Question

in △pqr, pq = qr. if m∠p = (3x - 120)° and m∠r = (x - 30)°, classify △pqr. select all that apply
□ a) acute
□ b) obtuse
□ c) right
□ d) equilateral
□ e) isosceles
□ f) scalene

Explanation:

Step1: Use the property of isosceles triangle

In \(\triangle PQR\), since \(PQ = QR\), then \(\angle P=\angle R\) (angles opposite equal sides are equal). So, \(3x - 120=x - 30\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(3x-x-120=x - x- 30\), which gives \(2x-120=-30\).
Add \(120\) to both sides: \(2x-120 + 120=-30+120\), so \(2x = 90\).
Divide both sides by \(2\): \(x=\frac{90}{2}=45\).

Step3: Find the measures of \(\angle P\) and \(\angle R\)

Substitute \(x = 45\) into \(\angle P=(3x - 120)^{\circ}\): \(\angle P=(3\times45-120)^{\circ}=(135 - 120)^{\circ}=15^{\circ}\), and \(\angle R=(x - 30)^{\circ}=(45-30)^{\circ}=15^{\circ}\).

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

Using the angle - sum property of a triangle (\(\angle P+\angle Q+\angle R = 180^{\circ}\)), we have \(15^{\circ}+\angle Q+15^{\circ}=180^{\circ}\).
So, \(\angle Q=180-(15 + 15)=150^{\circ}\).

Answer:

B. obtuse, E. isosceles