Question

point p is in the interior of ∠mno. if m∠mno = 15x + 6, m∠mnp = 14x − 4, and m∠pno = 20°, determine m∠mno. show your algebraic thinking to earn full credit.

Explanation:

Step1: Apply Angle Addition Postulate

Since point \( P \) is in the interior of \( \angle MNO \), we know that \( m\angle MNO = m\angle MNP + m\angle PNO \). Substituting the given angle measures, we get:
\[
15x + 6 = (14x - 4) + 20
\]

Step2: Solve for \( x \)

Simplify the right - hand side of the equation:
\[
15x + 6 = 14x + 16
\]
Subtract \( 14x \) from both sides:
\[
15x - 14x+6 = 14x - 14x + 16
\]
\[
x + 6 = 16
\]
Subtract 6 from both sides:
\[
x+6 - 6=16 - 6
\]
\[
x = 10
\]

Step3: Find \( m\angle MNO \)

Substitute \( x = 10 \) into the expression for \( m\angle MNO \), which is \( 15x + 6 \):
\[
m\angle MNO=15\times10 + 6
\]
\[
m\angle MNO = 150+6
\]
\[
m\angle MNO=156^{\circ}
\]

Answer:

\( 156^{\circ} \)