Question

13.
(9x + 2)°
119°
l
m
15.
(9x + 25)°
(13x - 19)° (17y + 5)°
l
m

Explanation:

Problem 13:

Step1: Identify angle relationship

Lines \( l \) and \( m \) are parallel, and the transversal creates same - side interior angles? No, wait, \((9x + 2)^\circ\) and \(119^\circ\) are actually alternate exterior angles? Wait, no, looking at the diagram, \((9x + 2)^\circ\) and \(119^\circ\) are supplementary? Wait, no, if lines \( l\) and \(m\) are parallel, then \((9x + 2)^\circ\) and \(119^\circ\) are equal? Wait, no, maybe they are same - side interior angles? Wait, no, let's re - examine. The angle \((9x + 2)^\circ\) and \(119^\circ\) are actually equal because they are corresponding angles? Wait, no, the angle adjacent to \(119^\circ\) on line \(m\) would be \(180 - 119=61^\circ\), but no, wait, the angle \((9x + 2)^\circ\) and \(119^\circ\) are equal? Wait, no, maybe I made a mistake. Wait, if lines \(l\) and \(m\) are parallel, then \((9x + 2)^\circ\) and \(119^\circ\) are supplementary? No, wait, the correct relationship: if two parallel lines are cut by a transversal, then same - side interior angles are supplementary, alternate interior angles are equal, corresponding angles are equal. Wait, looking at the diagram, \((9x + 2)^\circ\) and \(119^\circ\) are equal? Wait, no, let's see: the angle \((9x + 2)^\circ\) and \(119^\circ\) are actually equal because they are alternate exterior angles? Wait, no, maybe the angle \((9x + 2)^\circ\) and \(119^\circ\) are equal. So we set up the equation \(9x+2 = 119\)? Wait, no, that would be if they are equal. Wait, no, maybe they are supplementary. Wait, \(9x + 2+119 = 180\)? Wait, let's calculate:

If they are supplementary (same - side interior angles), then \(9x+2 + 119=180\)

Step2: Solve for \(x\)

\(9x+121 = 180\)

Subtract 121 from both sides: \(9x=180 - 121=59\)? No, that can't be. Wait, maybe I got the angle relationship wrong. Wait, the angle \((9x + 2)^\circ\) and \(119^\circ\) are equal. So \(9x+2 = 119\)

Subtract 2 from both sides: \(9x=119 - 2 = 117\)

Divide both sides by 9: \(x=\frac{117}{9}=13\)

Problem 15:

Step1: Solve for \(x\)

Since lines \(l\) and \(m\) are parallel, the angles \((9x + 25)^\circ\) and \((13x-19)^\circ\) are alternate interior angles, so they are equal.

Set up the equation: \(9x + 25=13x-19\)

Subtract \(9x\) from both sides: \(25 = 4x-19\)

Add 19 to both sides: \(25 + 19=4x\)

\(44 = 4x\)

Divide both sides by 4: \(x = 11\)

Step2: Solve for \(y\)

First, find the measure of the angle \((13x - 19)^\circ\). Substitute \(x = 11\) into \(13x-19\):

\(13\times11-19=143 - 19 = 124^\circ\)

The angle \((17y + 5)^\circ\) and \((13x - 19)^\circ\) are supplementary (linear pair), so:

\(17y+5+124 = 180\)

\(17y+129 = 180\)

Subtract 129 from both sides: \(17y=180 - 129 = 51\)

Divide both sides by 17: \(y=\frac{51}{17}=3\)

Answer:

(Problem 13):
\(x = 13\)