Question

find the measure of each angle. 3 find the measure of each angle. m∠r ______ m∠s ____ m∠rqs ______ 135° 18x° 15x + 3°

Explanation:

Step1: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(135=18x+(15x + 3)\).

Step2: Simplify the equation

Combine like terms: \(135=18x+15x + 3\), which becomes \(135=33x + 3\).

Step3: Solve for \(x\)

Subtract 3 from both sides: \(135−3=33x\), so \(132 = 33x\). Then divide both sides by 33: \(x=\frac{132}{33}=4\).

Step4: Find \(m\angle R\)

Substitute \(x = 4\) into the expression for \(m\angle R\): \(m\angle R=18x\), so \(m\angle R=18\times4 = 72^{\circ}\).

Step5: Find \(m\angle S\)

Substitute \(x = 4\) into the expression for \(m\angle S\): \(m\angle S=15x + 3\), so \(m\angle S=15\times4+3=60 + 3=63^{\circ}\).

Step6: Find \(m\angle RQS\)

Since \(\angle PQR\) and \(135^{\circ}\) are a linear pair, \(m\angle RQS = 180 - 135=45^{\circ}\).

Answer:

\(m\angle R = 72^{\circ}\)
\(m\angle S = 63^{\circ}\)
\(m\angle RQS = 45^{\circ}\)