Similar Shapes Worksheet - Determine if each pair of geometric figures is similar based on size, angles, and side lengths.

To determine if two shapes are similar, they must meet two conditions:

1. Corresponding angles are equal.
2. Corresponding sides are in proportion (same ratio).

Let’s go through each pair one by one.

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1) Two Squares

- Left: 2cm × 2cm
- Right: 4cm × 4cm

✔ All angles are 90° — so corresponding angles are equal.

✔ Side ratios: 2:4 = 1:2 — all sides scale by the same factor.

✔️ Similar

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2) Two Triangles

Both triangles have angles:
- 23°, 50°, 50° → so angles match.

Check side ratios:

Left triangle sides: 6cm, 3cm
Right triangle sides: 2cm, 1cm

Ratio of corresponding sides:
- 6cm / 2cm = 3
- 3cm / 1cm = 3

✔ Angles equal, sides proportional.

✔️ Similar

*(Note: The triangles are isosceles with base angles 50°, so the sides opposite equal angles should be equal — which they are. The side between the 50° angles is the base, and the other two are legs. So we matched the legs and bases correctly.)*

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3) Two Pentagons

Left pentagon: sides 5m, 5m, 6m, 6m, 3m
Right pentagon: all sides 4m

This is a problem.

The left pentagon is not regular — it has different side lengths.
The right pentagon is regular — all sides equal.

Even if angles were somehow the same (which we can’t assume without being told), the side ratios would not be consistent.

For example:
- One side 5m vs 4m → ratio 5/4 = 1.25
- Another side 6m vs 4m → ratio 6/4 = 1.5 → not the same

✘ Sides are not in proportion.

✔️ Not similar

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4) Two Irregular Hexagons? (Actually pentagons)

Left shape: sides 12mm, 6mm, 3mm; angles 135°, 135°, 90°
Right shape: sides 4mm, 2mm, 1mm; angles 135°, 135°, 90°

Wait — both shapes appear to have 5 sides? Let's count:

Left: top edge, right vertical, bottom right diagonal, bottom horizontal, left diagonal → 5 sides.
Same for right.

So they are pentagons.

Angles shown: both have two 135° and one 90° — but we need all five angles to correspond.

We’re only given 3 angles per shape — so we cannot confirm that all corresponding angles are equal.

But let’s check side ratios anyway:

Assume correspondence based on position:

- Long vertical: 12mm vs 4mm → ratio 3
- Bottom slant: 6mm vs 2mm → ratio 3
- Bottom horizontal: 3mm vs 1mm → ratio 3

✔ So the three labeled sides are in proportion.

BUT — since we don’t know the other two sides or angles, we cannot guarantee similarity.

However, in many school-level problems, if the visible angles match and sides are proportional, and the shapes “look” the same, they are considered similar.

But strictly speaking, we lack information about the other angles and sides.

➡️ Technically, NOT enough info — but likely intended to be similar.

Let’s look at the image again — the shapes are oriented the same way, and the labeled angles and sides correspond.

In educational contexts, this is often accepted as similar.

✔ Assume Similar (with caveat)

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5) Two Triangles

Yellow triangle: angles 80°, 90°, 60° → Wait! 80+90+60 = 230° ✘ Impossible!

That’s a mistake.

Actually, looking closely:

Yellow triangle: 80° at top, 90° at bottom left, so third angle = 180 - 80 - 90 = 10°, not 60°.

The 60° label is probably miswritten — or maybe it’s meant to be 10°.

Green triangle: angles 35°, 90°, 50° → 35+90+50=175° ✘ Also impossible!

Wait — 35+90+50=175? That’s wrong too.

Actually, green triangle: 35° at top, 90° at bottom left → third angle = 180 - 35 - 90 = 55°, not 50°.

So both triangles have incorrect angle sums — meaning the labels are wrong or misprinted.

But let’s ignore the angle labels for a moment and check side ratios.

Yellow triangle sides: 35cm, 45cm, 20cm
Green triangle sides: 25cm, 30cm, 20cm

Try matching:

Suppose we match 20cm to 20cm → ratio 1:1

Then 35cm vs 25cm → 35/25 = 1.4
45cm vs 30cm → 45/30 = 1.5 → not equal

Try matching 35cm to 25cm → 35/25 = 1.4
45cm to 30cm → 45/30 = 1.5 → still not same

Try 35cm to 30cm → 35/30 ≈ 1.167
45cm to 25cm → 45/25 = 1.8 → nope

No consistent ratio.

Also, angles don’t add up — so likely not similar.

✘ Not similar

*(Probably a typo in the problem — but based on given data, not similar.)*

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6) Two Irregular Shapes

Left: looks like a house shape — pentagon?
Right: rectangle with a cut-off corner — hexagon?

They have different numbers of sides — one has 5 sides, the other has 6.

✔ Similar shapes must have the same number of sides and corresponding angles/sides.

✘ Different number of sides → not similar

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## ✔ Final Answers:

1) Yes — similar squares
2) Yes — similar triangles (same angles, proportional sides)
3) No — pentagons with unequal side ratios
4) Yes — (assuming missing info is consistent; sides proportional, angles match where shown)
5) No — angles don’t add up, sides not proportional
6) No — different number of sides

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## 📝 Summary:

| Pair | Similar? |
|------|----------|
| 1 | ✔ Yes |
| 2 | ✔ Yes |
| 3 | ✘ No |
| 4 | ✔ Yes* |
| 5 | ✘ No |
| 6 | ✘ No |

> *For #4: Technically incomplete info, but in context, likely intended to be similar.

Let me know if you want diagrams or further clarification!