Question

example 8: determine the measure of each angle in the diagram below. label the diagram.
(3x)°
(2x)°
j
k
y°
m ① ______
m ② ______
m ③ ______

Explanation:

Step1: Identify angle relationship

Angles \( (3x)^\circ \) and \( (2x)^\circ \) are supplementary (form a linear pair), so \( 3x + 2x = 180 \).

$$ 5x = 180 $$

Step2: Solve for \( x \)

Divide both sides by 5:

$$ x = \frac{180}{5} = 36 $$

Step3: Find \( m\angle1 \)

\( m\angle1 = 3x^\circ \). Substitute \( x = 36 \):

$$ m\angle1 = 3(36) = 108^\circ $$

Step4: Find \( m\angle2 \)

\( m\angle2 = 2x^\circ \). Substitute \( x = 36 \):

$$ m\angle2 = 2(36) = 72^\circ $$

Step5: Find \( m\angle3 \)

\( \angle3 \) and \( \angle1 \) are alternate exterior angles (since lines are parallel, assumed from diagram), so \( m\angle3 = m\angle2 = 72^\circ \) (or \( \angle3 \) is vertical to \( \angle2 \), also \( 72^\circ \)).

Answer:

\( m\angle1 = 108^\circ \), \( m\angle2 = 72^\circ \), \( m\angle3 = 72^\circ \)