Similar and Congruent Triangles — Properties and Examples

By MathPal Editorial Team·Published Apr 21, 2026

Grade: 8-9 | Topic: Geometry

What You Will Learn

This guide explains the difference between similar and congruent triangles, the tests used to prove each, and how to apply proportions to find unknown side lengths in similar figures — a skill used throughout geometry and trigonometry.

Theory

Congruent triangles

Two triangles are congruent (symbol: $\cong$) when they are the same shape and size. All corresponding sides are equal and all corresponding angles are equal.

Congruence tests (sufficient to prove congruence):

Test	Meaning
SSS	All three pairs of corresponding sides equal
SAS	Two sides and the included angle equal
ASA	Two angles and the included side equal
AAS	Two angles and a non-included side equal
RHS	Right angle, hypotenuse, and one side equal

Note: SSA is not a valid congruence test (two sides and a non-included angle can produce two different triangles).

Similar triangles

Two triangles are similar (symbol: $\sim$) when they have the same shape but can be different sizes. All corresponding angles are equal and corresponding sides are proportional.

Similarity tests:

Test	Meaning
AA	Two pairs of equal angles (the third follows automatically)
SSS similarity	All three pairs of sides in proportion
SAS similarity	Two sides in proportion with the included angle equal

Scale factor and corresponding sides

When triangles ABC and DEF are similar with scale factor $k$:

$\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k$

This proportion lets you find any unknown side length.

Identifying corresponding parts

Always match vertices in the order named. If $\triangle ABC \sim \triangle DEF$:

• Angle A corresponds to angle D
• Angle B corresponds to angle E
• Side AB corresponds to side DE

Worked Examples

Example 1 — Proving congruence (SAS)

Triangle PQR has PQ = 5, QR = 8, angle Q = 60°.

Triangle XYZ has XY = 5, YZ = 8, angle Y = 60°.

Step 1: Two sides match: PQ = XY = 5 and QR = YZ = 8.

Step 2: The included angle (between those two sides) matches: angle Q = angle Y = 60°.

Conclusion: $\triangle PQR \cong \triangle XYZ$ by SAS.

Example 2 — Proving similarity (AA)

Triangle ABC has angle A = 50° and angle B = 70°.

Triangle DEF has angle D = 50° and angle E = 70°.

Step 1: Two pairs of angles match: A = D and B = E.

Step 2: The third angles must also match: $C = F = 180 - 50 - 70 = 60°$.

Conclusion: $\triangle ABC \sim \triangle DEF$ by AA.

Example 3 — Finding a missing side using proportions

$\triangle ABC \sim \triangle DEF$ with AB = 4, BC = 6, AC = 5, and DE = 10.

Find EF and DF.

Step 1: Find the scale factor: $k = \dfrac{DE}{AB} = \dfrac{10}{4} = 2.5$.

Step 2: Multiply each corresponding side by $k$:

$EF = BC \times k = 6 \times 2.5 = 15$ $DF = AC \times k = 5 \times 2.5 = 12.5$

Example 4 — Shadow problem (real-world similarity)

A tree casts a shadow 9 m long. At the same time, a 1.5 m tall post casts a shadow 3 m long. How tall is the tree?

Step 1: The sun's rays create similar triangles. Set up a proportion:

$\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{post height}}{\text{post shadow}}$

$\frac{h}{9} = \frac{1.5}{3}$

Step 2: Cross-multiply: $3h = 9 \times 1.5 = 13.5$.

Step 3: Solve: $h = 4.5$ m.

Common Mistakes

Mistake 1 — Mixing up corresponding vertices

❌ $\triangle ABC \sim \triangle DEF$: matching AB with EF instead of DE.

✅ The order of letters defines the correspondence. AB corresponds to DE, BC to EF, AC to DF.

Mistake 2 — Using SSA as a congruence test

❌ Concluding two triangles are congruent because two sides and a non-included angle match.

✅ SSA is not a valid test — it can produce two different triangle shapes. Use SAS, SSS, ASA, or AAS instead.

Mistake 3 — Forgetting to check which angles are included

❌ For SAS similarity, using any two sides and any angle.

✅ The angle must be the included angle — the one between the two sides you are comparing.

Practice Problems

Problem 1: Two triangles share angle A = 45°. Triangle 1 also has angle B = 80°. Triangle 2 also has angle E = 80°. Are they similar?

Show Answer

Yes — two pairs of equal angles (A = A and B = E), so similar by AA.

Problem 2: $\triangle PQR \sim \triangle STU$, PQ = 6, QR = 9, ST = 4. Find TU.

Show Answer

Scale factor: $k = \dfrac{ST}{PQ} = \dfrac{4}{6} = \dfrac{2}{3}$

$TU = QR \times k = 9 \times \dfrac{2}{3} = 6$

Problem 3: Triangle ABC has AB = 12, BC = 12, AC = 12. Triangle DEF has DE = 7, EF = 7, DF = 7. Are they congruent, similar, or neither?

Show Answer

Both are equilateral triangles (all angles 60°). They have equal angles but different side lengths.

Similar (AA or SSS similarity), but not congruent.

Problem 4: A 6 m ladder leans against a wall reaching 5 m high. A similar ladder is twice as long. How high does it reach?

Show Answer

Scale factor = 2. Height = $5 \times 2 = \mathbf{10}$ m.

Problem 5: Name the congruence test: two triangles share the same hypotenuse and one leg is equal in each.

Show Answer

RHS (Right angle, Hypotenuse, Side) — valid for right triangles.

Summary

• Congruent triangles ($\cong$): same shape and size. Tests: SSS, SAS, ASA, AAS, RHS.
• Similar triangles ($\sim$): same shape, proportional sizes. Tests: AA, SSS similarity, SAS similarity.
• To find missing sides: set up a proportion using corresponding sides and the scale factor.
• Always match corresponding parts in the order vertices are listed.

Related Topics

• Angles in a Triangle — Properties and Sum Rule — angle properties used in similarity proofs
• Pythagorean Theorem — applied to congruent and similar right triangles
• Ratios and Proportions — the proportion method used to find unknown sides

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