Similar and Congruent Triangles

Congruent and similar triangles appear in almost every geometry chapter, from Matric theorems to O Level and FSc proofs. Once you know the tests for each and how ratios work, a lot of "find the missing side" questions become quick and almost mechanical.

Congruent vs Similar: The Core Idea

Two triangles are congruent if they have exactly the same shape and the same size. One can be placed on top of the other perfectly, possibly after a rotation or reflection. We write \( \triangle ABC \cong \triangle PQR \).

Two triangles are similar if they have the same shape but not necessarily the same size. One is an enlargement (or reduction) of the other. We write \( \triangle ABC \sim \triangle PQR \).

So every pair of congruent triangles is also similar (with scale factor 1), but similar triangles are not always congruent.

Why the order of letters matters

In \( \triangle ABC \cong \triangle PQR \), the letters match up in order: \( A \leftrightarrow P \), \( B \leftrightarrow Q \), \( C \leftrightarrow R \). This tells you that side \( AB \) corresponds to \( PQ \), \( BC \) to \( QR \), and so on. Writing the correspondence wrongly is one of the most common sources of lost marks.

Tests for Congruence

You do not need to check all six parts (three sides, three angles). Any one of these conditions is enough:

1. SSS – all three pairs of sides are equal.
2. SAS – two sides and the included angle (the angle between them) are equal.
3. ASA or AAS – two angles and one corresponding side are equal.
4. RHS – both triangles are right-angled, and the hypotenuse and one other side are equal.

Note that SSA (two sides and a non-included angle) is not a valid test in general, because it can produce two different triangles. AAA is not a congruence test either: it only proves similarity.

Tests for Similarity

Two triangles are similar if any of the following is true:

1. AA – two pairs of angles are equal (the third pair then must be equal, since angles add up to \( 180^\circ \)).
2. SSS similarity – all three pairs of corresponding sides are in the same ratio.
3. SAS similarity – two pairs of sides are in the same ratio and the included angles are equal.

The scale factor

If \( \triangle ABC \sim \triangle PQR \), then

$$ \frac{PQ}{AB} = \frac{QR}{BC} = \frac{PR}{AC} = k $$

where \( k \) is the scale factor. Lengths multiply by \( k \), areas multiply by \( k^2 \), and (for similar solids) volumes multiply by \( k^3 \).

Worked Example 1: Proving Congruence

Question: In quadrilateral \( ABCD \), \( AB = CD \) and \( AB \parallel CD \). Prove that \( \triangle ABD \cong \triangle CDB \).

Solution:

• Step 1: \( AB = CD \) (given).
• Step 2: Since \( AB \parallel CD \) and \( BD \) is a transversal, \( \angle ABD = \angle CDB \) (alternate angles).
• Step 3: \( BD = DB \) (common side).
• Step 4: In \( \triangle ABD \) and \( \triangle CDB \) we have two sides and the included angle equal: \( AB = CD \), \( \angle ABD = \angle CDB \), \( BD = DB \).

Therefore \( \triangle ABD \cong \triangle CDB \) by SAS. As a bonus, this gives \( AD = CB \), which proves that \( ABCD \) is a parallelogram.

Worked Example 2: Finding a Missing Side

Question: In \( \triangle ABC \), point \( D \) lies on \( AB \) and point \( E \) lies on \( AC \) so that \( DE \parallel BC \). Given \( AD = 4 \) cm, \( DB = 6 \) cm and \( DE = 5 \) cm, find \( BC \).

Solution:

• Step 1: Show the triangles are similar. Because \( DE \parallel BC \), \( \angle ADE = \angle ABC \) (corresponding angles), and \( \angle A \) is common to both triangles. So \( \triangle ADE \sim \triangle ABC \) by AA.
• Step 2: Find the full side. \( AB = AD + DB = 4 + 6 = 10 \) cm.
• Step 3: Write the ratio of corresponding sides.

$$ \frac{BC}{DE} = \frac{AB}{AD} $$

• Step 4: Substitute and solve.

$$ \frac{BC}{5} = \frac{10}{4} \quad \Rightarrow \quad BC = 5 \times 2.5 = 12.5 \text{ cm} $$

So \( BC = 12.5 \) cm.

Worked Example 3: Areas of Similar Triangles

Question: Two similar triangles have corresponding sides of 6 cm and 9 cm. The area of the smaller triangle is \( 24 \text{ cm}^2 \). Find the area of the larger triangle.

Solution:

• Step 1: Scale factor for length.

$$ k = \frac{9}{6} = \frac{3}{2} $$

• Step 2: Scale factor for area.

$$ k^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} $$

• Step 3: Multiply.

$$ \text{Area} = 24 \times \frac{9}{4} = 54 \text{ cm}^2 $$

A frequent error is to multiply the area by \( \frac{3}{2} \) instead of \( \frac{9}{4} \), which gives the wrong answer of \( 36 \text{ cm}^2 \).

Common Mistakes

• Mismatching vertices. Always identify equal angles first, then write the triangles in matching order before setting up ratios.
• Using SSA or AAA for congruence. Neither guarantees congruent triangles.
• Forgetting the included angle in SAS. The angle must be between the two given sides.
• Using the part instead of the whole. In Example 2, the ratio uses \( AB = 10 \), not \( DB = 6 \).
• Applying the length ratio to area. Areas scale by \( k^2 \), volumes by \( k^3 \).

Tips for Exam Proofs

• Write the reason next to every statement: "given", "common", "vertically opposite angles", "alternate angles" and so on.
• State the test clearly at the end, for example "\( \therefore \triangle ABD \cong \triangle CDB \) (SAS)".
• Mark equal sides and angles on your diagram before you begin. It makes the correct test obvious.
• In similarity questions, redraw the two triangles separately in the same orientation. This avoids pairing the wrong sides.

Key Takeaways

• Congruent triangles have the same shape and size; similar triangles have the same shape only.
• Congruence tests: SSS, SAS, ASA/AAS and RHS. Similarity tests: AA, SSS ratio and SAS ratio.
• In similar triangles, all corresponding sides share one scale factor \( k \).
• Areas scale by \( k^2 \) and volumes by \( k^3 \).
• Matching the vertices in the correct order is the key to every correct ratio.

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