Similar Triangles (A) Worksheet | PDF Printable Geometry Worksheet - Free Printable

Let's solve each question in Section A of the "Similar Triangles (A)" worksheet. We are to determine which two triangles are similar in each case and state the reason for similarity.

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🔷 Key Concepts:

Two triangles are similar if:
- Their corresponding angles are equal (AA - Angle-Angle criterion), or
- Their corresponding sides are in proportion (SSS - Side-Side-Side), or
- Two pairs of sides are proportional and the included angles are equal (SAS - Side-Angle-Side).

We’ll use these criteria.

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## ✔ Question 1

Triangles A, B, and C:

- Triangle A: Angles = 100°, 30° → Third angle = 180° – 100° – 30° = 50°
- Triangle B: Angles = 100°, 40° → Third angle = 180° – 100° – 40° = 40°
- Triangle C: Angles = 100°, 50° → Third angle = 180° – 100° – 50° = 30°

So:
- A: 100°, 30°, 50°
- B: 100°, 40°, 40°
- C: 100°, 50°, 30°

Now compare:
- A and C both have angles: 100°, 50°, 30° → Same angles!

✔ Triangles A and C are similar by AA (Angle-Angle) similarity.

> Answer: A and C are similar by AA.

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## ✔ Question 2

Triangles A, B, C:

- Triangle A: Right angle (90°), 55° → Third angle = 180° – 90° – 55° = 35°
- Triangle B: Right angle (90°), 45° → Third angle = 180° – 90° – 45° = 45°
- Triangle C: 55°, 35° → Third angle = 180° – 55° – 35° = 90°

So:
- A: 90°, 55°, 35°
- B: 90°, 45°, 45°
- C: 90°, 55°, 35°

A and C both have angles: 90°, 55°, 35° → Same angles!

✔ Triangles A and C are similar by AA.

> Answer: A and C are similar by AA.

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## ✔ Question 3

Triangles with side lengths:

- Triangle A: Sides = 6, 8, 10
- Triangle B: Sides = 9, 12, 15
- Triangle C: Sides = 4, 5, 6

Check ratios:

Compare A and B:

- A: 6, 8, 10
- B: 9, 12, 15

Ratios:
- 6/9 = 2/3
- 8/12 = 2/3
- 10/15 = 2/3

All ratios = 2/3 → Sides are in proportion

✔ A and B are similar by SSS (Side-Side-Side) similarity.

Now check others:
- A vs C: 6/4 = 1.5, 8/5 = 1.6, 10/6 ≈ 1.67 → Not proportional
- B vs C: 9/4 = 2.25, 12/5 = 2.4, 15/6 = 2.5 → Not proportional

So only A and B are similar.

> Answer: A and B are similar by SSS.

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## ✔ Question 4

Triangles A, B, C:

- Triangle A: Sides = 6, 11; angle = 25°
- Triangle B: Sides = 6, 11; angle = 30°
- Triangle C: Sides = 10.2, 18.7; angle = 25°

Note: In triangle A, we have side 6, side 11, and angle 25° between them?
But the diagram shows the 25° angle adjacent to side 6 and side 11 — so it's the included angle.

Similarly, in triangle C, we have sides 10.2 and 18.7 with a 25° angle between them?

Wait — let’s look carefully:

- Triangle A: Side 6, side 11, angle 25° → between them?
- Triangle B: Side 6, side 11, angle 30° → but this is not the included angle? Wait, the angle is labeled at the vertex opposite side 6? Let's clarify.

Actually, from the diagram:
- In A: The 25° angle is at the vertex where sides 6 and 11 meet → so it’s the included angle.
- In B: The 30° angle is at the vertex where sides 11 and 6 meet → so also included angle.
- But angles are different: 25° vs 30° → so not same.

Now check C:
- Sides: 10.2 and 18.7, with 25° angle between them.

Compare A and C:

- A: sides 6 and 11, angle 25°
- C: sides 10.2 and 18.7, angle 25°

Are the sides proportional?

Check ratio:
- 6 / 10.2 = 60 / 102 = 10 / 17 ≈ 0.588
- 11 / 18.7 = 110 / 187 = 10 / 17 ≈ 0.588

✔ Ratios are equal! And the included angle is 25° in both.

So, two sides proportional and included angle equal → SAS similarity

Thus, A and C are similar by SAS.

Now check B:
- B has sides 6 and 11, but angle 30° — different from A and C.

So B is not similar to either.

> Answer: A and C are similar by SAS.

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✔ Final Answers:

| Question | Similar Triangles | Reason |
|--------|-------------------|--------|
| 1 | A and C | AA (Angles: 100°, 50°, 30°) |
| 2 | A and C | AA (Angles: 90°, 55°, 35°) |
| 3 | A and B | SSS (Sides in ratio 2:3) |
| 4 | A and C | SAS (Sides proportional and included angle 25°) |

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