Question

find the measure of each numbered angle and name the theorems that you used to justify your work.
m∠6 = (2x − 21)°
m∠7 = (3x − 34)°
a) m∠6 = m∠8 = 63°, m∠7 = 107° (supplements theorem and vertical ∠s theorem)
b) m∠6 = m∠8 = 73°, m∠7 = 107° (supplements theorem and vertical ∠s theorem)

Explanation:

Step1: Identify Angle Relationship

∠6 and ∠7 are supplementary (linear pair), so \( m\angle6 + m\angle7 = 180^\circ \). Substitute \( m\angle6=(2x - 21)^\circ \) and \( m\angle7=(3x - 34)^\circ \):
\( (2x - 21)+(3x - 34)=180 \)

Step2: Solve for \( x \)

Combine like terms: \( 5x - 55 = 180 \)
Add 55 to both sides: \( 5x = 235 \)
Divide by 5: \( x = 47 \)

Step3: Calculate Angle Measures

• \( m\angle6 = 2(47)-21 = 94 - 21 = 73^\circ \)
• \( m\angle7 = 3(47)-34 = 141 - 34 = 107^\circ \)

∠6 and ∠8 are vertical angles (Vertical Angles Theorem), so \( m\angle6 = m\angle8 = 73^\circ \). ∠7 and its vertical angle (not shown) would also be equal, but here we use supplementary and vertical angle relationships.

Answer:

B) \( m\angle6 = m\angle8 = 73^\circ \), \( m\angle7 = 107^\circ \) (Supplements Theorem and Vertical ∠s Theorem)