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Lesson 14 ~25 min Unit 3 · Geometry +85 XP

Introduction to Congruent Figures

Two figures are congruent when one can be slid, flipped or rotated to land exactly on top of the other — same shape AND same size. Four classic tests prove triangles are congruent: SSS, SAS, AAS, RHS.

Today's hook: If two triangles share THREE matching ingredients, they HAVE to be twins. But not just any three — pick wisely.
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Think First
warm-up

Trace a triangle onto tracing paper. Slide it across the desk. Flip the paper over. Rotate it. Do those moves change the triangle's shape or size? If you placed your traced copy back on top of the original, would it fit exactly? What word describes such "identical" figures?

Record your answer in your workbook.
1
What Is Congruence?
+5 XP

Two figures are congruent if one can be moved — by translation (slide), reflection (flip) or rotation (turn) — to fit exactly onto the other. "Exactly" means: same shape AND same size. Every matching pair of sides has equal length; every matching pair of angles has equal measure. The symbol is $\equiv$ (or sometimes $\cong$).

Congruent figures have equal corresponding sides AND equal corresponding angles. The order of vertices in a congruence statement matters: if $\triangle ABC \equiv \triangle DEF$, then $A$ matches $D$, $B$ matches $E$, $C$ matches $F$. Side $AB$ equals side $DE$, and $\angle A = \angle D$. Congruence is stronger than "similar" — similar shapes are the same shape but can be different sizes; congruent shapes must be the same size too.

Congruent triangles: ABC = DEF A B C D E F Slide / flip / rotate one onto the other -> perfect fit
$\triangle ABC \equiv \triangle DEF$ — same shape AND same size.
Same shape AND size
Both must match. Size matters — that's what separates congruent from similar.
$\equiv$ is the symbol
"$\triangle ABC \equiv \triangle DEF$" reads "triangle ABC is congruent to triangle DEF".
Vertex order matters
$A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$ tells you which parts match.
2
What You'll Master
objectives

Know

  • Definition of congruence (same shape AND size)
  • The four congruence tests: SSS, SAS, AAS, RHS
  • The congruence symbol $\equiv$ and statement convention

Understand

  • Why vertex order in the congruence statement matches corresponding parts
  • Why three correct pieces are enough (and which three)
  • Why SSA (two sides + non-included angle) is NOT a test

Can Do

  • Identify which test (SSS, SAS, AAS or RHS) applies to a pair of triangles
  • Write a congruence statement with vertices in matching order
  • State which sides and angles are equal as a consequence
3
Words You Need
vocabulary
CongruentSame shape AND same size; one fits exactly on the other.
CorrespondingMatching: corresponding sides face matching vertices.
SSSSide-Side-Side: three pairs of sides equal.
SASSide-Angle-Side: two sides AND the INCLUDED angle equal.
AASAngle-Angle-Side: two angles AND one matching side equal.
RHSRight-angle-Hypotenuse-Side: right-angled triangle with hypotenuse + one side equal.
Included angleThe angle BETWEEN two given sides.
HypotenuseThe longest side of a right-angled triangle, opposite the right angle.
4
SSS and SAS
+5 XP

SSS (Side-Side-Side): if all THREE pairs of corresponding sides are equal, the triangles are congruent. SAS (Side-Angle-Side): if TWO pairs of sides AND the angle BETWEEN them are equal, the triangles are congruent. The included angle is critical — SSA does NOT prove congruence.

SSS: $AB = DE$, $BC = EF$, $CA = FD$ $\Rightarrow$ $\triangle ABC \equiv \triangle DEF$ (SSS).
SAS: $AB = DE$, $\angle B = \angle E$, $BC = EF$ $\Rightarrow$ $\triangle ABC \equiv \triangle DEF$ (SAS). The angle MUST be between the two given sides — that's why it's "included". SSA (two sides plus a NON-included angle) is ambiguous — sometimes two different triangles fit, so SSA is not a valid test.

SAS: two sides + INCLUDED angle A B C angle D E F angle AB=DE, angle B = angle E, BC=EF -> triangles congruent (SAS)
SSS or SAS — both prove congruence of triangles.
SSS = three sides
All three sides equal — no angles needed.
SAS needs INCLUDED
The angle has to be between the two given sides.
SSA is NOT a test
Side-Side-non-included-Angle can produce two different triangles.
Quick book-notes · SSS and SAS
  • SSS: three pairs of sides equal $\Rightarrow$ congruent.
  • SAS: two sides + INCLUDED angle equal.
  • SSA does NOT prove congruence.
Micro-check: Two triangles have $AB = DE = 6$ cm, $\angle A = \angle D = 70^{\circ}$, $AC = DF = 9$ cm. Which test applies?
5
AAS and RHS
+5 XP

AAS (Angle-Angle-Side): if two angles AND a corresponding side are equal, the triangles are congruent. Because angles in a triangle sum to $180^{\circ}$, knowing two angles tells you the third — so it doesn't matter whether the side is between the two angles or not, as long as it's the CORRESPONDING side. RHS (Right-Hypotenuse-Side) is the special test for RIGHT-angled triangles: equal hypotenuses plus one equal non-hypotenuse side proves congruence.

AAS: $\angle A = \angle D$, $\angle B = \angle E$, $AB = DE$ (or any matching pair of sides) $\Rightarrow$ $\triangle ABC \equiv \triangle DEF$ (AAS).
RHS: both triangles right-angled, hypotenuse $AB = $ hypotenuse $DE$, and one other side equal $\Rightarrow$ congruent (RHS).
RHS is the only test that works just for right-angled triangles — in general triangles you'd need SAS/SSS/AAS. AAA (three angles) is NOT a congruence test — it shows similarity, not congruence (the triangles can be different sizes).

RHS: Right angle + Hypotenuse + Side A B C hyp D E F hyp Right angles + equal hypotenuse + equal side -> congruent (RHS)
Four tests: SSS, SAS, AAS, RHS (NOT AAA, NOT SSA).
AAS = ASA in NSW
Some texts call it ASA when the side is between the two angles. NSW uses AAS for either.
RHS needs right angle
Without right angles in both triangles, you can't use RHS.
AAA $\neq$ congruent
Same angles can be different sizes — only proves SIMILARITY.
Quick book-notes · AAS and RHS
  • AAS: two angles + a matching side equal.
  • RHS: right angle + hypotenuse + one other side equal.
  • AAA is NOT a congruence test (similarity only).
True or false: Two triangles with all three pairs of angles equal must be congruent.

Equal angles only mean SIMILAR — they can be different sizes. Congruence needs at least one matching SIDE.

6
Writing Congruence Statements
+5 XP

A congruence statement like $\triangle ABC \equiv \triangle DEF$ encodes which vertex matches which. The first vertex on the left ($A$) matches the first vertex on the right ($D$). $B$ matches $E$. $C$ matches $F$. Get the order WRONG and you've written a false statement — even if the triangles really are congruent.

If $\triangle ABC \equiv \triangle DEF$ then:
• Sides match: $AB = DE$, $BC = EF$, $CA = FD$.
• Angles match: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.
So before writing a congruence statement, identify which vertex of one triangle lies at the same "role" as a vertex of the other. Look at angle markings or equal-side tick marks to figure out matches.

Vertex order in a congruence statement Triangle ABC = Triangle DEF means A <-> D, B <-> E, C <-> F So AB = DE, BC = EF, CA = FD And angle A = angle D, etc. Wrong order = wrong matching = lost marks.
Vertex order encodes which parts equal which.
Match before writing
Look at tick marks and angle marks to pair vertices.
Add the reason
"$\triangle ABC \equiv \triangle DEF$ (SAS)" — name the test used.
Then use it
Once congruent, all matching parts are equal — cite "(matching $\angle$s of $\equiv$ $\triangle$s)".
Quick book-notes · Congruence statements
  • $\triangle ABC \equiv \triangle DEF$: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$.
  • Order encodes matching sides AND angles.
  • Always cite the test (SSS, SAS, AAS, RHS) after the statement.
Micro-check: If $\triangle PQR \equiv \triangle XYZ$, which statement is correct?

Watch Me Solve It · 3 examples

Watch Me Solve It · Identify the test
+15 XP per step
Q1
PROBLEM
In $\triangle ABC$ and $\triangle DEF$: $AB = DE = 7$ cm, $BC = EF = 5$ cm, $\angle B = \angle E = 60^{\circ}$. Are the triangles congruent? Which test?
  1. 1
    List the equal parts
    $AB = DE$ (side), $\angle B = \angle E$ (angle), $BC = EF$ (side)
  2. 2
    Check the angle's position
    $\angle B$ is BETWEEN sides $AB$ and $BC$ — it's the included angle.
  3. 3
    Apply the test
    $\triangle ABC \equiv \triangle DEF$ (SAS)
    Two sides + included angle — classic SAS.
Answer$\triangle ABC \equiv \triangle DEF$ (SAS).
Watch Me Solve It · AAS test
+15 XP per step
Q2
PROBLEM
In $\triangle PQR$ and $\triangle XYZ$: $\angle P = \angle X = 50^{\circ}$, $\angle Q = \angle Y = 70^{\circ}$, $PQ = XY = 8$ cm. Prove the triangles are congruent.
  1. 1
    List the equal parts
    $\angle P = \angle X$, $\angle Q = \angle Y$, $PQ = XY$
  2. 2
    Identify the test
    Two angles + a corresponding side $\Rightarrow$ AAS.
  3. 3
    Write the statement
    $\triangle PQR \equiv \triangle XYZ$ (AAS)
    $P \leftrightarrow X$, $Q \leftrightarrow Y$, $R \leftrightarrow Z$ — sides $PQ$ and $XY$ are corresponding.
Answer$\triangle PQR \equiv \triangle XYZ$ (AAS).
Watch Me Solve It · RHS test
+15 XP per step
Q3
PROBLEM
$\triangle KLM$ and $\triangle NOP$ are both right-angled at $L$ and $O$ respectively. $KM = NP = 13$ cm (hypotenuses). $LM = OP = 5$ cm. Prove the triangles are congruent.
  1. 1
    Note the right angles
    $\angle L = \angle O = 90^{\circ}$ (given)
  2. 2
    Hypotenuses equal
    $KM = NP$ (hypotenuses) and $LM = OP$ (matching sides)
  3. 3
    Apply RHS
    $\triangle KLM \equiv \triangle NOP$ (RHS)
    Right angle + hypotenuse + one other side — RHS test satisfied.
Answer$\triangle KLM \equiv \triangle NOP$ (RHS).

Common Pitfalls

8
Common Pitfalls
heads-up
Using SSA as a congruence test
Side-Side-non-included-Angle is NOT enough — two different triangles can fit the data.
Fix: The angle must be BETWEEN the two given sides (SAS). If it isn't, look for another test.
Treating AAA as congruence
Three pairs of equal angles only prove SIMILARITY — the triangles can be different sizes.
Fix: Need at least one matching SIDE to prove congruence.
Wrong vertex order in the statement
Writing $\triangle ABC \equiv \triangle EDF$ when $A$ actually matches $D$ encodes the WRONG pairs.
Fix: Identify matches first; then write the statement with vertices in correct order.
Copy Into Your Books

Definition

  • Same shape AND size
  • Symbol: $\equiv$
  • "$\triangle ABC \equiv \triangle DEF$"

The four tests

  • SSS — three sides
  • SAS — two sides + INCLUDED angle
  • AAS — two angles + matching side
  • RHS — right angle + hypotenuse + side

NOT tests

  • SSA (ambiguous)
  • AAA (similar, not congruent)

Vertex order

  • $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$
  • Sides $AB = DE$, etc.
  • $\angle A = \angle D$, etc.

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Pick the Test
4 problems

Four drills on identifying the congruence test. Solve, then reveal.

  1. 1 Two triangles have three pairs of equal sides. Which test?

    Three sides equal.SSS

  2. 2 Two right-angled triangles have equal hypotenuses and one equal short side. Which test?

    Right + hypotenuse + side.RHS

  3. 3 Two triangles share two angles and the side between them. Which test?

    Two angles + matching side.AAS (also called ASA)

  4. 4 If $\triangle ABC \equiv \triangle DEF$, which side equals $BC$?

    $B \leftrightarrow E$, $C \leftrightarrow F$.$EF$
Complete in your workbook.

Quick Check · 5 questions

1
Two figures are congruent if they have:
+10 XP
2
Two triangles match on two sides and the angle between them. Which test?
+10 XP
3
Which is NOT a valid congruence test?
+10 XP
4
$\triangle PQR \equiv \triangle XYZ$. Which is true?
+10 XP
5
Which test applies ONLY to right-angled triangles?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Recall Easy 3 MARKS

Q6. For each pair of triangles, name the congruence test that applies. (a) All three sides equal. (b) Two angles equal, plus one matching side. (c) Both right-angled, hypotenuses equal, one other side equal.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. In $\triangle ABC$ and $\triangle DEF$: $AB = DE = 9$ cm, $\angle A = \angle D = 65^{\circ}$, $AC = DF = 12$ cm.
(a) Identify the test and explain why the included-angle condition is satisfied.
(b) Write the congruence statement with correct vertex order.
(c) State the size of $\angle B$ given $\angle E = 80^{\circ}$.

Answer in your workbook.
Reason Hard 3 MARKS

Q8. Decide whether each is TRUE or FALSE and give a one-sentence reason.
(a) If two triangles have all three pairs of equal angles, they must be congruent.
(b) SSA is a valid congruence test.
(c) If $\triangle ABC \equiv \triangle DEF$ then $\angle C = \angle F$.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. C — Same shape AND same size.

2. B — SAS (two sides + included angle).

3. D — AAA proves similarity, not congruence.

4. A — $\angle P = \angle X$.

5. D — RHS.

Show Your Working Model Answers

Q6 (3 marks): (a) SSS [1]. (b) AAS [1]. (c) RHS [1].

Q7 (3 marks): (a) The test is SAS. $\angle A$ is the angle between sides $AB$ and $AC$, so it is the INCLUDED angle [1]. (b) $\triangle ABC \equiv \triangle DEF$ (SAS) [1]. (c) $\angle B = \angle E = 80^{\circ}$ (matching $\angle$s of $\equiv$ $\triangle$s) [1].

Q8 (3 marks): (a) FALSE — equal angles only force SIMILARITY; the triangles can be different sizes [1]. (b) FALSE — SSA is ambiguous and can produce two different triangles, so it cannot guarantee congruence [1]. (c) TRUE — in $\triangle ABC \equiv \triangle DEF$, vertex $C$ matches vertex $F$, so $\angle C = \angle F$ [1].

Stretch Challenge · +25 XP, +10 coins

The Ambiguous Case

Consider triangles with $AB = 10$ cm, $BC = 7$ cm and $\angle A = 35^{\circ}$. Notice that $\angle A$ is NOT between $AB$ and $BC$ — it's at vertex $A$, but side $BC$ doesn't touch it. This is "SSA". (a) Sketch two different triangles that satisfy these conditions. (b) Explain in words why SSA does NOT guarantee congruence. (c) What single extra piece of information would let you pin down a unique triangle (give two different valid choices)?

Reveal solution

(a) Imagine $AB$ fixed, with $\angle A = 35^{\circ}$ marking a ray from $A$. Then place a 7 cm segment from $B$ — the other endpoint can land on the ray in TWO different positions (one acute triangle, one obtuse), giving two non-congruent triangles. (b) SSA is ambiguous: with two sides plus a non-included angle, the third side can swing in two ways and create different triangles. (c) Pin it down by adding either: (i) the angle at $B$ or $C$ (giving AAS), or (ii) the length of the third side $AC$ (giving SSS), or (iii) specifying the included angle $\angle B$ between $AB$ and $BC$ (giving SAS).

R
Quick Review

Congruent

Same shape AND size. Symbol $\equiv$.

SSS

Three pairs of sides equal.

SAS

Two sides + INCLUDED angle equal.

AAS

Two angles + matching side equal.

RHS

Right + Hypotenuse + Side.

Order matters

$\triangle ABC \equiv \triangle DEF$: $A \leftrightarrow D$ etc.

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