Question

in the figure below, r || s, m∠3 = 3x - 105 and m∠6 = 2x + 10, find m∠6 and m∠8

a. what is the angle pair relationship?

b. select the correct equation

c. x =

d. m∠6 =

e. m∠8 =

Explanation:

Step1: Use the property of parallel lines and transversals

Since \( r\parallel s \), \( \angle3 \) and \( \angle6 \) are alternate - interior angles. So \( m\angle3=m\angle6 \).
Given \( m\angle3 = 3x-105 \) and \( m\angle6=2x + 10 \).
Set up the equation \( 3x-105=2x + 10 \).

Step2: Solve the equation for \( x \)

Subtract \( 2x \) from both sides of the equation:
\( 3x-2x-105=2x-2x + 10 \), which simplifies to \( x-105=10 \).
Add \( 105 \) to both sides: \( x=10 + 105=115 \).

Step3: Find \( m\angle6 \)

Substitute \( x = 115 \) into the formula for \( m\angle6 \).
\( m\angle6=2x+10=2\times115 + 10=230+10=240\). But this is wrong. Wait, no, actually, we made a mistake above. Since \( r\parallel s \), \( \angle3 \) and \( \angle6 \) are same - side interior angles. So \( m\angle3+m\angle6 = 180^{\circ}\).
Set up the equation \( 3x-105+2x + 10=180 \).
Combine like terms: \( 5x-95 = 180 \).
Add \( 95 \) to both sides: \( 5x=180 + 95=275 \).
Divide both sides by \( 5 \): \( x = 55 \).

Step4: Find \( m\angle6 \)

Substitute \( x = 55 \) into \( m\angle6=2x + 10 \).
\( m\angle6=2\times55+10=110 + 10=120^{\circ}\).

Step5: Find \( m\angle8 \)

Since \( \angle6 \) and \( \angle8 \) are vertical angles (or \( \angle3 \) and \( \angle8 \) are corresponding angles as \( r\parallel s \)).
\( m\angle8=m\angle6 = 120^{\circ}\).

Answer:

\( m\angle6 = 120^{\circ}\), \( m\angle8=120^{\circ}\)