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Lesson 16 ~25 min Unit 3 · Geometry +85 XP

Introduction to Similar Figures

Two figures are similar if they have exactly the same shape but possibly different sizes. Corresponding angles are equal and corresponding sides are in the same ratio — that ratio is called the scale factor.

Today's hook: A photo on your phone and the same photo blown up to a poster — they look identical, just bigger. That's similarity!
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Think First
warm-up

Imagine you take a photo of a friend, then enlarge it on a printer to 200%. The new picture is twice as wide and twice as tall, but your friend still looks like your friend — same shape, same proportions. What stayed the same? What changed? Write down 2 things in each category.

Record your answer in your workbook.
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What "Similar" Means
+5 XP

Two figures are similar if one is an enlargement (or reduction) of the other. The two figures have the same shape but can be different sizes. Two conditions must hold:

  • All pairs of corresponding angles are equal.
  • All pairs of corresponding sides are in the same ratio.

If $\triangle ABC$ is similar to $\triangle DEF$, we write $\triangle ABC \sim \triangle DEF$. The symbol $\sim$ means "is similar to". The order of letters matters — vertex $A$ matches vertex $D$, $B$ matches $E$, $C$ matches $F$. That means $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$, and the side opposite $A$ (which is $BC$) is in ratio with the side opposite $D$ (which is $EF$).

Similar triangles (same shape, different size) A B C D E F Sides in ratio 1:2 — angles identical
$\triangle ABC \sim \triangle DEF$  ⇒  equal angles, sides in ratio
Same shape
If you can scale (zoom) one to match the other, they're similar.
Angles match exactly
Corresponding angles are EQUAL — not "in ratio".
Sides in ratio
Every pair of corresponding sides is in the SAME ratio.
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What You'll Master
objectives

Know

  • Definition of similar figures
  • The symbol $\sim$ ("is similar to")
  • What "corresponding" sides and angles are
  • Definition of scale factor

Understand

  • Why congruent figures are a special case of similar
  • How to identify corresponding parts from the order of letters
  • Why scale factor > 1 enlarges and < 1 reduces

Can Do

  • Determine if two figures are similar from a diagram
  • Write the similarity statement with correct vertex order
  • Calculate the scale factor between two similar figures
3
Words You Need
vocabulary
SimilarSame shape, possibly different size. Symbol: $\sim$.
CongruentSame shape AND same size. Symbol: $\equiv$.
Corresponding anglesAngles that match between two similar figures.
Corresponding sidesSides that match between two similar figures.
Scale factorThe ratio between corresponding sides of similar figures.
EnlargementA bigger similar copy (scale factor > 1).
ReductionA smaller similar copy (scale factor < 1).
RatioA comparison of two quantities by division.
4
The Scale Factor
+5 XP

The scale factor is the multiplier that turns the smaller figure into the larger one. It is calculated as:

$\text{Scale factor} = \dfrac{\text{side of NEW figure}}{\text{corresponding side of ORIGINAL figure}}$

If the scale factor is $2$, the new figure is twice as big. If the scale factor is $\frac{1}{2}$, the new figure is half the size. A scale factor of $1$ means the figures are congruent (identical).

Example: a small triangle has sides $3, 4, 5$ cm. A similar triangle has sides $6, 8, 10$ cm. Check the ratios: $\frac{6}{3} = \frac{8}{4} = \frac{10}{5} = 2$. All three ratios match, so the figures ARE similar and the scale factor is $2$.

Scale factor = 2 3 4 5 Original 6 8 10 Enlarged x2
Scale factor $= \dfrac{6}{3} = \dfrac{8}{4} = \dfrac{10}{5} = 2$
All ratios must match
If even ONE pair of corresponding sides has a different ratio, the figures are NOT similar.
Bigger or smaller?
SF > 1 = enlargement. SF < 1 = reduction. SF = 1 = congruent.
Always new over old
For consistency: new figure side divided by original figure side.
Book notes · Scale factor
  • Scale factor = new side ÷ corresponding old side.
  • The same value works for EVERY pair of corresponding sides.
  • SF $> 1$ enlarges; SF $< 1$ reduces; SF $= 1$ means congruent.
Two similar rectangles: small has sides $4$ cm and $6$ cm; large has sides $12$ cm and $18$ cm. The scale factor (large ÷ small) is:
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Spot the Corresponding Parts
+5 XP

When two triangles are similar, every part has a "match". To find matching pairs:

  • Use the order of letters. In $\triangle ABC \sim \triangle PQR$, $A \leftrightarrow P$, $B \leftrightarrow Q$, $C \leftrightarrow R$.
  • Match angles first. Equal angles sit OPPOSITE corresponding sides.
  • Match shortest with shortest, longest with longest. In similar triangles the order of side lengths is preserved.

If $\triangle ABC \sim \triangle PQR$: $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$. The side opposite $A$ is $BC$; the side opposite $P$ is $QR$, so $BC$ and $QR$ are corresponding. Similarly $AB \leftrightarrow PQ$ and $AC \leftrightarrow PR$. The ratio $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$.

Matching corresponding parts A B C P Q R A→P, B→Q, C→R (order matters in the similarity statement)
$\dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{AC}{PR}$
Order is everything
$\triangle ABC \sim \triangle PQR$ tells you exactly which vertex matches which.
Opposite-side trick
Sides "across from" equal angles are corresponding.
Shortest↔shortest
The shortest side of one triangle matches the shortest side of the other.
Book notes · Corresponding parts
  • Use the order of letters in $\triangle ABC \sim \triangle PQR$ to find matches.
  • Equal angles sit opposite corresponding sides.
  • Shortest matches shortest, longest matches longest.
True or false?

In two similar figures, every pair of corresponding angles is EQUAL (not just in ratio).

6
Similar vs Congruent
+5 XP

Congruent figures are EXACTLY the same — same shape AND same size. Similar figures only need the same shape — the size can differ. So:

  • If two figures are congruent, they are automatically similar (with scale factor 1).
  • If two figures are similar, they are NOT necessarily congruent — they could be different sizes.

Symbols matter: $\equiv$ means "congruent" (identical). $\sim$ means "similar" (same shape). Congruent triangles have ALL sides equal AND all angles equal. Similar triangles have all angles equal AND all sides in the same ratio.

Congruent (identical) vs Similar (same shape) CONGRUENT ≡ Same size + same shape SIMILAR ~ Same shape, different size
Congruent ⇒ Similar, but NOT the other way around
$\equiv$ vs $\sim$
Congruent = identical. Similar = same shape, any size.
Congruent = special similar
Congruent figures have scale factor exactly 1.
Similar ≠ congruent
Two similar shapes can be very different sizes.
Book notes · Similar vs Congruent
  • $\equiv$ means congruent (identical); $\sim$ means similar (same shape).
  • Congruent figures are similar with SF $= 1$.
  • Similar figures may have any positive SF.
Fill in the blank: Two figures that are the same shape AND the same size are called ____________.

Watch Me Solve It · 3 examples

Watch Me Solve It · Find the scale factor
+15 XP per step
Q1
PROBLEM
Triangle $ABC$ has sides $5, 7, 9$ cm. The similar triangle $DEF$ has corresponding sides $15, 21, 27$ cm. Find the scale factor from $ABC$ to $DEF$.
  1. 1
    Set up ratio (new ÷ old)
    $\text{SF} = \dfrac{DE}{AB} = \dfrac{15}{5}$
  2. 2
    Check the other pairs match
    $\dfrac{15}{5} = 3, \; \dfrac{21}{7} = 3, \; \dfrac{27}{9} = 3$
  3. 3
    State the scale factor
    All three ratios equal $3$, so SF $= 3$.
    $DEF$ is three times the size of $ABC$.
AnswerScale factor $= 3$.
Watch Me Solve It · Are these similar?
+15 XP per step
Q2
PROBLEM
Rectangle $A$ has sides $4$ cm by $6$ cm. Rectangle $B$ has sides $10$ cm by $12$ cm. Are they similar?
  1. 1
    Match corresponding sides
    Short↔short: $4 \to 10$; long↔long: $6 \to 12$.
  2. 2
    Check the ratios
    $\dfrac{10}{4} = 2.5$; $\dfrac{12}{6} = 2$.
  3. 3
    Compare
    $2.5 \neq 2$, so ratios are NOT equal.
    Therefore the rectangles are NOT similar.
AnswerNot similar — the ratios of corresponding sides are different.
Watch Me Solve It · Write the similarity statement
+15 XP per step
Q3
PROBLEM
In $\triangle KLM$, $\angle K = 50^{\circ}, \angle L = 60^{\circ}, \angle M = 70^{\circ}$. In $\triangle XYZ$, $\angle X = 70^{\circ}, \angle Y = 50^{\circ}, \angle Z = 60^{\circ}$. Write the similarity statement.
  1. 1
    Match the equal angles
    $\angle K = 50^{\circ} = \angle Y$ — so $K \leftrightarrow Y$.
  2. 2
    Match the others
    $\angle L = 60^{\circ} = \angle Z$; $\angle M = 70^{\circ} = \angle X$.
  3. 3
    Write in matching order
    $\triangle KLM \sim \triangle YZX$
    Vertex order must match: $K \to Y, L \to Z, M \to X$.
Answer$\triangle KLM \sim \triangle YZX$.

Common Pitfalls

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Common Pitfalls
heads-up
Assuming all rectangles are similar
A $4 \times 6$ rectangle is NOT similar to a $4 \times 10$ rectangle — the ratios differ. Equal angles alone are not enough for quadrilaterals.
Fix: Check that ALL pairs of corresponding sides have the same ratio.
Writing the similarity statement out of order
$\triangle ABC \sim \triangle DEF$ means $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$. If you write the wrong order, your side ratios will mix up.
Fix: Match by equal angles first, then write vertices in that order.
Confusing $\sim$ with $\equiv$
$\sim$ means similar (same shape only). $\equiv$ means congruent (same shape AND size).
Fix: Use $\sim$ unless the figures are exactly identical.
Copy Into Your Books

Definition

  • Similar = same shape, possibly different size
  • Symbol: $\sim$
  • Corresponding angles equal
  • Corresponding sides in same ratio

Scale factor

  • SF $= \frac{\text{new}}{\text{old}}$
  • SF $> 1$: enlargement
  • SF $< 1$: reduction
  • SF $= 1$: congruent

Similarity statement

  • $\triangle ABC \sim \triangle DEF$
  • $A \to D, B \to E, C \to F$
  • Order MUST match

Similar vs Congruent

  • $\equiv$ same shape AND size
  • $\sim$ same shape, any size
  • Congruent ⇒ Similar (SF = 1)

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Spotting Similarity
4 problems

Four quick drills on similarity and scale factor.

  1. 1 Two similar triangles. The smaller has shortest side $4$; the larger has shortest side $20$. Find the scale factor.

    $\frac{20}{4} = 5$.SF = 5

  2. 2 $\triangle ABC \sim \triangle XYZ$. Name the side corresponding to $AB$.

    Letters match in order.$XY$

  3. 3 If two figures are congruent, what is their scale factor?

    Congruent = identical.SF = 1

  4. 4 A photo $6$ cm wide is enlarged to $24$ cm wide. What is the scale factor?

    $\frac{24}{6} = 4$.SF = 4
Complete in your workbook.

Quick Check · 5 questions

1
Two figures are similar if they have:
+10 XP
2
Rectangle $A$ ($4 \times 6$) is similar to rectangle $B$ ($12 \times 18$). The scale factor from $A$ to $B$ is:
+10 XP
3
The symbol $\sim$ means:
+10 XP
4
Given $\triangle ABC \sim \triangle PQR$, which side corresponds to $AC$?
+10 XP
5
Two congruent figures have a scale factor of:
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Recall Easy 3 MARKS

Q6. Triangle $ABC$ has sides $6$ cm, $8$ cm, $10$ cm. Similar triangle $DEF$ has corresponding sides $9$ cm, $12$ cm, $15$ cm.
(a) Find the scale factor from $ABC$ to $DEF$.
(b) Confirm all three pairs of sides give the same ratio.
(c) Is $DEF$ an enlargement or reduction?

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Decide whether each pair of rectangles is similar. Justify each.
(a) $3 \times 5$ and $9 \times 15$.
(b) $4 \times 7$ and $8 \times 12$.
(c) $2 \times 6$ and $5 \times 15$.

Answer in your workbook.
Reason Hard 3 MARKS

Q8. $\triangle ABC \sim \triangle DEF$. Given $\angle A = 50^{\circ}$, $\angle B = 65^{\circ}$, and side $AB = 4$ corresponds to side $DE = 12$:
(a) What is $\angle C$?
(b) What is $\angle D$?
(c) What is the scale factor from $ABC$ to $DEF$?

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — Same shape, possibly different size.

2. C — SF $= \frac{12}{4} = 3$.

3. A — $\sim$ means "is similar to".

4. D — $AC \to PR$ (letters match in order).

5. C — Congruent figures are identical: SF = 1.

Show Your Working Model Answers

Q6 (3 marks): (a) SF $= \frac{9}{6} = \frac{3}{2} = 1.5$ [1]. (b) $\frac{9}{6}=1.5$, $\frac{12}{8}=1.5$, $\frac{15}{10}=1.5$ ✓ [1]. (c) Enlargement since SF $> 1$ [1].

Q7 (3 marks): (a) $\frac{9}{3}=3$, $\frac{15}{5}=3$ — similar [1]. (b) $\frac{8}{4}=2$, $\frac{12}{7} \neq 2$ — NOT similar [1]. (c) $\frac{5}{2}=2.5$, $\frac{15}{6}=2.5$ — similar [1].

Q8 (3 marks): (a) $\angle C = 180 - 50 - 65 = 65^{\circ}$ [1]. (b) $\angle D = \angle A = 50^{\circ}$ (corresponding angles equal) [1]. (c) SF $= \frac{DE}{AB} = \frac{12}{4} = 3$ [1].

Stretch Challenge · +25 XP, +10 coins

Are They Similar?

Two pentagons have sides (in order around the perimeter): pentagon $A$: $2, 3, 4, 3, 2$ cm; pentagon $B$: $6, 9, 12, 9, 6$ cm. (a) Calculate the ratios of each pair of corresponding sides. (b) Based ONLY on those ratios, can we conclude the pentagons are similar? Explain why corresponding sides in ratio is NECESSARY but not SUFFICIENT — what else would we need to check?

Reveal solution

(a) Each ratio is $\frac{6}{2}=\frac{9}{3}=\frac{12}{4}=\frac{9}{3}=\frac{6}{2}=3$. All equal to $3$. (b) For triangles, equal side ratios is enough. For polygons with 4+ sides, we ALSO need corresponding angles to be equal — otherwise the pentagon could be "squashed" into a different shape with the same side lengths. So sides in ratio is necessary but not sufficient for similarity.

R
Quick Review

Similar

Same shape, possibly different size. Symbol $\sim$.

Conditions

Equal corresponding angles + sides in same ratio.

Scale factor

SF = new ÷ old. Same for every pair of sides.

Statement order

$\triangle ABC \sim \triangle DEF$ means $A \to D$, $B \to E$, $C \to F$.

Congruent

Same shape AND size. Special similar with SF $= 1$.

Enlarge / reduce

SF $> 1$ enlarges; SF $< 1$ reduces.

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