The given triangle PQR is acute and equilateral. Hence, options A and D are correct.
We are given that triangle PQR has PQ = QR, which means it is an isosceles triangle. Additionally, we are given the measures of two angles:
* m∠P = 2x - 26 degrees
* m∠R = x + 17 degrees
To classify the triangle completely, we need to determine the measure of the third angle, ∠Q.
In any triangle, the sum of the angles is 180 degrees. Therefore:
m∠P + m∠Q + m∠R = 180 degrees
Substitute the given expressions for the measures of ∠P and ∠R:
(2x - 26 degrees) + m∠Q + (x + 17 degrees) = 180 degrees
Combine like terms:
3x - 9 degrees + m∠Q = 180 degrees
Move the constant term to the right side of the equation:
3x + m∠Q = 189 degrees
Isolate m∠Q:
m∠Q = 189 degrees - 3x
Now, we can use the fact that ∠Q and ∠R are equal in an isosceles triangle:
m∠Q = m∠R = x + 17 degrees
Substitute this expression into the equation we obtained for m∠Q:
x + 17 degrees = 189 degrees - 3x
Solve for x:
4x = 172 degrees
x = 43 degrees
Now, substitute the value of x back into the expressions for m∠P, m∠Q, and m∠R:
m∠P = 2(43) - 26 = 86 - 26 = 60 degrees
m∠Q = 43 + 17 = 60 degrees
m∠R = 43 + 17 = 60 degrees
Therefore, all three angles in triangle PQR measure 60 degrees, making it an **equilateral triangle**.
Therefore, the correct options are:
* a) Acute: Since all angles are 60 degrees, they are all less than 90 degrees, making the triangle acute.
* d) Equilateral: All three sides are equal, and all angles are equal to 60 degrees, making the triangle equilateral.
The incorrect options are:
* b) Obtuse: None of the angles are greater than 90 degrees, so the triangle cannot be obtuse.
* c) Right: None of the angles are equal to 90 degrees, so the triangle cannot be right.
* f) Scalene: Since two sides are equal (PQ and QR), the triangle cannot be scalene.