Question

in the triangle below, ∠s is a right - angle. suppose that m∠q=(3x - 17)° and m∠r=(5x - 13)°. (a) write an equation to find x. make sure you use an “=” sign in your answer. (b) find the degree measure of each angle.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Since ∠S is a right - angle (90°), and ∠Q=(3x - 17)° and ∠R=(5x - 13)°, we have the equation (3x−17)+(5x - 13)+90 = 180.
(3x−17)+(5x - 13)+90=180

Step2: Simplify the left - hand side of the equation

Combine like terms: 3x+5x−17−13 + 90=180, which simplifies to 8x+60 = 180.
8x+60=180

Step3: Solve for x

Subtract 60 from both sides: 8x=180 - 60, so 8x=120. Then divide both sides by 8: x = 15.
8x=120, x = 15

Step4: Find the measure of ∠Q

Substitute x = 15 into the expression for ∠Q: m∠Q=(3x - 17)°=(3×15−17)°=(45 - 17)° = 28°.
m∠Q=(3×15−17)°=28°

Step5: Find the measure of ∠R

Substitute x = 15 into the expression for ∠R: m∠R=(5x - 13)°=(5×15−13)°=(75 - 13)° = 62°.
m∠R=(5×15−13)°=62°

Step6: Recall the measure of ∠S

Since ∠S is a right - angle, m∠S = 90°.

Answer:

(a) Equation: (3x−17)+(5x - 13)+90 = 180
(b)
m∠Q = 28°
m∠R = 62°
m∠S = 90°