Question

18)
(2x + 1)°
125°
19)
4x°
128°
20)
111°
3x°
find the measure of angle b.
21)
b
60°
60°
22)
b
25°
42°
23)
38°
b
find the value of x.
24)
57°
(2x + 31)°
30°

Explanation:

Problem 18:

Step1: Identify relationship (supplementary angles)

The two angles \( (2x + 1)^\circ \) and \( 125^\circ \) are supplementary (they form a linear pair with a transversal between parallel lines, so they should add up to \( 180^\circ \) if they are same - side interior or supplementary, but wait, actually, looking at the diagram, the two horizontal lines are parallel, and the transversal creates a corresponding or supplementary situation. Wait, no, if the two horizontal lines are parallel, and the angle \( 125^\circ \) and \( (2x + 1)^\circ \) are same - side interior? Wait, no, actually, if we consider the vertical line, the two horizontal lines are parallel, so the angle \( (2x + 1)^\circ \) and \( 125^\circ \) should be equal? Wait, no, maybe I made a mistake. Wait, no, if the two horizontal lines are parallel, and the transversal (the slanted line) creates a corresponding angle. Wait, no, the angle \( 125^\circ \) and \( (2x + 1)^\circ \): actually, if the two horizontal lines are parallel, and the vertical line is perpendicular? Wait, no, the vertical line is a transversal? Wait, no, the two horizontal lines are parallel, and the slanted line is a transversal. Wait, the angle \( 125^\circ \) and \( (2x + 1)^\circ \): if they are same - side interior angles, they should be supplementary. Wait, \( 125+(2x + 1)=180 \)

Step2: Solve for x

\( 125+2x + 1 = 180 \)
\( 2x+126 = 180 \)
\( 2x=180 - 126 \)
\( 2x = 54 \)
\( x=\frac{54}{2}=27 \)

Step1: Identify relationship (vertical angles or corresponding angles)

The angle \( 128^\circ \) and \( 4x^\circ \): since the two vertical lines are parallel, the angle \( 128^\circ \) and \( 4x^\circ \) are corresponding angles? Wait, no, the angle \( 128^\circ \) and \( 4x^\circ \): actually, the angle \( 128^\circ \) and \( 4x^\circ \) are equal because of parallel lines (corresponding angles). Wait, no, \( 128^\circ \) and \( 4x^\circ \): if the two vertical lines are parallel, and the transversal (the slanted line) creates corresponding angles. So \( 4x=128 \)

Step2: Solve for x

\( x=\frac{128}{4}=32 \)

Step1: Identify relationship (vertical angles or supplementary)

The angle \( 111^\circ \) and \( 3x^\circ \): since the lines are parallel (the two slanted lines and the other lines), the angle \( 111^\circ \) and \( 3x^\circ \) are supplementary (they form a linear pair with a transversal between parallel lines). So \( 3x+111 = 180 \)

Step2: Solve for x

\( 3x=180 - 111 \)
\( 3x = 69 \)
\( x=\frac{69}{3}=23 \)

Answer:

\( x = 27 \)

Problem 19: