Congruence and similarity worksheet with exercises on identifying congruent and similar shapes.

Let's go through each question on the worksheet step by step, explaining the concepts and solutions.

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1. Tick the shapes which are congruent.

Congruent shapes are identical in shape and size. They can be rotated or reflected, but their side lengths and angles must be exactly the same.

We have 8 triangles shown in two rows. We need to find pairs that are exactly the same size and shape.

Looking closely:

- The first triangle (top-left) is a right-angled triangle with a long base.
- The second triangle is a smaller right-angled triangle — different size → not congruent.
- The third triangle is a larger scalene triangle.
- The fourth triangle is small and upright.

Now compare:

- Top-left triangle vs bottom-left triangle: Same shape and size? No — bottom one is taller and narrower.
- Look at triangle 2 (top row) and triangle 7 (bottom row): Both are right-angled, with the same proportions and orientation — they look identical.
- Triangle 4 (top-right) and triangle 8 (bottom-right): These are both small, right-angled triangles with the same angle and side ratios — likely congruent.
- But wait — let’s check carefully.

Actually, from visual inspection:

- Triangle 1 (top-left) and Triangle 6 (bottom row, second from left): These appear to be same size and shape, just rotated. So they are congruent.
- Triangle 2 (top row, second) and Triangle 5 (bottom row, third from left): These are both small right-angled triangles — same size and shape → congruent.
- Triangle 3 (top row, third) and Triangle 7 (bottom row, sixth): Large scalene triangle — seems same as top one → congruent.
- Triangle 4 (top row, fourth) and Triangle 8 (bottom row, last): Small right-angled triangle — same as top-right one → congruent.

But wait — there are only 8 triangles total. Let’s number them for clarity:

| Top Row | Bottom Row |
|--------|------------|
| 1 | 5 |
| 2 | 6 |
| 3 | 7 |
| 4 | 8 |

Now compare:

- Triangle 1 and Triangle 5: Both are large, right-angled, similar orientation — but not the same. Triangle 5 is taller.
- Triangle 2 and Triangle 6: Both small right-angled — yes, same size and shape → congruent.
- Triangle 3 and Triangle 7: Large scalene triangle — appears same → congruent.
- Triangle 4 and Triangle 8: Small right-angled — same → congruent.

Wait — Triangle 1 looks like it might be congruent to Triangle 5? No — triangle 5 is upside-down and longer.

Actually, upon close inspection:

- Triangle 1 (top-left) and Triangle 5 (bottom-left): Are they the same? No — triangle 1 has a horizontal base, triangle 5 is vertical. But if you rotate triangle 5, does it match? It's slightly different — probably not congruent.

Let’s re-evaluate:

Best matches:

- Triangle 2 (top row, second) and Triangle 6 (bottom row, second): Same shape and size → congruent
- Triangle 3 (top row, third) and Triangle 7 (bottom row, sixth): Both large scalene triangles — same shape and size → congruent
- Triangle 4 (top row, fourth) and Triangle 8 (bottom row, last): Both small right-angled — same → congruent

So tick these three pairs.

✔ Answer: Tick the following:
- Triangle 2 and 6
- Triangle 3 and 7
- Triangle 4 and 8

(But since the instruction says "tick the shapes", you should mark each of those six triangles.)

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2. Tick the shapes which are similar.

Similar shapes have the same shape but may differ in size. Angles are equal; sides are in proportion.

We have 8 triangles again.

Look at the triangles:

- All are triangles, so we compare angles.

Many of them are equilateral or isosceles.

From visual inspection:

- The smaller triangles in the middle seem to be all equilateral — same angles (60°), so they are similar.
- The larger ones on the right also seem to be equilateral → similar to the others.

But some are right-angled.

So group them:

- Equilateral triangles: All have 60° angles → similar to each other.
- Right-angled triangles: Some have different angles → may not be similar.

So:

- All equilateral triangles (including the large ones) are similar.
- The right-angled triangles (like the big one on the far right) have different angles → not similar to equilateral ones.

So tick all the equilateral-looking triangles.

That includes:
- Top row: first, second, third, fourth?
Wait — no:

Look:

- Top row: first triangle — looks like acute, maybe equilateral?
- Second — right-angled → not equilateral
- Third — large, acute → possibly equilateral
- Fourth — small, right-angled

Bottom row:
- First — small, acute → could be equilateral
- Second — tiny → equilateral
- Third — large, acute → equilateral
- Fourth — very large, acute → equilateral

So likely, the non-right-angled ones are equilateral → all similar.

So tick:
- Top-left, top-middle-left, top-middle-right, top-right? Wait — top-right is right-angled.

No — actually, the right-angled ones are not equilateral.

So only the acute-angled ones (with no right angle) are equilateral → similar.

So tick:
- Top-left (if equilateral)
- Top-middle-left (small)
- Top-middle-right (large)
- Bottom-left (small)
- Bottom-middle-left (tiny)
- Bottom-middle-right (large)
- Bottom-right (very large)

Wait — but some of these are right-angled.

Actually, from the image:

- The large triangle on the far right (top row) is right-angled → not equilateral
- The one below it (bottom row, far right) is right-angled too

So the equilateral ones are:
- Top-left
- Top-middle-left
- Top-middle-right
- Bottom-left
- Bottom-middle-left
- Bottom-middle-right

And the right-angled ones (top-right and bottom-right) are not equilateral.

But are the right-angled ones similar to each other?

Yes — both are right-angled, and if they have same angles (e.g., 90°, 60°, 30°), then they are similar.

But from visual inspection, the top-right triangle is large and skinny, while the bottom-right is small and fat — probably not similar.

So best bet: All equilateral triangles are similar.

So tick:
- All triangles that are equilateral — i.e., all except the right-angled ones.

So tick:
- Top-left
- Top-middle-left
- Top-middle-right
- Bottom-left
- Bottom-middle-left
- Bottom-middle-right

But wait — top-middle-right is large and acute — yes
- Bottom-right is right-angled → do not tick

So do not tick the two right-angled triangles (top-right and bottom-right).

✔ Answer: Tick all six equilateral-like triangles (the ones without right angles).

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3. Which of the shapes on the bottom row are congruent to a shape on the top row? Join the pairs up.

We have two rows of shapes:

Top row:
1. Parallelogram
2. Up arrow
3. Pentagon
4. Trapezium
5. Cross (X)
6. Heart

Bottom row:
1. Pentagon
2. Down arrow
3. Cross (X)
4. Plus sign (+)
5. Heart
6. Parallelogram
7. Trapezium
8. Down arrow

Now, find congruent pairs — meaning same shape and size, regardless of rotation.

Let’s match:

- Parallelogram (top 1) ↔ Parallelogram (bottom 6) → same shape, size, orientation? Yes → congruent
- Up arrow (top 2) ↔ Down arrow (bottom 2) → same shape, just flipped → congruent (reflection is allowed)
- Pentagon (top 3) ↔ Pentagon (bottom 1) → same size and shape → congruent
- Trapezium (top 4) ↔ Trapezium (bottom 7) → same shape → congruent
- Cross (X) (top 5) ↔ Cross (X) (bottom 3) → same shape → congruent
- Heart (top 6) ↔ Heart (bottom 5) → same shape → congruent

Now what about:
- Plus sign (bottom 4): Not in top row → no match
- Down arrow (bottom 8): Already matched to up arrow?

Wait — bottom row has two down arrows: bottom 2 and bottom 8.

Top row has only one up arrow.

So:
- Up arrow (top 2) ↔ Down arrow (bottom 2) → congruent (mirror image)
- Down arrow (bottom 8): No match in top row → not congruent to any top shape

So the plus sign and down arrow (bottom 8) have no match.

So join:

- Top 1 (parallelogram) ↔ Bottom 6
- Top 2 (up arrow) ↔ Bottom 2
- Top 3 (pentagon) ↔ Bottom 1
- Top 4 (trapezium) ↔ Bottom 7
- Top 5 (cross) ↔ Bottom 3
- Top 6 (heart) ↔ Bottom 5

✔ Answer: Join these pairs:
- Parallelogram ↔ Parallelogram
- Up arrow ↔ Down arrow (bottom 2)
- Pentagon ↔ Pentagon
- Trapezium ↔ Trapezium
- Cross ↔ Cross
- Heart ↔ Heart

(Note: Down arrow at bottom 8 has no match.)

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4. This is triangle A. Which of these triangles are congruent to triangle A?

Triangle A:
- Angles: 60°, 70°, 50°
- One side: 6 cm between 60° and 70° angles

So side of 6 cm is between the 60° and 70° angles.

For congruence, we need same shape and size — so either:
- SSS, SAS, ASA, etc.

But here, we can use ASA or SAS.

We need triangles with:
- Two angles and included side = 6 cm, or
- Same angles and side length

Let’s examine each option:

#### Option 1:
- 6 cm side, angles 60° and 70° — same as A, and side between them → congruent ✔

#### Option 2:
- Side 6 cm, angles 70° and 50° — but the 6 cm side is not between them? Wait — it's labeled: 70° and 50° adjacent to 6 cm side.

But in triangle A, 6 cm is between 60° and 70°.

Here, 6 cm is between 70° and 50° → different → not congruent ✘

#### Option 3:
- 6 cm, angles 60° and 50° — but 6 cm is between them? Yes — but in triangle A, 6 cm is between 60° and 70°, not 60° and 50° → different → ✘

#### Option 4:
- 6 cm, angles 70° and 50° — but 6 cm is opposite 50°? Wait — label shows 6 cm side with 70° and 50° at ends → so it's between them → but 70° and 50° → missing 60° → sum = 120° → third angle = 60° → so angles are same: 60°, 70°, 50°

But is the 6 cm side between 70° and 50°? Then it's not the same as triangle A (where 6 cm is between 60° and 70°).

So unless the side is in the same position, it may not be congruent.

Wait — in triangle A, the 6 cm side is opposite the 50° angle? Let's see:

In triangle A:
- Angles: 60°, 70°, 50°
- Side of 6 cm is between 60° and 70° → so it is opposite the 50° angle

So the 6 cm side is opposite the 50° angle

Now check others:

#### Option 1:
- 6 cm side between 60° and 70° → so opposite 50° → same → congruent ✔

#### Option 2:
- 6 cm side between 70° and 50° → so opposite 60° → not same → ✘

#### Option 3:
- 6 cm side between 60° and 50° → opposite 70° → ✘

#### Option 4:
- 6 cm side between 70° and 50° → opposite 60° → ✘

#### Option 5:
- 6 cm side, angles 60° and 70° → but 6 cm is not between them — it's opposite one of them? Label shows 6 cm side with 60° and 70° at ends? Wait — it's drawn with 6 cm side and 60° and 70° at ends → so yes, between them → same as A → congruent ✔

Wait — is it?

Option 5:
- Triangle with 6 cm side, 60° and 70° at ends → so same as A → congruent ✔

But is it oriented differently? Yes — rotated — but still congruent.

#### Option 6:
- 6 cm side, angles 50° and 60° — but 6 cm is opposite 70° → no — label shows 6 cm side with 50° and 60° at ends → so between them → opposite 70° → not same → ✘

#### Option 7:
- 6 cm side, angles 50° and 60° — same as above → opposite 70° → ✘

#### Option 8:
- 6 cm side, angles 60° and 50° → but 6 cm is between them → opposite 70° → ✘

Wait — option 5 is a triangle with 6 cm side between 60° and 70° → same as A → congruent

But is it the same size? Yes — same angles, same side between them → congruent ✔

Now check option 1 — already said ✔

But option 1 and option 5 both have 6 cm between 60° and 70° → so both congruent.

But wait — option 1 is labeled with 60° and 70° at ends of 6 cm side → yes

Option 5 is also labeled with 60° and 70° at ends of 6 cm side → yes

But option 1 is smaller? No — same labels → same

But option 5 is rotated — but still congruent.

Now option 4:
- 6 cm side between 70° and 50° → opposite 60° → not same → ✘

But option 2:
- 6 cm side between 70° and 50° → opposite 60° → ✘

Wait — option 8:
- 6 cm side, 60° and 50° at ends → so between them → opposite 70° → ✘

But option 6:
- 6 cm side, 50° and 60° at ends → same → opposite 70° → ✘

Only option 1 and option 5 have 6 cm between 60° and 70° → so both congruent

But wait — option 5 is a triangle with 6 cm side, 60° and 70° at ends → yes

But is the side length correct? Yes.

Also, option 1 is the same.

But option 5 is drawn with the 6 cm side horizontal — same as A — but rotated.

So both option 1 and option 5 are congruent.

But wait — option 1 is labeled:
- 6 cm, 60°, 70° — with 6 cm between them → yes

Option 5:
- 6 cm, 60°, 70° — between them → yes

But option 5 has a different orientation — but still congruent.

So both are congruent.

Now option 2:
- 6 cm, 70°, 50° — between them → so angles 70°, 50°, 60° — same angles, but side between 70° and 50° → opposite 60° → whereas in A, 6 cm is opposite 50° → so not same → ✘

Similarly, option 4:
- 6 cm between 70° and 50° → opposite 60° → ✘

But option 6:
- 6 cm between 50° and 60° → opposite 70° → ✘

Wait — option 7:
- 6 cm, 50° and 60° — but 6 cm is not between them — it's opposite one? Label shows 6 cm side with 50° and 60° at ends → so between them → opposite 70° → ✘

So only option 1 and option 5 have the 6 cm side between 60° and 70° → congruent

But wait — option 5 is labeled:
- 6 cm, 60°, 70° — yes

But is it rotated? Yes — but still congruent.

So ✔ Option 1 and Option 5

But let’s double-check:

- Option 1: 6 cm between 60° and 70° → same as A → congruent
- Option 5: 6 cm between 60° and 70° → same → congruent

Others:
- Option 2: 6 cm between 70° and 50° → not same → ✘
- Option 3: 6 cm between 60° and 50° → ✘
- Option 4: 6 cm between 70° and 50° → ✘
- Option 6: 6 cm between 50° and 60° → ✘
- Option 7: 6 cm between 50° and 60° → ✘
- Option 8: 6 cm between 60° and 50° → ✘

Wait — option 8:
- 6 cm, 60°, 50° — but 6 cm is not between them — it's opposite 70°? Label shows 6 cm side with 60° and 50° at ends → so between them → opposite 70° → ✘

So only option 1 and option 5 are congruent.

But wait — option 5 is the same as option 1, just rotated.

So both are congruent.

✔ Answer: Triangles 1 and 5 are congruent to triangle A.

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✔ Final Answers Summary:

1. Congruent shapes:
- Tick:
- Triangle 2 and 6
- Triangle 3 and 7
- Triangle 4 and 8

2. Similar shapes:
- Tick all equilateral triangles (all except the two right-angled ones)

3. Join congruent pairs:
- Parallelogram ↔ Parallelogram
- Up arrow ↔ Down arrow (bottom 2)
- Pentagon ↔ Pentagon
- Trapezium ↔ Trapezium
- Cross ↔ Cross
- Heart ↔ Heart

4. Congruent to triangle A:
- Option 1 and Option 5

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