Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem on the "Similar Triangle Practice 2" worksheet. The goal is to determine if the triangles in each pair are similar, and if so, state the reason: SSS (Side-Side-Side), SAS (Side-Angle-Side), or AA (Angle-Angle). If not similar, write "no".

We'll go through each one step by step.

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1.

Two triangles with one angle labeled 87° in both.
- We only know one angle in each triangle.
- No other angles or sides are given.
- Not enough information to conclude similarity.

✔ Answer: no

> (We need at least two angles for AA, or proportional sides with included angle for SAS/SSS.)

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2.

Two triangles sharing a common vertex, with side lengths:
- Small triangle: 3
- Large triangle: 8

But we don’t have enough side info or angles.

Wait — the figure shows two triangles sharing an angle, and two sides are marked: one side of small triangle = 3, one side of large = 8.

But we cannot see if the included angle is shared or if sides are proportional.

Looking closely: It appears the triangles share an angle, and there’s a side ratio of 3:8.

But unless we know the included angle is the same and the sides are proportional, we can't use SAS.

Also, no other angles or side ratios are given.

So, not enough info.

✔ Answer: no

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3.

A large triangle with a smaller triangle inside it, with sides:
- Smaller triangle: 3 and 6
- Larger triangle: 5 and 10

Wait — the segment from the top to the base is split into 3 and 5? But the base is split into 6 and 10?

Wait — actually, the diagram shows:

- A large triangle with a line drawn from one side to another, forming a smaller triangle inside.
- On the left side: segments 3 and 5 → total 8
- Base: 6 and 10 → total 16

But the smaller triangle has sides 3 and 6, and the larger triangle has sides 5 and 10?

Wait — better interpretation:

It looks like two triangles formed by a line parallel to the base? But not clearly indicated.

But look: the small triangle has side 3 and 6, and the big triangle has side 5 and 10.

Wait — perhaps it's a triangle with a line drawn from a point on one side to a point on another, creating two triangles.

But the only values given:
- One side of small triangle: 3
- One side of big triangle: 5
- Base of small: 6
- Base of big: 10

If the small triangle has sides 3 and 6, and the big triangle has sides 5 and 10, then:

Check proportionality:
- 3/5 = 0.6
- 6/10 = 0.6 → same ratio

But do these sides correspond? Are they corresponding sides?

Assuming the triangles are sharing an angle at the top, and the sides are along the same rays, then yes — if the sides are proportional and the included angle is shared, then SAS applies.

But we don't have the included angle marked as equal.

Wait — but since they are parts of the same triangle, and the sides are along the same lines, the included angle is the same.

So:
- Side 3 and 6 are along the same ray?
Wait — actually, the small triangle has side 3 and 6, and the large triangle has 5 and 10.

But 3/5 = 6/10 = 0.6 → proportional.

And the included angle is the same because they share the vertex.

So SAS similarity applies.

✔ Answer: SAS

> (Because two sides are proportional and the included angle is shared.)

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4.

Right triangle with angles:
- One triangle: 90°, 42°, 48°
- Other triangle: also has 42° and 48°, and right angle implied?

Wait — the large triangle has a right angle and 48°, so third angle = 42°

The small triangle has 42° and 48°, and a right angle?

Yes — the small triangle has a right angle (marked), and 42°, so third angle = 48°

So both triangles have angles: 90°, 42°, 48°

Thus, AA similarity (two angles equal).

✔ Answer: AA

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5.

Two triangles sharing a vertex, with side lengths:
- Small triangle: 2, 3, 4
- Large triangle: 4.7, 3, 4?

Wait — labels:
- Small triangle: sides 2, 3, 4
- Large triangle: sides 4.7, 3, and 4?

Wait — the figure shows:
- One triangle has sides 4, 2, and 3
- The other has sides 4.7, 3, and ?

Wait — the large triangle has side 4.7, and the small has side 3?

Wait — the figure shows a triangle with sides:
- 4, 2, and 3 (small)
- And a larger triangle with sides: 4.7, 3, and 4?

Wait — actually, it looks like the small triangle has sides 2, 3, and 4.

The large triangle has sides: 4.7, 3, and ? — but 3 is shared?

Wait — perhaps the side of length 3 is common?

But the angles are not marked.

But look: the small triangle has sides 2, 3, 4

Large triangle has sides: 4.7, 3, and ?

Wait — maybe the side 3 is shared, and the other sides are 2 and 4.7?

But that doesn’t help.

Alternatively, check if sides are proportional.

Small: 2, 3, 4
Large: ?, 3, 4.7

But unless we know which sides correspond, we can’t compare.

But notice: the small triangle has side 3 and 4, large has 3 and 4.7 — so side 3 is common?

But 2 vs 4.7 — not proportional.

Alternatively, suppose the triangles are oriented such that:

- Small triangle: 2, 3, 4
- Large triangle: 4.7, 3, and ? — but no clear correspondence.

Wait — the figure shows a triangle with side 4.7 and 3, and the small triangle has 4, 2, 3 — but 3 is common?

But the angles are not marked.

No indication of angle equality.

So no way to confirm similarity.

Wait — but look: the small triangle has sides 2, 3, 4

The large triangle has sides 3, 4.7, and ?

Wait — maybe the side of length 3 is shared, and the other sides are 2 and 4.7?

But unless the included angle is the same, we can’t apply SAS.

No angle marked.

So insufficient information.

✔ Answer: no

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6.

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

Check proportions:
- 3 / 9 = 1/3
- 4.5 / 13.5 = 1/3 → same ratio

Now, are these corresponding sides? And is the included angle equal?

They appear to be sharing a vertex, and the sides are along the same lines.

So the included angle is the same.

Therefore, SAS similarity applies.

✔ Answer: SAS

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7.

Two triangles with angles:
- First triangle: 67°, 35° → third angle = 180 - 67 - 35 = 78°
- Second triangle: 77°, 35° → third angle = 180 - 77 - 35 = 68°

So angles: 67°, 35°, 78° vs 77°, 35°, 68°

Only one angle is the same: 35°

Not enough for AA.

So not similar.

✔ Answer: no

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8.

Two triangles with sides:
- Small triangle: 5.7, 7.3, 3.2
- Large triangle: 10.95, 8.55, 4.8

Check ratios:

Compare 5.7 / 10.95 = ?

5.7 ÷ 10.95 ≈ 0.5207

7.3 / 8.55 ≈ 0.854

3.2 / 4.8 ≈ 0.666...

Not equal.

Try different pairing.

Check if ratios are consistent.

Try 5.7 / 10.95 = 570 / 1095 = divide numerator and denominator by 15: 38 / 73 ≈ 0.5205

7.3 / 8.55 = 730 / 855 = 146 / 171 ≈ 0.8538

3.2 / 4.8 = 1/1.5 = 0.666...

No match.

Try: 5.7 / 8.55 = 570 / 855 = 114 / 171 ≈ 0.666...

7.3 / 10.95 ≈ 0.666...

3.2 / 4.8 = 0.666...

Oh! Wait!

Try:

- 5.7 / 8.55 = ?
- 5.7 ÷ 8.55 = 570 / 855 = 114 / 171 = 2 / 3 ≈ 0.666...
- 7.3 / 10.95 = 730 / 1095 = 146 / 219 ≈ 0.666...
- 3.2 / 4.8 = 1/1.5 = 2/3 ≈ 0.666...

All ratios = 2/3

So all three sides are in proportion.

Therefore, SSS similarity.

✔ Answer: SSS

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9.

Two triangles with sides:
- Small triangle: 8, 7
- Large triangle: 17.5, 19.5

Check ratios:
- 8 / 17.5 = 80 / 175 = 16 / 35 ≈ 0.457
- 7 / 19.5 = 70 / 195 = 14 / 39 ≈ 0.359

Not equal.

Try 8 / 19.5 = ? 7 / 17.5 = 70 / 175 = 2/5 = 0.4

8 / 19.5 ≈ 0.41 — not same.

So not proportional.

Also, the triangles appear to be intersecting, but no angle info.

So no.

✔ Answer: no

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10.

Two triangles with angles:
- First triangle: 78°, 48° → third angle = 54°
- Second triangle: 54°, 48° → third angle = 78°

So angles: 78°, 48°, 54° — same set.

So AA similarity (two angles equal).

✔ Answer: AA

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11.

A triangle with two arrows indicating a line parallel to the base.

This is a classic case: a line parallel to the base of a triangle cuts off a smaller triangle.

Then, by AA similarity (corresponding angles are equal due to parallel lines), the two triangles are similar.

✔ Answer: AA

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12.

Two triangles sharing a vertex, with sides:
- Small triangle: 4, 5
- Large triangle: 6, 13

Wait — the base is 13, and a segment of 6 is marked.

But the small triangle has side 4 and 5, large has 6 and 13?

Wait — the diagram shows:

- A large triangle with base 13
- A smaller triangle inside, with base 6
- Side of small triangle: 4
- Side of large triangle: 5

Wait — the side from vertex to base: small triangle has side 4, large has 5?

But the bases are 6 and 13.

Check ratios:
- 4 / 5 = 0.8
- 6 / 13 ≈ 0.4615 — not equal

Are the triangles similar?

Check if angles are equal.

But no angles marked.

But the triangles appear to share a vertex, and the sides are along the same lines.

So the included angle is the same.

Now check if sides are proportional.

Suppose:
- Small triangle: sides 4 and 6
- Large triangle: sides 5 and 13

Then:
- 4 / 5 = 0.8
- 6 / 13 ≈ 0.46 — not equal

Or:
- 4 / 13 = ~0.307
- 6 / 5 = 1.2 — no

No consistent ratio.

So not similar.

✔ Answer: no

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✔ Final Answers:

| # | Answer |
|---|--------|
| 1 | no |
| 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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🔍 Explanation Summary:

- AA: Two angles equal → triangles similar (always true).
- SAS: Two pairs of proportional sides with included angle equal → similar.
- SSS: All three sides proportional → similar.
- If not enough info or ratios don't match → no.

Let me know if you'd like this formatted for printing or with diagrams annotated!