# Congruent and Similar Figures

> Angle facts, congruence tests, proving properties of quadrilaterals, similar triangles and scale factors for length, area and volume.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/congruence-and-similarity/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The flagpole and the shadow
At 3 pm, Aiden notices that his 1.6 m tall shadow is 2 m long. The school flagpole's shadow is 15 m long. The sun's rays hit both at the same angle, so Aiden and the flagpole make two triangles with the same shape but different sizes.

Triangles with the same shape are **similar**. Their sides are in the same ratio, so the flagpole is $15 \times \dfrac{1.6}{2} = 12$ m tall. Aiden measured it with nothing but a tape measure and a bit of geometry.

**In short:** Congruent figures are the same shape and size, proved with the tests SSS, SAS, AAS or RHS. Similar figures are the same shape but a different size: with scale factor $k$, lengths are multiplied by $k$, areas by $k^2$ and volumes by $k^3$.

## Starting point: Years 9 and 10

### Angle facts to know

- Angles on a straight line add to $180^\circ$. Angles around a point add to $360^\circ$.
- Vertically opposite angles are equal.
- Parallel lines: **alternate** angles are equal (Z shape), **corresponding** angles are equal (F shape), **co-interior** angles add to $180^\circ$ (C shape).
- Angles in a triangle add to $180^\circ$. The exterior angle equals the sum of the two opposite interior angles.
- Angles in a polygon with $n$ sides add to $(n - 2) \times 180^\circ$.

### Congruent triangles

**Congruent** figures are exactly the same shape and size. We write $\triangle ABC \equiv \triangle DEF$, listing matching vertices in order.

### Key idea: Four congruence tests
- **SSS:** three pairs of sides equal.
- **SAS:** two sides and the angle **between** them equal.
- **AAS:** two angles and a matching side equal.
- **RHS:** right angle, hypotenuse and one other side equal.

### Watch out: AAA does not prove congruence
Three equal angles only prove the same **shape**. One triangle could be bigger. That is similarity, not congruence.

### Writing a congruence proof

State each fact with a reason, name the test, then use "matching sides (or angles) of congruent triangles are equal" to finish.

### Worked example: A parallelogram's diagonal
In parallelogram $ABCD$, draw diagonal $AC$. Then:
- $\angle BAC = \angle DCA$ (alternate angles, $AB \parallel DC$)
- $\angle BCA = \angle DAC$ (alternate angles, $BC \parallel AD$)
- $AC$ is common.

So $\triangle ABC \equiv \triangle CDA$ (AAS). Hence $AB = CD$ and $BC = DA$: the opposite sides of a parallelogram are equal.

Congruence proves the familiar quadrilateral facts: a rhombus has perpendicular diagonals, a rectangle has equal diagonals, and the diagonals of a parallelogram bisect each other.

### Similar figures

**Similar** figures have the same shape. Matching angles are equal and matching sides are in the same ratio, the **scale factor** $k$.

$$k = \frac{\text{image length}}{\text{original length}}$$

*Diagram: Two similar triangles, the second twice the size of the first*
*Every side doubles. The angles stay the same.*

### Key idea: Four similarity tests
- **AAA:** all matching angles equal (two is enough).
- **SSS:** all three pairs of sides in the same ratio.
- **SAS:** two pairs of sides in the same ratio and the angle between them equal.
- **RHS:** right angle, and hypotenuses and one other pair of sides in the same ratio.

### Worked example: Finding a side
Two triangles are similar with matching sides 6 and 9. Another side of the small one is 8. The matching side is $8 \times \dfrac{9}{6} = 12$.

**Mind map**

```text
Congruence and similarity
  Angle facts
    Parallel lines
    Triangle 180°
    Polygon (n - 2) × 180°
  Congruent
    Same shape and size
    SSS, SAS, AAS, RHS
    Proofs with reasons
  Similar
    Same shape
    Scale factor k
    AAA, SSS, SAS, RHS
  Scaling
    Length × k
    Area × k²
    Volume × k³
```

### Year 11: Area and volume scale factors
If lengths are multiplied by $k$:
- areas are multiplied by $k^2$,
- volumes are multiplied by $k^3$.

**Example.** Two similar jugs have heights 10 cm and 15 cm, so $k = 1.5$. The small jug holds 800 mL. The large one holds $800 \times 1.5^3 = 2700$ mL.

**Similar triangle proofs** follow the same pattern as congruence proofs. They often prove a ratio such as $\dfrac{AB}{AD} = \dfrac{AC}{AE}$, which is how the trig ratios themselves are justified: every right triangle with a $30^\circ$ angle is similar, so $\sin 30^\circ$ is always $\dfrac{1}{2}$.

### Year 12: Similar triangles with calculus
A cone-shaped funnel is 12 cm tall with a top radius of 4 cm. When the water is $h$ cm deep, similar triangles give the water's radius: $\dfrac{r}{h} = \dfrac{4}{12}$, so $r = \dfrac{h}{3}$. Then

$$V = \frac{1}{3}\pi r^2 h = \frac{\pi h^3}{27}$$

Writing $V$ in one variable like this is the key first step in Year 12 rate and optimisation problems.

## Quick summary

- Know the angle facts: straight line $180^\circ$, triangle $180^\circ$, polygon $(n - 2) \times 180^\circ$, and the parallel line rules.
- Congruent: same shape and size. Tests: SSS, SAS, AAS, RHS.
- Similar: same shape. Tests: AAA, SSS, SAS, RHS (ratios).
- Scale factor $k$: lengths $\times k$, areas $\times k^2$, volumes $\times k^3$.
- Proofs give a reason for every statement.

## Quiz

Try these questions from the chapter quiz.

1. Two angles in a triangle are $65^\circ$ and $45^\circ$. The third is:
   - A. $80^\circ$
   - B. $70^\circ$
   - C. $110^\circ$
   - D. $60^\circ$
   - Answer: B. $70^\circ$. $180 - 110 = 70$.

2. The interior angles of a hexagon add to:
   - A. $540^\circ$
   - B. $720^\circ$
   - C. $900^\circ$
   - D. $360^\circ$
   - Answer: B. $720^\circ$. $(6 - 2) \times 180^\circ$.

3. Co-interior angles between parallel lines:
   - A. are equal
   - B. add to $180^\circ$
   - C. add to $90^\circ$
   - D. add to $360^\circ$
   - Answer: B. add to $180^\circ$. C shape: they are supplementary.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/congruence-and-similarity/#quiz