Question

solve for x, then find the measure of the angle given:
7.
(3x + 23)°
c
4x°
b
a
m∠abc
8.
l
5x°
n
m
(3x + 50)°
p
q
m∠mpq
9.
m
n
(a + 28)°
2a°
p
m∠mnp
10.
w
5y°
z
x
(2y + 78)°
m∠wxz

Explanation:

Step1: Solve for \( x \) in problem 7

Since \( 3x + 23=4x \) (alternate interior angles are equal).
Subtract \( 3x \) from both sides: \( 23 = 4x-3x \), so \( x = 23 \).

Step2: Find \( m\angle ABC \)

Substitute \( x = 23 \) into \( 4x \). Then \( m\angle ABC=4\times23 = 92^{\circ} \).

Step3: Solve for \( x \) in problem 8

Since \( 5x=3x + 50 \) (alternate interior angles are equal).
Subtract \( 3x \) from both sides: \( 5x-3x=50 \), \( 2x = 50 \), so \( x = 25 \).

Step4: Find \( m\angle MPQ \)

Substitute \( x = 25 \) into \( 3x + 50 \). Then \( m\angle MPQ=3\times25+50=75 + 50=125^{\circ} \).

Step5: Solve for \( a \) in problem 9

Since \( a + 28=2a \) (alternate interior angles are equal).
Subtract \( a \) from both sides: \( 28=2a - a \), so \( a = 28 \).

Step6: Find \( m\angle MNP \)

Substitute \( a = 28 \) into \( a + 28 \). Then \( m\angle MNP=28+28 = 56^{\circ} \).

Step7: Solve for \( y \) in problem 10

Since \( 5y=2y + 78 \) (alternate interior angles are equal).
Subtract \( 2y \) from both sides: \( 5y-2y=78 \), \( 3y = 78 \), so \( y = 26 \).

Step8: Find \( m\angle WXZ \)

Substitute \( y = 26 \) into \( 5y \). Then \( m\angle WXZ=5\times26=130^{\circ} \).

Answer:

1. \( m\angle ABC = 92^{\circ} \)
2. \( m\angle MPQ = 125^{\circ} \)
3. \( m\angle MNP = 56^{\circ} \)
4. \( m\angle WXZ = 130^{\circ} \)