Question

find the measures of the interior angle b and the exterior angle c. ∠b = □° ∠c = □° (triangle with angle at a: 75°, angle at b: x°, exterior angle at c: (2x + 10)°)

Explanation:

Step1: Use 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, \(2x + 10=x + 75\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides of the equation: \(2x+10 - x=x + 75 - x\), which simplifies to \(x+10 = 75\). Then subtract 10 from both sides: \(x+10 - 10=75 - 10\), so \(x = 65\).

Step3: Find the measure of \(\angle B\)

Since \(x = 65\), \(\angle B=x^{\circ}=65^{\circ}\).

Step4: Find the measure of \(\angle C\)

Substitute \(x = 65\) into \(2x + 10\). \(2x+10=2\times65 + 10=130 + 10=140^{\circ}\).

Answer:

\(\angle B = 65^{\circ}\), \(\angle C=140^{\circ}\)