Lesson 2 ~25 min Unit 3 · Geometry +85 XP

Types of Triangles by Their Sides

Classify triangles as equilateral, isosceles or scalene by comparing side lengths. Use tick marks to show equal sides and connect side-classification to angle and symmetry properties.

Today's hook: Ancient Egyptians built the four faces of the Great Pyramid as triangles with three almost-equal sides. Why does using all-equal sides make a pyramid so strong? And why do bridge engineers still trust triangles more than any other shape?

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Think First

warm-up

Draw three triangles on paper. In one, make all three sides the same length. In another, make exactly two sides the same length. In the third, make every side a different length. What do you notice about the angles in each one? Could you fold any of them in half so the two halves match?

Record your answer in your workbook.

The Big Idea

+5 XP

A triangle is a three-sided closed shape. We can sort every triangle in the world into one of three groups by looking only at the lengths of its sides: equilateral (all three sides equal), isosceles (exactly two sides equal), or scalene (no sides equal). Small tick marks drawn on the sides tell you which sides have the same length.

An equilateral triangle has all three sides equal, all three angles equal to $60^{\circ}$, and three lines of symmetry. An isosceles triangle has two equal sides, two equal "base angles", and one line of symmetry running down the middle. A scalene triangle has three different side lengths, three different angles, and no lines of symmetry at all.

Equilateral: 3 equal sides | Isosceles: 2 equal sides | Scalene: 0 equal sides

"Equi" = equal

Equilateral has "equi" in the name, all parts equal.

"Iso" = same

Isosceles, "iso" means same. Two sides are the same.

Scalene = scattered

All three sides are scattered, no two match.

What to write in your book

Equilateral triangle: 3 equal sides and 3 equal angles ($60^{\circ}$ each).
Isosceles triangle: exactly 2 equal sides, 2 equal base angles, 1 line of symmetry.
Scalene triangle: no sides equal, no angles equal, 0 lines of symmetry.

Quick check, which triangle has exactly two equal sides?

What You'll Master

objectives

Know

Equilateral = 3 equal sides, 3 angles each $60^{\circ}$, 3 lines of symmetry
Isosceles = 2 equal sides, 2 equal base angles, 1 line of symmetry
Scalene = 0 equal sides, 0 equal angles, 0 lines of symmetry
Tick marks indicate sides of equal length
The triangle inequality: any two sides must add to more than the third

Understand

Why "two equal sides" automatically forces "two equal base angles"
How tick marks let you read a diagram without measuring
That side-classification and angle-classification are independent ways of describing the same triangle
Why three lengths sometimes cannot be joined into a closed triangle

Can Do

Classify a triangle as equilateral, isosceles or scalene from a diagram or side measurements
Add or read tick marks correctly on a triangle diagram
State the number of lines of symmetry for each type
Test three given lengths against the triangle inequality to decide if they form a valid triangle

Words You Need

vocabulary

EquilateralA triangle with all three sides of equal length (and therefore all angles $60^{\circ}$).

IsoscelesA triangle with exactly two sides of equal length.

ScaleneA triangle with no two sides of equal length (every side is different).

Base anglesIn an isosceles triangle, the two equal angles opposite the equal sides.

ApexThe "top" vertex of an isosceles triangle, where the two equal sides meet.

Tick marksSmall dashes on a side of a polygon, sides with the same number of ticks are equal in length.

Line of symmetryA line that folds a shape so both halves match exactly.

Triangle inequalityThe rule that any two sides of a triangle must add to more than the third side, otherwise no triangle can be formed.

Spot the Trap

heads-up

Wrong: "If a triangle looks like it has two equal sides, it must be isosceles." Not unless the tick marks (or stated measurements) say so, eyeballing isn't a proof.

Right: Read the tick marks. One tick on two sides → those two sides are equal → isosceles.

Wrong: "A scalene triangle has at most one equal side." That makes no sense, "scalene" means no sides are equal at all.

Right: An equilateral triangle could technically be called "isosceles" (it has two equal sides), but we always give it the most specific name: equilateral.

Reading Tick Marks

+5 XP

When mathematicians draw a triangle, they don't always write the side lengths. Instead, they put small tick marks on each side. Sides with the same number of ticks are equal in length. This shorthand works for any polygon, not just triangles.

One tick on every side → all three sides equal (equilateral). One tick on two of the sides → those two sides are equal (isosceles). Different numbers of ticks (or no ticks at all on any side) → no two sides equal (scalene). The trick is to count, not to eyeball.

Same number of ticks → sides are equal in length

Count, don't eyeball

Drawings aren't always to scale. Trust the ticks, not the look.

Add ticks when drawing

If you draw an isosceles triangle, draw the ticks too, otherwise readers can't tell.

Different ticks = different lengths

A side with one tick is NOT equal to a side with two ticks.

What to write in your book

Tick marks on sides indicate equal lengths, same number of ticks $=$ same length.
Three ticks (one each) on every side → equilateral; two ticks (one each) on two sides → isosceles.
Always trust the ticks, not the visual, diagrams aren't drawn to scale.

True or false?

A side with one tick mark is equal in length to a side with two tick marks.

Equal Sides → Equal Angles → Symmetry

+5 XP

Side-equality and angle-equality go together. If two sides are equal, the angles opposite them must also be equal, and that gives the shape a line of symmetry. So once you classify by sides, you immediately know things about the angles too.

Equilateral → all 3 angles must be $60^{\circ}$ (because $180 \div 3 = 60$) and there are 3 lines of symmetry. Isosceles → the two angles opposite the equal sides (the base angles) are equal, and there is 1 line of symmetry down the middle. Scalene → all three angles are different and there are 0 lines of symmetry.

Equal sides ↔ Equal opposite angles ↔ Line of symmetry

Look across, not next to

The equal base angles sit opposite the equal sides, not next to them.

Fold test

If you can fold the triangle so two halves match, you've found a line of symmetry.

Equilateral is special

Three lines of symmetry: one from each vertex straight to the opposite side.

What to write in your book

Equal sides force equal opposite angles, the angles opposite the equal sides are themselves equal.
Equilateral triangle: each angle is $60^{\circ}$ (because $180 \div 3 = 60$).
Lines of symmetry: equilateral $= 3$, isosceles $= 1$, scalene $= 0$.

Fill in the blank.

In an equilateral triangle, every interior angle measures °.

The Triangle Inequality

+5 XP

Not every set of three lengths can be joined into a triangle! There is a rule called the triangle inequality: the sum of any two sides must be greater than the third side. If this fails for even one pair, the two shorter sides simply cannot stretch far enough to meet and close up the shape.

Check sides $5, 6, 8$: $5+6=11>8$, $5+8=13>6$, $6+8=14>5$, all three checks pass, so this triangle closes and exists. Now check sides $3, 4, 9$: $3+4=7$, which is NOT greater than $9$, so this check fails. The two shortest sides ($3$ and $4$) combined only reach $7$ units, they can never stretch across a $9$-unit gap, so no triangle can be formed.

Sides $a, b, c$ form a triangle only if $a+b>c$ AND $a+c>b$ AND $b+c>a$.

Check all three sums

Add each pair of sides and compare to the third, not just the two smallest.

Shortcut, shortest pair

If the two SHORTEST sides beat the LONGEST side, the other two checks always pass too.

Equal is not enough

If two sides add to EXACTLY the third (e.g. $3+4=7$), the shape flattens into a straight line, not a triangle.

What to write in your book

Triangle inequality: for sides $a, b, c$, every pair must add to more than the third, $a+b>c$, $a+c>b$, $b+c>a$.
Shortcut: add the two SHORTEST sides and compare to the LONGEST, if that sum is bigger, a triangle exists.
If the two shortest sides add to exactly the longest (or less), no triangle can be formed.

Quick check, can a triangle be formed with sides $4\text{ cm}, 5\text{ cm}$ and $10\text{ cm}$?

Watch Me Solve It · 3 examples

Watch Me Solve It · Classify by side lengths

+15 XP per step

PROBLEM

A triangle has sides of length $7\text{ cm}, 7\text{ cm}$ and $4\text{ cm}$. Classify it by its sides, and state how many lines of symmetry it has.

1
Count equal sides
Two sides equal $7\text{ cm}$, one side is $4\text{ cm}$
Exactly two sides match.
2
Name the type
Two equal sides → isosceles
3
Lines of symmetry
Isosceles → 1 line of symmetry
It runs from the apex (where the two $7\text{ cm}$ sides meet) down to the middle of the $4\text{ cm}$ base.

AnswerIsosceles; 1 line of symmetry.

Watch Me Solve It · Reading tick marks

+15 XP per step

PROBLEM

A triangle has one tick on every side, all three sides carry a single tick mark. Classify the triangle, state the size of each angle, and give the number of lines of symmetry.

1
Read the ticks
All three sides have one tick → all three sides are equal
2
Name the type
Three equal sides → equilateral
3
Angles and symmetry
Each angle $= 180 \div 3 = 60^{\circ}$; 3 lines of symmetry
Angles in a triangle sum to $180^{\circ}$, and three equal sides force three equal angles.

AnswerEquilateral; each angle $60^{\circ}$; 3 lines of symmetry.

Watch Me Solve It · Real-world shape

+15 XP per step

PROBLEM

A road-sign triangle (a yield sign) has sides $90\text{ cm}, 90\text{ cm}$ and $90\text{ cm}$. A racing sail has sides $4\text{ m}, 6\text{ m}$ and $7\text{ m}$. Classify each by its sides.

1
Yield sign, count equal sides
$90 = 90 = 90$ → all three sides equal
That makes it equilateral.
2
Sail, count equal sides
$4 \ne 6$, $6 \ne 7$, $4 \ne 7$ → no two sides equal
3
Name the sail
Zero equal sides → scalene
A racing sail is deliberately scalene so wind hits each side differently.

AnswerYield sign: equilateral. Sail: scalene.

Common Pitfalls

heads-up

Eyeballing instead of reading the ticks

Diagrams are rarely drawn perfectly to scale. A triangle that "looks isosceles" might actually be scalene.

Fix: Always classify from tick marks or stated side lengths, not from the visual.

Confusing "isosceles" with "equilateral"

Students often swap which one means "two equal" vs "all equal".

Fix: Equi lateral = equi=equal sides (all). Iso sceles = iso=same (just two).

Saying scalene means "one equal side"

Scalene means NO equal sides at all, every side is different.

Fix: "Scattered scalene", three different sides scattered all over.

Copy Into Your Books

Equilateral

3 equal sides
All angles $60^{\circ}$
3 lines of symmetry

Isosceles

2 equal sides
2 equal base angles
1 line of symmetry

Scalene

0 equal sides
0 equal angles
0 lines of symmetry

Reading Marks

Same ticks → equal sides
Trust ticks, not the eye
Add ticks when you draw

Triangle Inequality

Any two sides > the third
Check all three pairs
Shortest pair vs longest side is the quick test

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Sides & Symmetry

4 problems

Four quick drills to lock in side-classification. Try each yourself first, then reveal.

1 Classify the triangle with sides $5\text{ cm}, 12\text{ cm}, 13\text{ cm}$.

No two sides are equal.Scalene
2 A triangle has all sides $8\text{ cm}$. How many lines of symmetry does it have?

All sides equal → equilateral.3 lines of symmetry
3 An isosceles triangle has one base angle of $50^{\circ}$. What is the other base angle?

Base angles of an isosceles triangle are equal.$50^{\circ}$
4 A triangle has sides $6\text{ cm}, 6\text{ cm}, 6\text{ cm}$. Classify it.

All three sides equal.Equilateral

Complete in your workbook.

Quick Check · 6 questions

A triangle has sides $9\text{ cm}, 9\text{ cm}, 5\text{ cm}$. Classify it.

+10 XP

Which statement is TRUE for every equilateral triangle?

+10 XP

Classify a triangle with sides $4\text{ m}, 6\text{ m}, 7\text{ m}$.

+10 XP

How many lines of symmetry does a scalene triangle have?

+10 XP

A diagram shows a triangle with two ticks on each of two sides and no ticks on the third. What type is it?

+10 XP

Which set of side lengths CAN form a triangle?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Classify each triangle by its sides: (a) sides $6, 6, 6$, (b) sides $3, 4, 5$, (c) sides $7, 7, 10$, (d) sides $8, 11, 11$, (e) sides $5, 5, 5$, (f) sides $9, 12, 15$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. For each triangle below, state the type AND the number of lines of symmetry:
(a) An isosceles triangle with equal sides of $12\text{ cm}$ and a base of $7\text{ cm}$.
(b) An equilateral triangle with each side $4\text{ cm}$.
(c) A scalene triangle with sides $5, 8, 10\text{ cm}$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. An isosceles triangle has two equal sides of length $x\text{ cm}$ and a base of $6\text{ cm}$. The perimeter is $20\text{ cm}$. Find $x$, then state whether the triangle could also be equilateral. Justify your answer.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B Isosceles. Two sides of $9\text{ cm}$, one of $5\text{ cm}$.

2. C Every angle is $60^{\circ}$. Three equal sides force three equal angles.

3. A Scalene. $4, 6, 7$ are all different.

4. D0. No equal sides means no fold line.

5. B Isosceles. Two ticks on two sides → those two sides are equal.

6. C$6, 7, 10$: every pairwise sum ($13, 16, 17$) is greater than the remaining side.

Show Your Working Model Answers

Q6 (3 marks): (a) equilateral, (b) scalene, (c) isosceles, (d) isosceles, (e) equilateral, (f) scalene. [1 mark per 2 correct]

Q7 (3 marks): (a) Isosceles; 1 line of symmetry [1]. (b) Equilateral; 3 lines of symmetry [1]. (c) Scalene; 0 lines of symmetry [1].

Q8 (3 marks): Perimeter $= 2x + 6 = 20$ [1]. So $2x = 14$, $x = 7\text{ cm}$ [1]. The triangle has sides $7, 7, 6$, only two sides equal, so it is isosceles but NOT equilateral (the base $6 \ne 7$) [1].

Stretch Challenge · +25 XP, +10 coins

The Triangle Inequality Puzzle

Can you build a triangle with sides $2\text{ cm}, 3\text{ cm}, 7\text{ cm}$? Try drawing it. Now try $5\text{ cm}, 5\text{ cm}, 5\text{ cm}$, and finally $4\text{ cm}, 7\text{ cm}, 9\text{ cm}$. Which sets work, which don't, and can you spot the rule about when three lengths CAN make a triangle?

Reveal solution

Sides $2, 3, 7$: impossible$2 + 3 = 5 < 7$, the two shorter sides can't reach across. Sides $5, 5, 5$: equilateral. Sides $4, 7, 9$: scalene (works because $4 + 7 = 11 > 9$). The rule is the triangle inequality: the sum of any two sides must be greater than the third.

Quick Review

Equilateral

3 equal sides, all angles $60^{\circ}$.

Isosceles

2 equal sides, 2 equal base angles.

Scalene

0 equal sides, 0 lines of symmetry.

Tick marks

Same number of ticks = equal sides.

Symmetry

Equilateral: 3 Isosceles: 1 Scalene: 0.

Apex / base

The apex is where equal sides meet; base is opposite.

Triangle inequality

Any two sides must add to more than the third.

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