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

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

We'll use the following criteria for triangle similarity:

- AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- SSS (Side-Side-Side): If all three sides of one triangle are proportional to the corresponding sides of another triangle, the triangles are similar.
- SAS (Side-Angle-Side): If two sides are proportional and the included angle is equal, the triangles are similar.

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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°

Now compare:
- Triangle A: 100°, 30°, 50°
- Triangle B: 100°, 40°, 40°
- Triangle C: 100°, 50°, 30°

So, Triangle A and Triangle C both have angles: 100°, 50°, 30° → same angles.

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

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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°

Thus, Triangle A and Triangle C both have angles: 90°, 55°, 35° → same angles.

✔ Answer: A and C are similar by AA similarity.

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3)

Triangles A, B, C 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:
- 6/9 = 2/3
- 8/12 = 2/3
- 10/15 = 2/3

All sides are in ratio 2:3 → proportional.

✔ So, A and B are similar by SSS similarity.

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

✔ Answer: A and B are similar by SSS similarity.

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4)

Triangles A, B, C:

- Triangle A: Side 6, 11, angle 25° between them? Let’s see:
- Side 6 and 11 with included angle 25°?
- Actually, the 25° is at the vertex between sides 6 and 11 → yes, it's included angle.

- Triangle B: Sides 11 and 6, angle 30° — but this is not the included angle between them? Wait, let’s examine:
- It shows side 11, side 6, and angle 30° — but is 30° between them?

Looking closely:
- In Triangle B, the angle shown is 30°, and it's between the side of length 11 and the side of length 6? No — actually, from the diagram, the 30° angle appears to be opposite the side of length 6, not between the 11 and 6.

But wait: Triangle A has:
- Two sides: 6 and 11
- Included angle: 25°

Triangle C has:
- Two sides: 10.2 and 18.7
- Angle: 25° — and from diagram, it seems to be between these two sides.

Let’s check if A and C are similar via SAS:

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

Now check ratio of sides:
- 10.2 / 6 = 1.7
- 18.7 / 11 = 1.7

Yes! Both ratios = 1.7 → proportional, and included angle = 25°

So, A and C satisfy SAS similarity.

Now check B:
- B has sides 11 and 6, but angle 30° — not 25°, so not matching.

Also, the 30° angle is not between the 11 and 6 — it might be adjacent or opposite.

But since only A and C have a 25° included angle and proportional adjacent sides, they are similar.

✔ Answer: A and C are similar by SAS similarity.

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✔ Final Answers:

1) A and C — AA similarity
2) A and C — AA similarity
3) A and B — SSS similarity
4) A and C — SAS similarity

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Let me know if you'd like a visual explanation or help writing these in a worksheet format!