Question

list the side lengths of δpqr in order from shortest to longest.
(image of triangle pqr with angles at p: z - 11°, at r: z + 17°, at q: z + 21°)
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Explanation:

Step1: Find the sum of angles

The sum of angles in a triangle is \(180^\circ\). So, \((z - 11^\circ)+(z + 17^\circ)+(z + 21^\circ)=180^\circ\).
Simplify: \(3z + 27^\circ = 180^\circ\).

Step2: Solve for z

Subtract \(27^\circ\) from both sides: \(3z = 153^\circ\).
Divide by 3: \(z = 51^\circ\).

Step3: Find each angle

• \(\angle P = z - 11^\circ = 51^\circ - 11^\circ = 40^\circ\)
• \(\angle R = z + 17^\circ = 51^\circ + 17^\circ = 68^\circ\)
• \(\angle Q = z + 21^\circ = 51^\circ + 21^\circ = 72^\circ\)

Step4: Relate angles to sides

In a triangle, the larger the angle, the longer the opposite side.

• Opposite \(\angle P\) (40°) is \(QR\)
• Opposite \(\angle R\) (68°) is \(PQ\)
• Opposite \(\angle Q\) (72°) is \(PR\)

So, order of angles: \(\angle P < \angle R < \angle Q\), so order of sides: \(QR < PQ < PR\)

Answer:

\(QR < PQ < PR\)