Congruence and similarity in geometric proofs | Year 11

01 / LEARN

Understand the idea.

Match triangles before matching lengths

Congruent triangles have matching lengths and angles; similar triangles have matching angles and proportional lengths. Use a valid test before taking corresponding parts: SSS, SAS, ASA/AAS or RHS for congruence, and AA or proportional-side tests for similarity.

Write triangle names in corresponding order. From $△ A B C \sim △ D E F$ , match A to D, B to E and C to F. Keep every ratio in the same direction; three equal angles establish shape but do not fix size.

Turn the triangle fact into the target result

Build a geometric proof in four stages: name the triangles, justify enough equal sides or angles, state a congruence or similarity test, then extract the target fact. Equal radii, shared sides, vertical angles and angles on one chord often provide the links.

For a proportional equation, check which endpoints correspond before selecting lengths. Do not assume a right angle or midpoint from appearance. State the given facts and derived relations so that the argument applies to every non-degenerate configuration meeting the hypotheses.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A median in an isosceles triangle

In triangle ABC, AB $= A C$ and D is the midpoint of BC. Prove AD is perpendicular to BC.

Worked solution

1
List the three matched side equalities.
$A B = A C, B D = D C, A D = A D$
The first equality is given, the second follows from D being the midpoint, and AD is shared.
2
Establish congruence and equal angles.
$△ A D B ≅ △ A D C \Rightarrow ∠ A D B = ∠ A D C$
SSS proves the triangles congruent, so corresponding angles are equal.
3
Use the straight line through B, D and C.
$\begin{aligned} ∠ A D B + ∠ A D C & = 180^{\circ} \\ ∠ A D B = ∠ A D C & = 90^{\circ} \end{aligned}$
Equal adjacent angles on a straight line must both be right angles.
4
State the perpendicularity.
$A D ⟂ B C$
AD meets BC at a right angle at D.

Answer

AD is perpendicular to BC.

EXAMPLE 02

An altitude in a right triangle

A right triangle ABC has right angle A. The altitude AH meets hypotenuse BC at H. Prove $A H^{2} = B H \times H C$ .

Worked solution

1
Identify the right-angle pair.
$∠ A H B = ∠ C H A = 90^{\circ}$
AH is an altitude to BC, and B, H and C are collinear.
2
Match another angle pair.
$∠ A B H = 90^{\circ} - ∠ A C B = ∠ C A H$
Both angles are complementary to angle C: one in triangle ABC and the other in triangle CAH.
3
Establish similarity in matching order.
$△ A B H \sim △ C A H$
AA gives the correspondence A to C, B to A and H to H.
4
Compare corresponding sides and multiply.
$\begin{aligned} \frac{B H}{A H} & = \frac{A H}{C H} \\ A H^{2} & = B H \times C H \end{aligned}$
The positive altitude permits cross-multiplication; CH and HC denote the same segment.

Answer

$A H^{2} = B H \times H C$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Similar triangles have matching side lengths 4 and 6. What is their large-to-small area ratio?

Show answer & reasoning

$9 : 4$ .

The large-to-small length ratio is $6 : 4 = 3 : 2$ . Areas scale as the square of the length factor, so the area ratio is $9 : 4$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Triangles $A B C$ and $D E F$ have $A B = D E$ , $B C = E F$ and $A C = D F$ . State a valid conclusion with a test.
2 marks
A student proves all three corresponding angles equal and claims congruence. Give a concrete counterexample.
2 marks
Triangles $A B C$ and $D E F$ are similar in that order. $A B = 6$ , $B C = 9$ and $D E = 10$ . Find $E F$ .
2 marks
$A B$ and $A C$ are tangents from $A$ to a circle with centre $O$ . Prove $A O$ bisects angle $B A C$ .
4 marks
An altitude to the hypotenuse divides it into lengths 4 and 9. Find the altitude and explain the theorem used.
3 marks
In the same altitude construction, the hypotenuse pieces are $p$ and $q$ . Explain why $p + q$ is not the altitude in general.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Triangles $A B C$ and $D E F$ have $A B = D E$ , $B C = E F$ and $A C = D F$ . State a valid conclusion with a test.
2 marks
All three corresponding sides are equal.
$△ A B C ≅ △ D E F$ by SSS.
A student proves all three corresponding angles equal and claims congruence. Give a concrete counterexample.
2 marks
All corresponding angles are $60^{\circ}$ , but their side lengths differ.
Equilateral triangles with side lengths 2 and 5.
Triangles $A B C$ and $D E F$ are similar in that order. $A B = 6$ , $B C = 9$ and $D E = 10$ . Find $E F$ .
2 marks
Scale from $A B C$ to $D E F$ is $\frac{10}{6} = \frac{5}{3}$ ; $E F = 9 (\frac{5}{3}) = 15$ .
15 units.
$A B$ and $A C$ are tangents from $A$ to a circle with centre $O$ . Prove $A O$ bisects angle $B A C$ .
4 marks
$O B = O C$ as radii; $A O$ is a shared hypotenuse; tangent-radius angles $B$ and $C$ are right.
Corresponding angles $B A O$ and $O A C$ are equal.
Triangles $A B O$ and $A C O$ are congruent by RHS.
An altitude to the hypotenuse divides it into lengths 4 and 9. Find the altitude and explain the theorem used.
3 marks
By similarity, $h^{2} = 4 \times 9 = 36$ . Take $h = 6 > 0$ .
6 units.
In the same altitude construction, the hypotenuse pieces are $p$ and $q$ . Explain why $p + q$ is not the altitude in general.
3 marks
The geometric-mean theorem gives $h^{2} = p q$ .
For example, $p = 4$ , $q = 9$ give hypotenuse 13 and altitude 6, not 13.
It is the full hypotenuse, whereas the altitude is $\sqrt{p q}$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

In this lesson

Understand the idea.

Match triangles before matching lengths

Turn the triangle fact into the target result

See the method in action.

A median in an isosceles triangle

List the three matched side equalities.

Establish congruence and equal angles.

Use the straight line through B, D and C.

State the perpendicularity.

An altitude in a right triangle

Identify the right-angle pair.

Match another angle pair.

Establish similarity in matching order.

Compare corresponding sides and multiply.

Try it yourself.

Check your understanding

Keep practising