3.) according to the triangle inequality theorem for angles, which angle would be the largest? (triangle with vertices p, q, r; sides: pr=26, pq=17, rq=21)

Question

Accepted Answer

# Explanation:
## Step1: Recall Triangle Inequality Theorem for Angles
The theorem states that in a triangle, the largest angle is opposite the longest side.

## Step2: Identify the Longest Side
In triangle $ PQR $, the side lengths are $ PR = 26 $, $ PQ = 17 $, and $ RQ = 21 $. Comparing these lengths: $ 26>21>17 $, so the longest side is $ PR $.

## Step3: Find the Angle Opposite the Longest Side
The side $ PR $ is opposite angle $ Q $ (since in triangle $ PQR $, side $ PR $ is between $ P $ and $ R $, so the angle opposite is $ \angle Q $). Wait, no, correction: Let's label the triangle properly. Let's see the vertices: $ P $, $ Q $, $ R $. Side $ PR $: connects $ P $ and $ R $, so the angle opposite $ PR $ is $ \angle Q $. Side $ PQ $: connects $ P $ and $ Q $, opposite angle $ \angle R $. Side $ RQ $: connects $ R $ and $ Q $, opposite angle $ \angle P $. Wait, no, let's re - check:

- Side $ PQ $ has length 17, opposite angle $ \angle R $.
- Side $ QR $ has length 21, opposite angle $ \angle P $.
- Side $ PR $ has length 26, opposite angle $ \angle Q $. Wait, no, I think I mixed up. Let's use the standard notation: in triangle $ ABC $, side $ a $ is opposite angle $ A $, side $ b $ opposite angle $ B $, side $ c $ opposite angle $ C $. So in triangle $ PQR $, let's denote:

- Vertex $ P $, $ Q $, $ R $.
  - Side opposite $ P $: $ QR = 21 $
  - Side opposite $ Q $: $ PR = 26 $
  - Side opposite $ R $: $ PQ = 17 $

Since the longest side is $ PR = 26 $, which is opposite angle $ Q $, so angle $ Q $ should be the largest? Wait, no, wait, maybe I made a mistake in labeling. Wait, the side between $ P $ and $ R $ is $ PR = 26 $, the side between $ P $ and $ Q $ is $ PQ = 17 $, and the side between $ Q $ and $ R $ is $ QR = 21 $. So:

- Angle at $ Q $: between $ PQ $ and $ QR $, so the side opposite angle $ Q $ is $ PR $ (length 26).
- Angle at $ R $: between $ PR $ and $ QR $, side opposite is $ PQ $ (length 17).
- Angle at $ P $: between $ PQ $ and $ PR $, side opposite is $ QR $ (length 21).

Since $ PR = 26 $ is the longest side, the angle opposite to it (angle $ Q $) is the largest? Wait, no, wait, let's re - do:

Wait, the side lengths:

- $ PQ = 17 $ (between $ P $ and $ Q $)
- $ QR = 21 $ (between $ Q $ and $ R $)
- $ PR = 26 $ (between $ P $ and $ R $)

So:

- Angle opposite $ PQ $ (17) is $ \angle R $
- Angle opposite $ QR $ (21) is $ \angle P $
- Angle opposite $ PR $ (26) is $ \angle Q $

Since $ 26 $ is the longest side, the angle opposite to it ($ \angle Q $) is the largest? Wait, no, wait, maybe I had the sides wrong. Wait, looking at the diagram: the side from $ R $ to $ P $ is 26, from $ P $ to $ Q $ is 17, from $ Q $ to $ R $ is 21. So:

- Side $ RP = 26 $
- Side $ PQ = 17 $
- Side $ QR = 21 $

So the angles:

- $ \angle Q $ is between $ PQ $ and $ QR $, so the side opposite $ \angle Q $ is $ RP = 26 $
- $ \angle R $ is between $ QR $ and $ RP $, side opposite is $ PQ = 17 $
- $ \angle P $ is between $ RP $ and $ PQ $, side opposite is $ QR = 21 $

Since $ RP = 26 $ is the longest side, the angle opposite to it ($ \angle Q $) is the largest. Wait, but maybe I made a mistake. Wait, no, the correct approach: in a triangle, longer side → larger opposite angle. So the longest side is 26 (side $ PR $), so the angle opposite to $ PR $ is $ \angle Q $. So angle $ Q $ is the largest. Wait, but let's confirm the side - angle correspondence again. Let's name the triangle: vertices $ P $, $ Q $, $ R $.

- Side $ PQ $: connects $ P $ and $ Q $, length 17. Opposite angle: $ \angle R $ (because angle $ R $ is at vertex $ R $, between $ PR $ and $ QR $, so the side not connected to $ R $ is $ PQ $).
- Side $ QR $: connects $ Q $ and $ R $, length 21. Opposite angle: $ \angle P $ (angle at $ P $, between $ PQ $ and $ PR $, side not connected to $ P $ is $ QR $).
- Side $ PR $: connects $ P $ and $ R $, length 26. Opposite angle: $ \angle Q $ (angle at $ Q $, between $ PQ $ and $ QR $, side not connected to $ Q $ is $ PR $).

Since $ PR = 26 $ is the longest side, $ \angle Q $ (opposite $ PR $) is the largest angle.

# Answer:
The largest angle is $ \angle Q $ (the angle at vertex $ Q $).

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Triangle Inequality Theorem for Angles

Step2: Identify the Longest Side

Step3: Find the Angle Opposite the Longest Side

Answer: