Question 9 of 14intermediate🔧 ApplyLong Answer3 marks

S and T are points on sides PR and QR of Δ\Delta PQR such that ∠P=∠RTS\angle P = \angle RTS. Show that Δ\Delta RPQ ∼\sim Δ\Delta RTS.

Correct Answer

In Δ\Delta RPQ and Δ\Delta RTS,

∠R=∠R\angle R = \angle R (Common angle)

∠P=∠RTS\angle P = \angle RTS (Given)

Therefore, by AAA similarity criterion, Δ\Delta RPQ ∼\sim Δ\Delta RTS.

Exercise: EXERCISE 6.3 | Q: 5 | (Page Page 23)
For More Understanding

Explanation

This question tests the AAA similarity criterion. When two triangles share a common angle and have another pair of equal angles, all three corresponding angles become equal, making the triangles similar.

The key insight is recognizing that ∠R\angle R is common to both triangles Δ\Delta RPQ and Δ\Delta RTS, and combining this with the given condition ∠P=∠RTS\angle P = \angle RTS establishes the similarity.

Solution Steps

  1. Step 1: Identify that ∠R\angle R is common to both triangles Δ\Delta RPQ and Δ\Delta RTS.

  2. Step 2: Note the given condition: ∠P=∠RTS\angle P = \angle RTS.

  3. Step 3: Apply AAA similarity criterion — since two angles are equal, the third angle is also equal (by angle sum property).

  4. Step 4: Conclude that Δ\Delta RPQ ∼\sim Δ\Delta RTS by AAA similarity criterion.

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