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 an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \((2x + 10)+(5x - 10)=140\).

Step2: Simplify the equation

Combine like terms: \(2x+5x+10 - 10 = 140\), which simplifies to \(7x=140\).

Step3: Solve for \(x\)

Divide both sides of the equation \(7x = 140\) by \(7\): \(x=\frac{140}{7}=20\).

Step4: Find \(\angle R\)

Substitute \(x = 20\) into the expression for \(\angle R\): \(\angle R=(2x + 10)^{\circ}\). So, \(\angle R=(2\times20 + 10)^{\circ}=(40 + 10)^{\circ}=50^{\circ}\).

Step5: Find \(\angle S\)

Substitute \(x = 20\) into the expression for \(\angle S\): \(\angle S=(5x - 10)^{\circ}\). So, \(\angle S=(5\times20-10)^{\circ}=(100 - 10)^{\circ}=90^{\circ}\).

Answer:

\(x = 20\), \(\angle R=50^{\circ}\), \(\angle S = 90^{\circ}\)