Question

1. ∠p and ∠q are supplementary angles. if the m∠p=(4x + 1)° and m∠q=(9x - 3)°, find m∠q.
2. ∠u and ∠w are vertical angles. if m∠u=(5x + 1)° and m∠w=(20x - 9)°, find m∠u.

Explanation:

Step1: Recall supplementary - angle property

Supplementary angles add up to 180°. So, \(m\angle P+m\angle Q = 180^{\circ}\).
Substitute \(m\angle P=(4x + 1)^{\circ}\) and \(m\angle Q=(9x - 3)^{\circ}\) into the equation: \((4x + 1)+(9x - 3)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(4x+9x+1 - 3=180\), which gives \(13x-2 = 180\).

Step3: Solve for \(x\)

Add 2 to both sides of the equation: \(13x-2 + 2=180 + 2\), so \(13x=182\).
Divide both sides by 13: \(x=\frac{182}{13}=14\).

Step4: Find \(m\angle Q\)

Substitute \(x = 14\) into the expression for \(m\angle Q\): \(m\angle Q=(9x - 3)^{\circ}\).
\(m\angle Q=9\times14-3=126 - 3=123^{\circ}\).

Answer:

\(123^{\circ}\)