Congruence And Similarity

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Notes

Congruence

Two shapes are **congruent** if they are identical in shape and size.
One shape may be a **reflection**, **rotation**, or **translation** of the other.
If one shape is an enlargement of the other, they are **not congruent**.
To prove congruence, show that **corresponding sides** are equal in length and **corresponding angles** are equal in size.
Tracing paper can help check congruence if shapes are drawn to scale.

Similarity

Two shapes are **similar** if they have the same shape and their corresponding sides are **in proportion**.
One shape is an **enlargement** of the other; similarity does not imply congruence.
For triangles, prove similarity by showing **corresponding angles are equal** (e.g., using vertically opposite angles, alternate angles on parallel lines).
For non-triangular shapes, show that **all corresponding sides** are in the same ratio (scale factor).
If all angles are equal, the shapes are similar (but not necessarily congruent).

Similar Lengths

Equivalent lengths on similar shapes are linked by a **scale factor** (k).
If the second shape is larger, $k >$ 1; if smaller, $0 < k < 1$ .
To find k: divide a length on the second shape by the corresponding length on the first shape.
To find a missing length: multiply the corresponding length by k (if going to larger) or divide by k (if going to smaller).
Redraw overlapping similar shapes separately to avoid confusion.

Similar Areas & Volumes

If length scale factor $= k$ , then **area scale factor = k²** and **volume scale factor = k³**.
Given area scale factor, length scale factor $= √(area$ SF); volume scale factor $= (\sqrt{area SF})^{3}$ .
Given volume scale factor, length scale factor = ∛(volume SF); area scale factor = (∛(volume $SF))^{2}$ .
To find missing area or volume: identify known quantities, find the relevant scale factor, then multiply or divide accordingly.
Always check whether the result should be larger or smaller than the given quantity.

Worked Examples (Lengths)

Example: Two similar rectangles have sides 6 cm and 3 cm. Scale factor $= \frac{6}{3} = 2 (larger$ to smaller) or $\frac{3}{6} = 0.5 (smaller$ to larger).
To find missing side: if $AD = 15$ cm on larger, corresponding PS on smaller $= 15 \times 0.5 = 7.5$ cm.
Always identify corresponding sides correctly before applying scale factor.

Worked Examples (Areas & Volumes)

Example: Solid A volume $32 cm^{3}$ , solid B volume $108 cm^{3}$ . $k^{3} = \frac{108}{32} = \frac{27}{8}$ , so $k =$ ∛(27/8) $= \frac{3}{2}$ .
If height of $A = 10$ cm, height of $B = 10 \times (\frac{3}{2}) = 15$ cm.
For areas: if area scale factor $= 4$ , then length scale factor $= \sqrt{4} = 2$ .

Congruent vs Similar Shapes

Similar Triangles (Parallel Lines)

Scale Factors for Length, Area, Volume

Finding Missing Length in Similar Shapes

Practice questions

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1.Two shapes are congruent if they are identical in shape and size. Which of the following transformations does NOT change congruence?
Easy
- AEnlargement
- BReflection
- CRotation
- DTranslation
2.Triangle ABC is similar to triangle PQR. $AB = 6$ cm, $PQ = 9$ cm. What is the length scale factor from triangle ABC to triangle PQR?
Easy
- A1.5
- B0.666...
- C2
- D3
3.Two rectangles are similar. The smaller rectangle has width 4 cm and length 6 cm. The larger rectangle has width 10 cm. What is the length of the larger rectangle?
Easy
- A15 cm
- B12 cm
- C20 cm
- D8 cm
4.Solid A and solid B are mathematically similar. The volume of solid A is $32 cm^{3}$ and the volume of solid B is $108 cm^{3}$ . The height of solid A is 10 cm. Find the height of solid B.
Medium
- A15 cm
- B20 cm
- C12 cm
- D18 cm
5.Two cones are mathematically similar. The total surface area of the smaller cone is $80 cm^{2}$ and of the larger cone is $180 cm^{2}$ . The volume of the smaller cone is $168 cm^{3}$ . Calculate the volume of the larger cone.
Medium
- A $567 cm^{3}$
- B $378 cm^{3}$
- C $756 cm^{3}$
- D $283.5 cm^{3}$
6.A model of a car has a scale 1:20. The volume of the actual car is $12 m^{3}$ . Find the volume of the model in cubic centimetres.
Medium
- A $1500 cm^{3}$
- B $1500000 cm^{3}$
- C $3000 cm^{3}$
- D $600 cm^{3}$
7.Two mathematically similar containers have heights of 30 cm and 75 cm. The larger container has a capacity of 5.5 litres. Calculate the capacity of the smaller container in millilitres.
Hard
- A352 ml
- B880 ml
- C220 ml
- D550 ml
8.The diagram shows two mathematically similar solid metal prisms. The volume of the smaller prism is $648 cm^{3}$ and the volume of the larger prism is $2187 cm^{3}$ . The area of the cross-section of the smaller prism is $36 cm^{2}$ . Calculate the area of the cross-section of the larger prism.
Hard
- A $81 cm^{2}$
- B $72 cm^{2}$
- C $108 cm^{2}$
- D $54 cm^{2}$

Summary

## Congruence & Similarity ### Congruence - **Congruent**: identical in shape and size. One may be a reflection, rotation, or translation of the other. - To prove: show corresponding sides equal and corresponding angles equal. - Not congruent if one is an enlargement. ### Similarity - **Similar**: same shape, corresponding sides in proportion (one is an enlargement of the other). - For triangles: prove all corresponding angles equal → sides automatically in proportion. - For other shapes: show scale factor same for all corresponding sides. ### Similar Lengths - **Length scale factor (k)**: divide a length on second shape by corresponding length on first. - To find missing length: multiply by k if going to larger shape; divide by k if going to smaller. ### Similar Areas & Volumes - If length scale factor = k, then: - Area scale factor = k² - Volume scale factor = k³ - To convert between scale factors: - k = √(area SF) = ∛(volume SF) - area SF = k² = (∛(volume SF))² - volume SF = k³ = (√(area SF))³ **Exam Tips**: - Sketch triangles separately if they overlap. - Use tracing paper to check congruence (if drawn to scale). - Label shapes clearly to avoid mixing up.

Key terms

Congruent: Two shapes are congruent if they are identical in shape and size; one may be a reflection, rotation, or translation of the other.
Similar: Two shapes are similar if they have the same shape and their corresponding sides are in proportion; one is an enlargement of the other.
Scale factor (length): The ratio of corresponding lengths on two similar shapes; found by dividing a length on the second shape by the corresponding length on the first.
Area scale factor: The square of the length scale factor; used to relate areas of similar shapes.
Volume scale factor: The cube of the length scale factor; used to relate volumes of similar shapes.

Slides

Congruence

Two shapes are congruent if they are identical in shape and size.
One shape may be a reflection, rotation, or translation of the other.
If a shape is enlarged, it is not congruent because the size changes.

Which of the following transformations produces a shape that is always congruent to the original?

Enlargement
Rotation
Reflection
Both rotation and reflection

Proving Congruence

To prove two shapes are congruent, show that corresponding sides are equal in length and corresponding angles are equal in size.
You do not need to show they face the same direction.
Tracing paper can help check congruence if the diagram is drawn to scale.

Which of these are true about congruence?

Two congruent shapes have the same shape and size.
A shape and its enlargement are always congruent.
Corresponding angles must be equal for congruence.

Similarity

Two shapes are similar if they have the same shape and corresponding sides are in proportion.
One shape is an enlargement of the other.
Similarity does not imply congruence; similar shapes can be different sizes.

Two triangles have all corresponding angles equal. Which statement is correct?

They are congruent.
They are similar.
They are both similar and congruent.
They are neither similar nor congruent.

Proving Triangles Similar

To prove two triangles are similar, show that their corresponding angles are equal.
Look for equal angles using properties like vertically opposite angles, alternate angles on parallel lines, and base angles of isosceles triangles.
If all three angles match, the triangles are similar.

Triangle ABX and CDX are similar because angle AXB equals angle $CXD (____ angles)$ , angle ABC equals angle $BCD (____$ angles on parallel lines), and angle BAD equals angle $ADC (____$ angles on parallel lines).

Similar Lengths – Scale Factor

The scale factor (k) from one shape to another is found by dividing a length on the second shape by the corresponding length on the first.
If the second shape is larger, $k >$ 1; if smaller, $0 < k < 1$ .
To find a missing length, multiply the known length by k (if going to larger) or divide by k (if going to smaller).

Two similar rectangles have corresponding sides 6 cm and 9 cm. What is the scale factor from the smaller to the larger?

1.5
0.6667
2
3

Similar Areas & Volumes

If length scale factor $= k$ , then area scale factor $= k^{2}$ and volume scale factor $= k^{3}$ .
To find missing areas or volumes, first find the length scale factor, then square or cube it.
Alternatively, find the area or volume scale factor directly and take square or cube roots to find the length scale factor.

If the length scale factor is 3, then the area scale factor $is ____ and$ the volume scale factor $is ____$ .

Working with Similar Areas & Volumes

Identify known quantities as lengths, areas, or volumes.
Find the scale factor from two known quantities of the same type.
Use the relationships $k, k^{2}, k^{3}$ to find the required scale factor and then the missing quantity.

Two similar solids have volumes $32 cm^{3}$ and $108 cm^{3}$ . The height of the smaller is 10 cm. What is the height of the larger?

15 cm
12 cm
18 cm
20 cm

Example: Similar Lengths

Shapes ABCD and PQRS are similar. $AB = 6$ cm, $PQ = 3$ cm, $AD = 15$ cm. Find PS.
Scale factor from ABCD to PQRS $= \frac{3}{6} = 0.5$ .
$PS = AD \times 0.5 = 15 \times 0.5 = 7.5$ cm.

If the area scale factor between two similar shapes is 4, then the length scale factor $is ____$ .

Which of these are true about similar shapes?

Corresponding sides are in proportion.
Corresponding angles are always equal.
If the area scale factor is 9, the length scale factor is 3.

Summary

Congruent shapes are identical in size and shape; similar shapes have the same shape but different sizes.
For similar shapes, length scale factor k, area scale factor $k^{2}$ , volume scale factor $k^{3}$ .
Always identify corresponding sides or quantities before applying scale factors.

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