041i - Classifying Triangles

Lesson 41 — Classifying Triangles by Side and Angle Properties

Strand: Space | Descriptor: AC9M7SP02 | Duration: 45 minutes

Learning Intentions

• To classify triangles according to their side properties and their angle properties.
• To understand that these two classifications are independent of each other.

Success Criteria

I can:

1. Classify a triangle as scalene, isosceles or equilateral by its sides.
2. Classify a triangle as acute, right-angled or obtuse by its angles.
3. Give both classifications for the same triangle.
4. Explain which combinations are possible and which are impossible.

Warmup

(6 minutes — sorting, pairs)

Give each pair a set of paper triangles of varied shapes.

1. Sort them into groups. You choose the rule.
2. Write down the rule you used.
3. Compare with another pair — did you use the same rule?
4. How many different sensible rules did the class find?

Discussion: Collect the rules on the board. Expect “by size”, “by how pointy”, “by equal sides”. Steer towards two mathematically useful families: sides and angles.

Activities

Activity 1 — Explicit Instruction: Classifying by Sides (10 min)

The three types:

Name	Sides	Angles (a consequence)	Lines of symmetry
Scalene	all three different	all three different
Isosceles	exactly two equal	the two base angles equal
Equilateral	all three equal	all three equal ( each)

Notation to teach and enforce. Equal sides are marked with matching dashes (one dash, two dashes); equal angles with matching arcs. Insist students mark diagrams before reasoning about them.

A definitional point worth raising. Some texts define isosceles as “at least two equal sides”, which makes every equilateral triangle isosceles as well. Others say “exactly two”. State which convention the class will use — “at least two” is the more useful one, because it makes equilateral a special case of isosceles, and that nesting idea returns in Lesson 42 with quadrilaterals.

I do: Classify three drawn triangles by sides, marking equal sides as you go.

We do: Given side lengths, classify each.

1. cm, cm, cm
2. cm, cm, cm
3. cm, cm, cm
4. cm, cm, cm

(Answers: scalene; isosceles; equilateral (and isosceles); isosceles.)

Activity 2 — Explicit Instruction: Classifying by Angles (10 min)

The three types:

Name	Angles
Acute-angled	all three angles less than
Right-angled	exactly one angle of
Obtuse-angled	exactly one angle greater than

Key point to emphasise: it is the largest angle that decides the classification. Every triangle has at least two acute angles, so counting acute angles tells you nothing — look at the biggest one.

We do: Classify by angles.

1. , ,
2. , ,
3. , ,
4. , ,

(Answers: acute; right-angled; obtuse; acute.)

Forward link: each set of three angles above totals . Lesson 46 investigates why that is always true — for now, treat it as something you have noticed.

Activity 3 — Inquiry: the Double Classification Grid (14 min)

Pairs, ruler and protractor, grid paper.

Every triangle has a side name and an angle name. Complete this grid: for each cell, either draw a triangle that fits, or explain why none exists.

	Acute	Right-angled	Obtuse
Scalene
Isosceles
Equilateral

Socratic scaffolding:

Prompt	Purpose
Start with the easy cells. Can you draw a scalene right-angled triangle?	Yes — e.g. ––, with angles roughly , , .
Try isosceles right-angled.	Yes — angles , , .
Now equilateral. What are its angles?	All equal, and they total , so each.
Can an equilateral triangle be right-angled?	No — that would need a angle, but all its angles are .
Can it be obtuse?	No, for the same reason.
So which equilateral cells are possible?	Only acute.
Can a triangle have two right angles?	No — the two would already total , leaving nothing for the third.
Looking back — what does the grid show?	Side and angle classifications are almost independent, but the equilateral row is forced.

Completed grid:

	Acute	Right-angled	Obtuse
Scalene	✓ e.g.	✓ e.g.	✓ e.g.
Isosceles	✓ e.g.	✓	✓ e.g.
Equilateral	✓	✗ impossible	✗ impossible

Discussion: Seven of the nine cells are possible. What single fact rules out the other two? (All angles of an equilateral triangle are equal, and equal angles totalling must each be — neither nor obtuse.)

Checks for Understanding

(5 minutes — exit ticket)

1. Classify by sides: (a) cm (b) cm (c) cm.
2. Classify by angles: (a) (b) (c) .
3. Give both names for a triangle with angles .
4. Reasoning. Explain why a triangle cannot have two obtuse angles.
5. Sketch an obtuse isosceles triangle and mark its equal sides.

Answers: 1. (a) isosceles (b) scalene (c) equilateral; 2. (a) right-angled (b) obtuse (c) acute; 3. Right-angled isosceles; 4. Two obtuse angles would each exceed , totalling more than — leaving nothing for the third angle; 5. Any triangle with one angle over and two equal sides, e.g. angles .

Common Misconceptions

Misconception	How to pre-empt it
Believing a triangle can only have one classification.	Insist on giving both names — side and angle — for every triangle from the start.
Judging “acute” by counting acute angles.	Every triangle has at least two. Only the largest angle decides.
Thinking an equilateral triangle can be right-angled.	The grid task rules this out by reasoning, not assertion.
Assuming a triangle looks isosceles because it is drawn that way.	Require markings or measurements. Appearance is not evidence.
Believing scalene means “no special properties” rather than “no equal sides”.	Give the definition precisely and test with — scalene, yet right-angled.
Confusing “obtuse triangle” with “a triangle containing an obtuse angle drawn oddly”.	Measure with a protractor rather than eyeballing.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A triangle has two angles of . What is the third angle, and what are both classifications?

Answer

Third angle . It is isosceles (two equal angles means two equal sides) and acute-angled.

E2 (AMC Junior style). An isosceles triangle has one angle of . Find the other two angles.

Answer

The angle must be the unique one, since two of them would exceed . The remaining splits equally: and .

E3 (Challenge). An isosceles triangle has one angle of . Find all possible sets of angles.

Answer

Case 1 — the is the apex: the other two are equal, each. Angles .

Case 2 — the is one of the equal pair: the third is . Angles .

Two possibilities. A good reminder to check whether a problem has more than one answer.

E4 (Challenge). How many triangles with whole-number side lengths and a perimeter of cm exist? (Remember: any two sides must total more than the third.)

Answer

Listing with : , , . Also fails since is not greater than ; fails similarly; fails. So 3 triangles.

E5 (Reasoning challenge). Explain why the triangle inequality — any two sides must total more than the third — must hold.

Answer

The shortest path between two points is the straight line joining them. Travelling from one vertex to another via the third vertex is a detour, so it must be longer than going direct. Hence the two shorter sides together exceed the longest.

Homework

1. Classify by sides: (a) cm (b) cm (c) cm (d) cm.
2. Classify by angles: (a) (b) (c) (d) .
3. For each set of angles, give both the side name and the angle name: (a) (b) (c) (d) .
4. Sketch and label: (a) an acute scalene triangle (b) a right-angled isosceles triangle (c) an obtuse scalene triangle.
5. An isosceles triangle has one angle of . Find all possible sets of angles.
6. State how many lines of symmetry each has: (a) scalene (b) isosceles (c) equilateral.
7. Reasoning. Explain why an equilateral triangle must be acute-angled.
8. Reasoning. Can a right-angled triangle be equilateral? Justify your answer.
9. Challenge. How many triangles with whole-number sides have a perimeter of cm?

Answers: Q1 — (a) isosceles (b) scalene (c) equilateral (d) scalene. Q2 — (a) right-angled (b) obtuse (c) acute (d) acute. Q3 — (a) right-angled isosceles (b) obtuse isosceles (c) acute scalene (d) acute equilateral. Q5 — either or . Q6 — (a) (b) (c) . Q7 — all its angles are equal and total , so each is , which is acute. Q8 — no; it would need a angle, but all angles must be . Q9 — , , — three triangles.