of ∠5

Since ∠5 and ∠6 are supplementary angles (they form a linear pair), we use the formula \(m\angle5 + m\angle6=180^{\circ}\).
Substituting \(m\angle6 = 123.5^{\circ}\), we get \(m\angle5=180^{\circ}- 123.5^{\circ}=56.5^{\circ}\).

Step2: Use the property of parallel lines and transversal

Lines \(x\) and \(y\) are parallel and cut by a transversal. ∠1 and ∠5 are corresponding angles.
By the Corresponding Angles Postulate, if two parallel lines are cut by a transversal, then corresponding angles are congruent. So \(m\angle1=m\angle5\).

Answer:

\(56.5^{\circ}\)