Question

question 1-9
in △pqr, pq = qr, m∠p = (2x - 26)°, and m∠r = (x + 17)°. classify △pqr. select all that apply.
□ right
□ acute
□ obtuse
□ scalene
□ isosceles
□ equilateral

Explanation:

Step1: Identify Triangle Type (Isosceles)

Since \( PQ = QR \), \( \triangle PQR \) has two equal sides, so it's isosceles. Thus, \( \angle P=\angle R \) (base angles of isosceles triangle are equal).

Step2: Solve for \( x \)

Set \( m\angle P = m\angle R \):
\( 2x - 26 = x + 17 \)
Subtract \( x \) from both sides: \( x - 26 = 17 \)
Add 26 to both sides: \( x = 43 \).

Step3: Find Angle Measures

Calculate \( m\angle P \) and \( m\angle R \):
\( m\angle P = 2(43) - 26 = 86 - 26 = 60^\circ \)
\( m\angle R = 43 + 17 = 60^\circ \)
Find \( m\angle Q \): \( 180 - 60 - 60 = 60^\circ \).

Step4: Classify by Angles/Sides

• By sides: Isosceles (two equal sides), but all angles are \( 60^\circ \), so it's also equilateral (all sides equal, all angles \( 60^\circ \)).
• By angles: All angles \( 60^\circ \) (acute, since \( 60^\circ < 90^\circ \)), and equilateral triangles are also acute and isosceles.

Answer:

isosceles, acute, equilateral (select these options as they apply: isosceles because \( PQ = QR \), acute because all angles \( 60^\circ \), equilateral because all angles \( 60^\circ \) implies all sides equal).