Question

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

Explanation:

Step1: Identify Isosceles Triangle Angles

Since \( PQ = QR \), \( \triangle PQR \) is isosceles with \( \angle P=\angle R \)? Wait, no—wait, in \( \triangle PQR \), sides \( PQ = QR \), so the base is \( PR \), and the equal angles are \( \angle P \) and \( \angle R \)? Wait, no: in a triangle, equal sides are opposite equal angles. So \( PQ = QR \), so angle opposite \( PQ \) is \( \angle R \), and angle opposite \( QR \) is \( \angle P \). So \( \angle P=\angle R \)? Wait, no, \( PQ = QR \), so \( \angle P = \angle R \)? Wait, let's check: side \( PQ \) is opposite \( \angle R \), side \( QR \) is opposite \( \angle P \). So if \( PQ = QR \), then \( \angle R=\angle P \). Wait, but the problem gives \( m\angle P=(2x - 26)^\circ \) and \( m\angle R=(x + 17)^\circ \). Wait, that would mean \( 2x - 26=x + 17 \). Let's solve for \( x \).

Step2: Solve for \( x \)

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

Step3: Find Angle Measures

Now, \( m\angle P=2(43)-26=86 - 26 = 60^\circ \). \( m\angle R=43 + 17 = 60^\circ \). Then, \( m\angle Q=180 - 60 - 60 = 60^\circ \). Wait, but that would make it equilateral? Wait, no, wait—wait, maybe I made a mistake. Wait, \( PQ = QR \), so sides \( PQ \) and \( QR \) are equal, so angles opposite them: \( \angle R \) (opposite \( PQ \)) and \( \angle P \) (opposite \( QR \))? Wait, no, vertex \( Q \) is between \( P \) and \( R \), so \( PQ \) and \( QR \) meet at \( Q \), so the triangle has vertices \( P \), \( Q \), \( R \), with \( PQ = QR \). So sides: \( PQ \), \( QR \), \( PR \). So angle at \( P \) is \( \angle P \), at \( Q \) is \( \angle Q \), at \( R \) is \( \angle R \). So \( PQ = QR \), so angle opposite \( PQ \) is \( \angle R \), angle opposite \( QR \) is \( \angle P \). Therefore, \( \angle P=\angle R \). Wait, but the problem's angle measures: \( \angle P=(2x - 26) \), \( \angle R=(x + 17) \). So setting them equal: \( 2x - 26=x + 17 \), so \( x = 43 \), as above. Then \( \angle P = 60^\circ \), \( \angle R = 60^\circ \), so \( \angle Q = 60^\circ \). Wait, but then all angles are 60°, so it's equilateral (and thus isosceles, acute, equilateral). But let's check the options: right, acute, obtuse, scalene, isosceles, equilateral.

Wait, maybe I messed up the equal angles. Wait, \( PQ = QR \), so the equal sides are \( PQ \) and \( QR \), so the base is \( PR \), and the two equal angles are \( \angle P \) and \( \angle R \)? Wait, no—wait, in triangle \( PQR \), vertices are \( P \), \( Q \), \( R \). So side \( PQ \) is between \( P \) and \( Q \), side \( QR \) is between \( Q \) and \( R \), side \( PR \) is between \( P \) and \( R \). So \( PQ = QR \), so the angles opposite those sides: side \( PQ \) is opposite \( \angle R \), side \( QR \) is opposite \( \angle P \). Therefore, \( \angle R = \angle P \). So solving \( 2x - 26 = x + 17 \) gives \( x = 43 \), so \( \angle P = 60^\circ \), \( \angle R = 60^\circ \), so \( \angle Q = 60^\circ \). So the triangle is equilateral (all sides equal, since all angles are 60°), isosceles (at least two sides equal), acute (all angles < 90°), and equilateral. Wait, but let's check the angle sum: 60 + 60 + 60 = 180, correct.

Wait, but maybe I made a mistake in the angle opposite. Wait, \( PQ = QR \), so the sides adjacent to \( Q \) are equal, so \( \angle P = \angle R \), yes. So then all angles are 60°, so the triangle is equilateral, isosceles, acute, and equilateral. But let's check the options: the checkboxes are right, acute, obtuse, scalene, isosceles, equilateral.

So let's verify:

- **Isosceles**: Yes, because \( PQ = QR \).
- **Equilateral**: Yes, because all angles are 60°, so all sides are equal (since in a triangle, equal angles imply equal sides, and if all angles are 60°, all sides are equal).
- **Acute**: All angles (60°) are less than 90°, so acute.
- **Right**: No, 60°≠90°.
- **Obtuse**: No, 60°<90°.
- **Scalene**: No, at least two sides equal (actually all three).

Wait, but let's re-examine the problem. Wait, maybe I misread the side lengths. The problem says "in \( \triangle PQR \), \( PQ = QR \)". So \( PQ = QR \), so it's isosceles with \( \angle P = \angle R \). Then solving \( 2x - 26 = x + 17 \) gives \( x = 43 \), so \( \angle P = 60° \), \( \angle R = 60° \), so \( \angle Q = 60° \). So all angles are 60°, so it's equilateral (hence isosceles), acute, and equilateral.

So the correct classifications are: acute, isosceles, equilateral. Wait, but let's check the angle measures again. Wait, maybe I made a mistake in the angle opposite. Wait, \( PQ = QR \), so side \( PQ \) is opposite \( \angle R \), side \( QR \) is opposite \( \angle P \). So \( \angle R = \angle P \). So \( 2x - 26 = x + 17 \) → \( x = 43 \). Then \( \angle P = 2*43 -26 = 86 -26 = 60 \), \( \angle R = 43 +17 = 60 \), so \( \angle Q = 180 - 60 -60 = 60 \). So all angles 60°, so equilateral (so isosceles, acute, equilateral).

So the checkboxes to select are: acute, isosceles, equilateral. Wait, but let's confirm the definitions:

- **Isosceles**: At least two sides equal (here, \( PQ = QR \), so yes).
- **Equilateral**: All sides equal (since all angles 60°, so yes, because in a triangle, equal angles imply equal sides, and three equal angles mean three equal sides).
- **Acute**: All angles < 90° (60° < 90°, so yes).
- **Right**: No, no angle 90°.
- **Obtuse**: No, no angle > 90°.
- **Scalene**: No, at least two sides equal.

So the correct options are acute, isosceles, equilateral. But let's check the problem's checkboxes: the options are right, acute, obtuse, scalene, isosceles, equilateral (from the image: the checkboxes are labeled right, acute, obtuse, scalene, isosceles, equilateral? Wait, the image shows:

- right
- acute
- obtuse
- scalene
- isosceles
- equilateral

Wait, the order in the image: first checkbox is "right", then "acute", "obtuse", "scalene", "isosceles", "equilateral". Wait, maybe I misread the angle labels. Wait, maybe \( PQ = QR \), so \( \angle P = \angle R \), but maybe the problem has a typo, or I misread the angles. Wait, let's re-express:

Given \( PQ = QR \), so \( \angle P = \angle R \) (opposite equal sides). So \( m\angle P = m\angle R \). So \( 2x - 26 = x + 17 \) → \( x = 43 \). Then \( m\angle P = 2*43 -26 = 60° \), \( m\angle R = 43 +17 = 60° \), so \( m\angle Q = 180 - 60 -60 = 60° \). So all angles 60°, so the triangle is equilateral (hence isosceles), acute, and equilateral. So the checkboxes to select are acute, isosceles, equilateral.

Wait, but let's confirm the definitions:

- **Acute triangle**: All angles are acute (less than 90°). 60° is acute, so yes.
- **Isosceles triangle**: At least two sides equal. Since \( PQ = QR \), yes.
- **Equilateral triangle**: All sides equal (since all angles are 60°, by the converse of the isosceles triangle theorem, all sides are equal). So yes.

So the correct options are acute, isosceles, equilateral.

Answer:

The correct checkboxes are:

• acute
• isosceles
• equilateral

(In the image, these correspond to the checkboxes labeled "acute", "isosceles", and "equilateral".)