Question

1. in the diagram below \\( \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } \\) and \\( m \angle 4 = ( 5 x - 7 ) ^ { \circ } \\) and \\( m \angle 6 = ( 3 x + 9 ) ^ { \circ } \\). find the measure of \\( \angle 2 \\) in the diagram below.

Explanation:

Step1: Use the property of alternate - interior angles

Since \(AB\parallel CD\), \(\angle4\) and \(\angle6\) are same - side interior angles. So \(m\angle4 + m\angle6=180^{\circ}\) (because same - side interior angles are supplementary when two parallel lines are cut by a transversal).
We have \(m\angle4=(5x - 7)^{\circ}\) and \(m\angle6=(3x + 9)^{\circ}\). Then \((5x-7)+(3x + 9)=180\).
Simplify the left - hand side: \(5x-7+3x + 9=8x+2\). So the equation becomes \(8x+2 = 180\).
Subtract 2 from both sides: \(8x=180 - 2=178\). Then \(x=\frac{178}{8}=\frac{89}{4}=22.25\).

Step2: Find \(m\angle4\)

Substitute \(x = 22.25\) into \(m\angle4=(5x - 7)^{\circ}\).
\(m\angle4=5\times22.25-7=111.25 - 7=104.25^{\circ}\).

Step3: Use the property of vertical angles

\(\angle2\) and \(\angle4\) are vertical angles. Vertical angles are congruent. So \(m\angle2=m\angle4\).

Answer:

\(104.25^{\circ}\)