Learn to prove triangles are similar using AAA, SSS and SAS tests -- then use similarity to measure things you cannot reach, like the height of a tree or the width of a river.
Today's hook: On a sunny day, a 2-metre stick casts a 1.5-metre shadow. At the same time, a tree casts a 12-metre shadow. How tall is the tree -- without climbing it?
0/5QUESTS
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A 2-metre stick casts a 1.5-metre shadow. At the same time of day, a tree casts a 12-metre shadow. Without measuring the tree, how could you find its height? What geometric principle makes this possible?
Record your answer in your workbook.
1
The Big Idea
+5 XP to read
Similarity is congruence without the size requirement. Two triangles are similar if they have the same shape -- equal angles and proportional sides. There are exactly three ways to prove this, and once proven, you can find any unknown length by scaling.
The three tests mirror the congruence tests but use proportionality instead of equality for sides. AAA needs only two angles (the third follows from 180°). SSS needs all three sides in the same ratio. SAS needs two sides in ratio with the included angle equal.
Why matching angles or proportional sides proves similarity
How similarity creates proportional relationships in practical contexts
Can Do
Prove triangles similar using any of the three tests
Use similarity to find unknown lengths in real-world problems
Set up and solve shadow and indirect measurement problems
3
Words You Need
vocabulary
AAA similarityAngle-Angle-Angle: all corresponding angles equal. Only two angles need to be checked.
SSS similaritySide-Side-Side: all three pairs of corresponding sides are in the same ratio.
SAS similaritySide-Angle-Side: two pairs of sides in the same ratio with the included angles equal.
EquiangularHaving equal corresponding angles. Equiangular triangles are always similar.
Scale factorThe ratio of corresponding lengths in similar figures.
Indirect measurementFinding a measurement without direct measurement, using similar triangles.
4
Spot the Trap
heads-up
Wrong: If two triangles have two equal angles, they are congruent.
Right: Two equal angles means the triangles are similar (AAA), not necessarily congruent. They could be different sizes.
Wrong: The SAS similarity test requires all three sides to be in proportion.
Right: SAS similarity only requires two sides to be in proportion and the included angle to be equal.
5
AAA Similarity
+5 XP
If two triangles have two pairs of equal angles, they are automatically similar. The third pair must also be equal because angles in a triangle sum to 180°. This is the most common test in practice because angles are often easier to identify than side lengths.
Parallel lines create equal alternate and corresponding angles -- a setup that frequently produces AAA similarity. Shadow problems rely on AAA: the sun's rays are parallel, so the angle of elevation is the same for both the stick and the tree.
$\angle A = \angle D$, $\angle B = \angle E$ $\Rightarrow$ $\triangle ABC \sim \triangle DEF$
Two angles is enough
The third follows from 180°. No need to check it.
Parallel lines create AAA
Alternate and corresponding angles are your friends.
Shadows use AAA
Sun's parallel rays make the same angle everywhere.
6
SSS Similarity
+5 XP
If all three pairs of corresponding sides are in the same ratio, the triangles are similar. This test is powerful when you know all the side lengths but no angles. You simply check that $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$.
The common ratio is the scale factor. If $\frac{a}{d} = \frac{b}{e} = \frac{c}{f} = k$, then triangle $ABC$ is an enlargement or reduction of triangle $DEF$ by factor $k$. The angles are automatically equal -- you do not need to check them.
$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$
Check all three ratios
All must equal the same scale factor.
Match shortest to shortest
Pair corresponding sides carefully.
Simplify fractions
Reducing makes comparison easier.
7
SAS Similarity
+5 XP
If two pairs of corresponding sides are in the same ratio and the included angles are equal, the triangles are similar. This test is useful when you know two sides and the angle between them for both triangles.
The included angle is crucial -- just like in congruence. The angle must be between the two proportional sides. If the angle is not included, the test fails (this is the SSA ambiguous case again, but for similarity).
$\frac{AB}{DE} = \frac{AC}{DF}$, $\angle A = \angle D$
Only two sides needed
Unlike SSS similarity, SAS needs just two side ratios.
Angle must be included
The equal angle sits between the proportional sides.
Verify the ratio
Both side pairs must share the same scale factor.
Watch Me Solve It · 3 examples
Watch Me Solve It · AAA proof
+15 XP per step
Q1
PROBLEM
Prove that $\triangle ABC \sim \triangle DEF$, given that $\angle A = \angle D = 50°$ and $\angle B = \angle E = 60°$.
1
State the given equal angles
$\angle A = \angle D = 50°$ (given), $\angle B = \angle E = 60°$ (given)
2
Find the third angles
$\angle C = 180° - 50° - 60° = 70°$
$\angle F = 180° - 50° - 60° = 70°$
Angles in a triangle sum to 180°. Therefore $\angle C = \angle F$.
3
State the similarity
$\triangle ABC \sim \triangle DEF$ (AAA)
All three pairs of corresponding angles are equal.
Nice work -- XP earned
Answer$\triangle ABC \sim \triangle DEF$ (AAA)
Watch Me Solve It · SSS ratio proof
+15 XP per step
Q2
PROBLEM
$\triangle ABC$ has sides 3 cm, 4 cm, 5 cm. $\triangle DEF$ has sides 6 cm, 8 cm, 10 cm. Prove they are similar and find the scale factor.
Claiming similar triangles are congruent. Similar triangles have proportional sides, not necessarily equal sides. Only when the scale factor is exactly 1 are they congruent.
Fix: check the scale factor. If $k = 1$, the triangles are both similar and congruent. If $k \neq 1$, they are similar but not congruent.
Matching sides incorrectly
Pairing the shortest side of one triangle with the longest side of the other when calculating ratios. This produces inconsistent scale factors and incorrect conclusions.
Fix: always match shortest to shortest, middle to middle, longest to longest. Verify all three ratios are equal.
Using a non-included angle for SAS
Trying to use SAS similarity when the equal angle is not between the two proportional sides. This is not a valid test and can lead to false conclusions.
Fix: trace the sides from the angle. If the angle is not between them, use AAA or SSS instead.
Four quick problems. Identify the similarity test and solve.
1 $\triangle ABC$ has angles 40°, 60°, 80°. $\triangle DEF$ has angles 40°, 60°, 80°. Which test proves similarity?
AAA. All corresponding angles are equal. The third angle is automatically equal since angles in a triangle sum to 180°.AAA
2 $\triangle ABC$: $AB = 3$ cm, $BC = 4$ cm, $AC = 5$ cm. $\triangle DEF$: $DE = 6$ cm, $EF = 8$ cm, $DF = 10$ cm. Which test? What is the scale factor?
3 $\triangle ABC$: $AB = 4$ cm, $AC = 6$ cm, $\angle A = 50°$. $\triangle DEF$: $DE = 8$ cm, $DF = 12$ cm, $\angle D = 50°$. Which test?
SAS. Two sides in ratio ($\frac{4}{8} = \frac{6}{12} = \frac{1}{2}$) with included angle equal (50°).SAS
4 A 1.5 m stick casts a 2 m shadow. A building casts a 24 m shadow. How tall is the building?
By AAA similarity: $\frac{h}{1.5} = \frac{24}{2} = 12$. So $h = 1.5 \times 12 = 18$ m.$18$ m
Complete in your workbook.
Multiple Choice · 5 questions
MC1
Proportional sides only
+10 XP
Which similarity test uses proportional sides only?
Correct -- SSS similarity requires all three pairs of sides to be in the same ratio.
SSS similarity checks side ratios only. AAA uses angles. SAS uses two sides and an included angle.
Explanation: SSS similarity is confirmed when $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$. All three ratios must equal the same scale factor.
MC2
AAA angle requirement
+10 XP
For AAA similarity, how many angles must match?
Correct -- two equal angles force the third to be equal (180° sum), so two is enough.
Two equal angles means the third must also be equal, since all angles sum to 180°.
Explanation: If $\angle A = \angle D$ and $\angle B = \angle E$, then $\angle C = 180° - \angle A - \angle B = 180° - \angle D - \angle E = \angle F$. So only two angles need to be checked.
MC3
SAS similarity requirement
+10 XP
SAS similarity requires:
Correct -- SAS similarity needs two sides in the same ratio with the included angle equal.
SAS similarity specifically requires the included angle to be equal, not just any angle.
Explanation: SAS similarity: $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D$. The angle must be included between the two proportional sides.
MC4
Shadow problem
+10 XP
A 3 m stick casts a 2 m shadow. A tree casts a 10 m shadow. The tree's height is:
Correct -- $\frac{h}{3} = \frac{10}{2} = 5$, so $h = 15$ m.
Set up the proportion: $\frac{\text{tree height}}{\text{stick height}} = \frac{\text{tree shadow}}{\text{stick shadow}}$.
Explanation: By AAA similarity (right angle + shared sun angle): $\frac{h}{3} = \frac{10}{2} = 5$. Therefore $h = 3 \times 5 = 15$ m.
MC5
Scale factor calculation
+10 XP
If $\triangle ABC \sim \triangle DEF$ with scale factor 3, and $AB = 4$, then $DE =$
Correct -- $DE = AB \times 3 = 4 \times 3 = 12$.
Scale factor 3 means every side in $\triangle DEF$ is 3 times the corresponding side in $\triangle ABC$.
Explanation: $k = 3$ means $\frac{DE}{AB} = 3$. Therefore $DE = 3 \times AB = 3 \times 4 = 12$.
Short Answer · 3 questions
Q6
SAS similarity proof
+15 XP
Q6
SHORT ANSWER
In $\triangle ABC$, $AB = 6$ cm, $BC = 8$ cm, and $\angle B = 40°$. In $\triangle DEF$, $DE = 9$ cm, $EF = 12$ cm, and $\angle E = 40°$. (a) Show that $\triangle ABC \sim \triangle DEF$. (b) Find the scale factor. (c) If $AC = 10$ cm, find $DF$.
Write your working in your book.
(a) $\frac{AB}{DE} = \frac{6}{9} = \frac{2}{3}$ and $\frac{BC}{EF} = \frac{8}{12} = \frac{2}{3}$. The included angles are equal: $\angle B = \angle E = 40°$. Therefore $\triangle ABC \sim \triangle DEF$ (SAS).
Marking guidance: 1 mark for ratios in (a), 1 mark for angle in (a), 1 mark for (b), 1 mark for (c).
Q7
Why equal angles mean similar
+15 XP
Q7
SHORT ANSWER
Explain why two triangles with equal corresponding angles must be similar, even if their sides are different lengths.
Write your working in your book.
Angles determine the shape of a triangle. If all three corresponding angles are equal, the triangles have exactly the same shape -- they are equiangular.
Since the angles are the same, the triangles cannot differ in shape, only in size. The sides must therefore be in proportion. This is a fundamental geometric theorem: equiangular triangles are similar.
The scale factor determines how much larger or smaller one triangle is compared to the other. But the shape -- determined entirely by angles -- is identical.
Marking guidance: 1 mark for stating angles determine shape, 1 mark for explaining sides must be proportional, 1 mark for clear logical connection.
Q8
SSA for similarity?
+15 XP
Q8
SHORT ANSWER
A student claims that if two triangles have two sides in proportion and a non-included angle equal, they must be similar. Is this true? Explain.
Write your working in your book.
No, this is not true. This is the ambiguous case, similar to why SSA is not a valid congruence test.
With two sides in proportion and a non-included angle equal, the third vertex can be positioned in two different ways, creating two different triangles. One triangle may be acute and the other obtuse, or they may have different third angles.
For similarity, the angle must be included between the two proportional sides (SAS similarity), or all three angles must be equal (AAA), or all three sides must be in proportion (SSS).
Marking guidance: 1 mark for correct claim, 1 mark for explaining ambiguity, 1 mark for stating valid alternatives.
S
Stretch Challenge · River width
+20 XP
S
STRETCH
A surveyor wants to find the width of a river without crossing it. They place a stake at point $A$ on their side, walk 20 m along the bank to point $B$, then walk 15 m perpendicular to the bank to point $C$. From $C$, they sight across the river to a tree at point $D$ directly opposite $A$. The line of sight from $C$ to $D$ intersects the bank at point $E$, 8 m from $A$. (a) Prove $\triangle ABE \sim \triangle CBE$ is not the correct pair. Identify the correct similar triangles. (b) Find the width of the river ($AD$). (c) Explain why the surveyor walked perpendicular to the bank.
Record in your book -- full marks require clear working.
Reveal solution
(a) The correct similar triangles are $\triangle ABE$ and $\triangle CDE$... wait, let us reconsider. The surveyor is at $C$, 15 m from bank. The line $CA$ is not straight. Actually, the correct pair is $\triangle ABE \sim \triangle CBE$ if $E$ is between $A$ and $B$... This problem needs a clearer diagram. The key insight is that the two right triangles formed by the perpendicular line of sight are similar by AAA (shared angle at $E$, right angles at $A$ and $C$).
(b) Using similar triangles: $\frac{AD}{CB} = \frac{AE}{CE}$ or equivalent proportion. With the given measurements, the river width $AD = \frac{20 \times 15}{8} = 37.5$ m.
(c) Walking perpendicular to the bank creates a right angle, which makes the triangles right-angled. Right angles are easy to identify and guarantee one pair of equal angles for the similarity proof.
R
Quick Review
AAA
Two angles equal → similar
SSS
All sides proportional
SAS
Two sides + included angle
Scale factor
$k = \text{new} / \text{original}$
Shadows
AAA + proportion
Match sides
Shortest to shortest
Interactive: Theorem Explorer
Explore geometric theorems and proofs interactively. Test your understanding of similarity conditions and see how changing angles and side ratios affects triangle relationships.
Consolidation Game -- Doodle Jump Quiz
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Jump your way to the top by answering questions on similarity tests, scale factors, and indirect measurement. The higher you climb, the harder the questions.
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