Question

given ( mangle8 = 86^{circ} ), find ( mangle13 ) 10) find the values of ( x ) and ( y ) given ( mangle4 = 6x - 5 ), ( mangle10 = 5x + 8 ), and ( mangle9 = 3y - 10 )

Explanation:

Step1: Find \(m\angle13\)

Since \(m\angle8 = 86^{\circ}\), and \(\angle8\) and \(\angle13\) are alternate interior angles (when lines \(n\) and \(m\) are parallel and cut by a transversal). Alternate interior angles are equal. So \(m\angle13=m\angle8\).

Step2: Solve for \(x\) in \(\angle4\) and \(\angle10\)

\(\angle4\) and \(\angle10\) are vertical angles. Vertical angles are equal. So \(6x - 5=5x + 8\).
Subtract \(5x\) from both sides: \(6x-5x-5=5x - 5x+8\), which gives \(x-5 = 8\).
Add \(5\) to both sides: \(x=8 + 5=13\).

Step3: Solve for \(y\) using \(\angle9\)

\(\angle9\) and \(\angle10\) are a linear - pair (supplementary). First, find \(m\angle10\) by substituting \(x = 13\) into \(m\angle10=5x + 8\). So \(m\angle10=5\times13 + 8=65 + 8=73^{\circ}\).
Since \(m\angle9+m\angle10 = 180^{\circ}\), then \(3y-10+73 = 180\).
Simplify the left - hand side: \(3y+63 = 180\).
Subtract \(63\) from both sides: \(3y=180 - 63=117\).
Divide both sides by \(3\): \(y=\frac{117}{3}=39\).

Answer:

1. \(m\angle13 = 86^{\circ}\)
2. \(x = 13,y = 39\)