Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s go through each problem one by one, applying the similarity criteria:

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

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1.

Two triangles sharing a vertex with vertical angles. Both have an 87° angle. Since vertical angles are equal, the other pair of angles must also be equal (by triangle sum = 180°). So we have two pairs of congruent angles → AA.

✔ Answer: AA

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2.

We see two triangles sharing an angle (marked with arcs), and adjacent sides labeled 3 and 8. But we don’t know if the sides are proportional or if the included angle is between corresponding sides. The diagram doesn’t give enough info to confirm SAS — the side lengths shown may not be corresponding sides, and no ratios are given. Also, no other angles or sides are labeled.

⚠️ Not enough information to prove similarity.

✔ Answer: no

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3.

Two triangles: one has sides 3, 5, 6; the other has sides 5, 10, and the base is 6 and 10? Wait — actually, it looks like a smaller triangle inside a larger one, sharing an angle. The sides around the shared angle are 3 & 5 in small triangle, and 6 & 10 in large triangle.

Check ratios:
- 3/6 = 1/2
- 5/10 = 1/2

So sides are proportional, and the included angle is the same (shared angle) → SAS.

✔ Answer: SAS

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4.

One triangle is right-angled (90°), with angles 42° and 48°. The other triangle (inside) shares the 42° angle and the right angle → so third angle must be 48°. So both triangles have angles 90°, 42°, 48° → AA.

✔ Answer: AA

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5.

Two triangles with sides: small triangle has sides 2, 3, 4.7? Wait — let's read carefully: the small triangle has sides 2, 3, and 4.7? Actually, looking at the diagram, it appears to be two triangles sharing a common side, but the side lengths are not clearly paired for proportionality. The sides are 4, 2, 4.7 on one, and 3, ? — it’s ambiguous which sides correspond. No clear proportional sides or matching angles.

⚠️ Not enough information.

✔ Answer: no

*(Note: If you assume the triangles share an angle and sides 2 & 3 vs 4 & 4.7 — 2/4=0.5, 3/4.7≈0.64 — not equal → not SAS. And no AA. So “no” is correct.)*

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6.

Two triangles with sides:
- Small: 3, 4.5
- Large: 9, 13.5

Check ratios:
- 3/9 = 1/3
- 4.5/13.5 = 1/3 → proportional!

And they share a vertical angle (between the sides) → SAS.

✔ Answer: SAS

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7.

Two triangles formed by intersecting lines. Angles given: 67°, 35°, and 77°.

In first triangle: angles 67°, 35° → third angle = 180 - 67 - 35 = 78°? Wait, but diagram shows 77° in the other triangle. Let’s check:

Actually, one triangle has angles 67° and 35° → third angle = 78°.

Other triangle has 77° and 35° (vertical angle) → third angle = 180 - 77 - 35 = 68°.

So angles are:
- Triangle 1: 67°, 35°, 78°
- Triangle 2: 77°, 35°, 68°

Only one angle matches (35°) → not AA.

No side lengths → can't use SSS or SAS.

✔ Answer: no

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8.

Two triangles with all sides labeled.

Small triangle: 3.2, 5.7, 7.3
Large triangle: 4.8, 8.55, 10.95

Check ratios:

- 3.2 / 4.8 = 2/3 ≈ 0.666...
- 5.7 / 8.55 = 570/855 = divide numerator and denominator by 285 → 2/3
- 7.3 / 10.95 = 730/1095 = divide by 365 → 2/3

All ratios = 2/3 → SSS

✔ Answer: SSS

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9.

Two triangles with sides:
- Top: 7, 8
- Bottom: 17.5, 19.5

They share a vertical angle.

Check ratios of sides adjacent to the angle:

- 7 / 17.5 = 70 / 175 = 2/5 = 0.4
- 8 / 19.5 = 80 / 195 = 16/39 ≈ 0.410 — NOT equal!

So not proportional → not SAS.

No other info → not similar.

✔ Answer: no

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10.

Two triangles with angles:
- One has 54°, 78° → third angle = 180 - 54 - 78 = 48°
- Other has 48°, 78° → third angle = 54°

So both triangles have angles 48°, 54°, 78° → AA

✔ Answer: AA

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11.

Triangle with a line segment parallel to the base (indicated by arrows). This creates a smaller triangle inside.

By the Basic Proportionality Theorem (or Triangle Similarity Theorem), if a line is drawn parallel to one side of a triangle, it cuts the other two sides proportionally and forms a similar triangle.

Also, corresponding angles are equal (corresponding angles with parallel lines) → AA.

✔ Answer: AA

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12.

Two triangles: small triangle has sides 4, 5, and part of base 6. Large triangle has base 13.

The small triangle is inside the large one, sharing the top angle. The sides from the vertex are 4 and 5, and the base segments are 6 and 13? Wait — total base is 13, and the small triangle’s base is 6? Not necessarily — the diagram shows a line cutting the two sides.

Actually, it’s a classic setup: the small triangle shares the top angle, and the sides are cut proportionally?

Check ratios:

Assume the sides from the vertex are divided into 4 and 5 (small triangle) and extended to 4 + x and 5 + y? But we’re not given the full side lengths — only that the base of small triangle is 6, and total base is 13.

Wait — actually, this is likely not similar because:

If the small triangle has sides 4, 5, 6, and the large triangle has sides proportional? But we don’t know the other two sides — only the base is 13. Unless the sides are proportional, we can’t say.

But here’s the key: the line is drawn from a point on one side to a point on the other, but unless it’s parallel, we can’t assume similarity. The diagram does not show parallel lines (no arrow marks like in #11). So we can’t assume AA or SAS.

Also, 4/13 ≠ 5/13 — not proportional unless we know the full side lengths.

So — insufficient info.

✔ Answer: no

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## ✔ Final Answers:

1. AA
2. no
3. SAS
4. AA
5. no
6. SAS
7. no
8. SSS
9. no
10. AA
11. AA
12. no

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Let me know if you’d like a visual explanation or step-by-step for any specific one!