Question

find the value of x and the angle measures using the exterior angle theorem.

x =
∠r =
∠s =

Explanation:

Step1: Apply the Exterior Angle Theorem

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \((5x - 3)+(18x + 3)=161\).
Simplify the left - hand side: \(5x-3 + 18x+3=23x\). So, \(23x = 161\).

Step2: Solve for \(x\)

Divide both sides of the equation \(23x = 161\) by \(23\): \(x=\frac{161}{23}=7\).

Step3: Find \(\angle R\)

Substitute \(x = 7\) into the expression for \(\angle R=(5x - 3)^{\circ}\). Then \(\angle R=(5\times7 - 3)^{\circ}=(35 - 3)^{\circ}=32^{\circ}\).

Step4: Find \(\angle S\)

Substitute \(x = 7\) into the expression for \(\angle S=(18x + 3)^{\circ}\). Then \(\angle S=(18\times7+3)^{\circ}=(126 + 3)^{\circ}=129^{\circ}\).

Answer:

\(x = 7\), \(\angle R=32^{\circ}\), \(\angle S = 129^{\circ}\)