Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Let's go through each problem on the "Similar Triangle Practice 2" worksheet and determine whether the triangles are similar using the AA (Angle-Angle), SSS (Side-Side-Side), or SAS (Side-Angle-Side) similarity criteria. If they are not similar, we'll write "no".

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🔍 Similarity Criteria Recap:

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

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Now let’s solve each problem:

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1.

Two triangles with one angle marked as 87° in both.

- Only one angle is given for each triangle.
- No other angles or side info.
- We cannot confirm AA (need two angles).
- No side info → can't use SSS or SAS.

✔ Answer: no

> *We need more information.*

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2.

Two triangles sharing a vertex, with one side labeled 3 and 8.

- The small triangle has side 3; larger triangle has side 8.
- They appear to share an angle at the vertex.
- But only one side is labeled per triangle.
- No other sides or angles given.

✔ Answer: no

> *Not enough info to apply any criterion.*

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3.

A large triangle with a smaller triangle inside it.
- Large triangle: sides 5, 6, 10? Wait — actually:
- Bottom side = 10
- A segment divides it into 6 and 4? Wait, labels: 3, 5, 6, 10
- It looks like a triangle with a line drawn from one side to the opposite side, forming a smaller triangle.

But look closely:
- Small triangle has side 3 and 6?
- Larger triangle has side 5 and 10?

Wait:
- The small triangle has one side 3, and the large triangle has side 5?
- Bottom base is split: 6 and 4? Wait, bottom is labeled 6 and 10?

Actually, the big triangle has base 10, and a point splits it into 6 and 4? But only 6 is labeled.

Wait — re-examining:
- The triangle has a line drawn from a point on one side to the opposite side.
- One side of the small triangle is 3, and the large triangle has a side of 5.
- The base of the large triangle is 10, and the small triangle's base is 6?

Wait — perhaps this is a case of parallel lines creating similar triangles?

But there is no indication of parallel lines.

Alternatively, maybe the two triangles share an angle, and have proportional sides?

Let’s assume:
- Small triangle: sides 3 and 6
- Large triangle: sides 5 and 10

Then ratio: 3/5 = 0.6, 6/10 = 0.6 → same ratio

But do these sides include the included angle?

If the angle between them is the same (they share a vertex), then SAS applies.

But are the sides adjacent to the shared angle?

Yes — if the 3 and 5 are along the same ray, and 6 and 10 are along the base, then yes.

So if:
- Side 3 corresponds to side 5
- Side 6 corresponds to side 10
- Angle between them is common

Then ratio = 3/5 = 6/10 = 0.6 → proportional

✔ Answer: SAS

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

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4.

Right triangle with angles: 90°, 42°, 48°
Another triangle has angles: 42°, 48°, so third angle = 90°

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

✔ Answer: AA

> Two angles are equal → triangles are similar.

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5.

Two triangles: one with sides 4, 2, 4.7; other with sides 3, 4.7, 3?

Wait — labeling:
- Left triangle: sides 4, 2, and 4.7
- Right triangle: sides 3, 4.7, 3?

Wait — no. The figure shows:
- Inner triangle: sides 4, 2, and 4.7?
- Outer triangle: sides 3, 4.7, and 3?

Wait — actually, it looks like a triangle with a smaller triangle cut out, but the sides are labeled:
- One triangle has sides: 4, 2, and 4.7
- Other has: 3, 3, and 4.7?

No — look again.

The figure shows:
- A triangle with side 4.7 and 3 on top, and 4 and 2 on the left and bottom.

Wait — perhaps it's a triangle divided by a line segment.

But better: check ratios.

Suppose:
- Smaller triangle: sides 2, 4, 4.7
- Larger triangle: sides 3, ?, ?

But no clear correspondence.

Wait — maybe the two triangles share a vertex and have sides:
- Small: 2 and 4
- Large: 3 and 4.7?

But 2/3 ≈ 0.67, 4/4.7 ≈ 0.85 → not proportional.

No clear proportionality.

Also, no angles given.

✔ Answer: no

> Not enough information to determine similarity.

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6.

Two triangles sharing a vertex, with sides:
- Small triangle: 3 and 4.5
- Large triangle: 9 and 13.5

Check ratios:
- 3 / 9 = 1/3
- 4.5 / 13.5 = 1/3

So two sides are proportional: 1:3

Do they share the included angle?

Yes — they appear to share the angle at the vertex.

So SAS applies.

✔ Answer: SAS

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7.

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

Wait — angles:
- First: 67°, 35°, 78°
- Second: 77°, 35°, 68°

Only 35° is common.

Other angles differ → not AA.

No side info.

✔ Answer: no

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8.

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

Check ratios:
- 5.7 / 10.95 = ? → 5.7 ÷ 10.95 = 0.5207
- 7.3 / 8.55 ≈ 0.854
- 3.2 / 4.8 ≈ 0.666

Not proportional.

Try different pairings.

Is there a consistent scale factor?

Try 5.7 / 10.95 ≈ 0.5207
7.3 / 8.55 ≈ 0.854 → not same

Wait — what if we try:

Check: 5.7 / 8.55 = ? → 5.7 ÷ 8.55 ≈ 0.667
7.3 / 10.95 ≈ 0.667
3.2 / 4.8 ≈ 0.667

Ah! So:
- 5.7 / 8.55 = 0.666...
- 7.3 / 10.95 = 0.666...
- 3.2 / 4.8 = 0.666...

So all sides are in ratio 2:3 (since 2/3 ≈ 0.666...)

So SSS applies.

✔ Answer: SSS

> All three sides are proportional.

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9.

Two triangles overlapping, forming an "X"

Sides:
- Top triangle: 8 and 7
- Bottom triangle: 17.5 and 19.5

Check ratios:
- 8 / 17.5 = 0.457
- 7 / 19.5 ≈ 0.359 → not equal

Wait — perhaps different correspondence.

Maybe:
- 8 / 19.5 ≈ 0.41
- 7 / 17.5 = 0.4 → close but not exact

Wait — 8 / 17.5 = 0.457
7 / 19.5 ≈ 0.359

Not proportional.

But wait — are the triangles formed by intersecting lines?

Possibly vertical angles.

But we need to see if the sides are proportional.

Alternatively, check if the segments are proportional.

Wait — the figure shows:
- One triangle has sides 8 and 7
- Other has 17.5 and 19.5

But unless we know which sides correspond, hard to say.

But notice: 8 / 17.5 = 16/35 = 0.457
7 / 19.5 = 14/39 ≈ 0.359 → not equal

Try: 8 / 19.5 ≈ 0.41, 7 / 17.5 = 0.4 → still not matching.

No consistent ratio.

Also, no angles given.

✔ Answer: no

> Sides not proportional.

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10.

Two triangles with angles:
- One: 54°, 78°, 48° → sum = 180 → good
- Other: 78°, 48°, and missing angle = 180 - 78 - 48 = 54°

So both triangles have angles: 54°, 78°, 48°

✔ Answer: AA

> Two angles match → triangles are similar.

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11.

Triangle with two arrows pointing inward — suggests parallel lines cutting the triangle.

This is a classic case of similar triangles due to parallel lines.

When a line is drawn parallel to one side of a triangle, it creates a smaller triangle similar to the original.

So the two triangles (small and large) are similar by AA because:
- Corresponding angles are equal (due to parallel lines and transversals)

✔ Answer: AA

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12.

Large triangle with a smaller triangle inside, sharing a vertex.

Sides:
- Large triangle: 5, 4, 6, 13?
- Wait: one side is 13, another is 6, and a smaller triangle has 5 and 4?

Wait — the base is labeled 13, and a segment divides it into 6 and 7? But only 6 is labeled.

Wait — the figure shows:
- Large triangle has base 13
- A line from a point on one side to the base, forming a smaller triangle
- Smaller triangle has side 5 and 4
- Larger triangle has side 6 and 13?

Wait — labels:
- One side of small triangle: 5
- Adjacent side: 4
- Base of large triangle: 13
- Segment on base: 6

So perhaps:
- Small triangle: sides 5, 4, ?
- Large triangle: sides 5, 6, 13?

No — not clear.

Wait — likely:
- The small triangle has sides 5 and 4
- The large triangle has corresponding sides 6 and 13?

But 5/6 ≈ 0.833, 4/13 ≈ 0.308 → not proportional.

Alternatively, maybe the small triangle has side 5 and the large triangle has side 6? Then 5/6 = 0.833

But no other side info.

Alternatively, if the line is parallel to the base, then the triangles would be similar.

But no indication of parallel lines.

So without that, we can’t assume.

Given only side lengths 5, 4, 6, 13 — no clear proportionality.

✔ Answer: no

> Not enough information or no evidence of proportionality.

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✔ Final Answers:

| Problem | 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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📝 Summary of Reasoning:

- AA: Used when two angles are known and equal (e.g., problems 4, 10, 11)
- SAS: Used when two sides are proportional and the included angle is shared (problems 3, 6)
- SSS: Used when all three sides are proportional (problem 8)
- no: When not enough info or proportions don’t match (problems 1, 2, 5, 7, 9, 12)

Let me know if you'd like this formatted as a printable answer key!