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

Let’s go through each question one by one. We’re looking for two triangles that are similar in each set, and we need to say why they’re similar.

Remember: Triangles are similar if their angles match (AA rule) or if their sides are in proportion (SSS or SAS rules). Since most of these show angles or side lengths, we’ll use those clues.

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Question 1:

Triangle A: angles = 30°, 100° → third angle = 180 - 30 - 100 = 50°

Triangle B: angles = 40°, 100° → third angle = 180 - 40 - 100 = 40°

Triangle C: angles = 100°, 50° → third angle = 180 - 100 - 50 = 30°

So:

- Triangle A has angles: 30°, 50°, 100°
- Triangle B has angles: 40°, 40°, 100°
- Triangle C has angles: 30°, 50°, 100°

→ So A and C have the same three angles! That means they are similar by AA (Angle-Angle) — actually AAA, but AA is enough since two matching angles mean the third must match too.

✔ Answer for Q1: Triangles A and C are similar because all corresponding angles are equal (AA similarity).

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Question 2:

Triangle A: right triangle with one angle 55° → other non-right angle = 90 - 55 = 35°

Triangle B: right triangle with one angle 45° → other non-right angle = 90 - 45 = 45°

Triangle C: angles given as 55° and 35° → third angle = 180 - 55 - 35 = 90° → so it’s also a right triangle!

So:

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

→ So A and C have the same angles: 55°, 35°, 90°

✔ Answer for Q2: Triangles A and C are similar because all corresponding angles are equal (AA similarity).

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Question 3:

We’re given side lengths. Let’s check ratios.

Triangle A: sides 6, 8, 10 → let’s order them: 6, 8, 10

Triangle B: sides 9, 12, 15 → ordered: 9, 12, 15

Triangle C: sides 4, 5, 6 → ordered: 4, 5, 6

Check if any two sets of sides are proportional.

Try A and B:

6/9 = 2/3
8/12 = 2/3
10/15 = 2/3

→ All ratios are 2/3 → so sides are proportional → SSS similarity

Now check A and C:

6/4 = 1.5
8/5 = 1.6 → not equal → not proportional

B and C:

9/4 = 2.25
12/5 = 2.4 → not equal

Only A and B have proportional sides.

Also note: Triangle A (6-8-10) is a scaled version of 3-4-5 right triangle → yes, 6² + 8² = 36+64=100=10² → right triangle.

Triangle B (9-12-15): 9²+12²=81+144=225=15² → also right triangle.

But even without knowing they’re right triangles, the side ratios prove similarity.

✔ Answer for Q3: Triangles A and B are similar because their corresponding sides are in proportion (SSS similarity).

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Question 4:

Given sides and some angles.

Triangle A: sides 6, 11, and included angle 25°? Wait — look at diagram: angle between sides 6 and 11 is 25°? Actually, from drawing, it looks like the 25° angle is between the two labeled sides? Let me assume based on standard labeling.

Actually, looking carefully:

In Triangle A: sides 6 and 11 with included angle 25°? Or is the 25° opposite one side? The diagram shows the 25° angle at the vertex where sides 6 and 11 meet? Probably yes — so it’s SAS situation.

Wait — better to compare what’s given.

Triangle A: sides 6 and 11, angle between them = 25°? (Assuming from diagram)

Triangle B: sides 6 and 11, angle between them = 30°? Diagram shows 30° angle near side 6 and 11 — probably included angle.

Triangle C: sides 10.2 and 18.7, angle between them = 25°? Diagram shows 25° angle between those two sides.

Wait — let’s list clearly:

Triangle A: two sides 6 and 11, included angle 25°

Triangle B: two sides 6 and 11, included angle 30° → different angle → not similar to A

Triangle C: two sides 10.2 and 18.7, included angle 25°

Compare A and C:

Side ratio: 6 / 10.2 = ? and 11 / 18.7 = ?

Calculate:

6 ÷ 10.2 ≈ 0.5882

11 ÷ 18.7 ≈ 0.5882 → same!

And included angle is 25° for both → so SAS similarity: two sides proportional and included angle equal.

Check exact fractions:

6 / 10.2 = 60 / 102 = 10 / 17

11 / 18.7 = 110 / 187 = divide numerator and denominator by 11 → 10 / 17 → same!

Yes! So 6:10.2 = 11:18.7 = 10:17, and included angle 25° matches.

So A and C are similar by SAS.

Triangle B has different included angle (30°), so doesn’t match.

✔ Answer for Q4: Triangles A and C are similar because two sides are proportional and the included angle is equal (SAS similarity).

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Final Answer:

1) Triangles A and C — AA similarity (all angles match: 30°, 50°, 100°)
2) Triangles A and C — AA similarity (all angles match: 35°, 55°, 90°)
3) Triangles A and B — SSS similarity (sides in ratio 2:3)
4) Triangles A and C — SAS similarity (two sides proportional and included angle equal)