A Tale of Three Intersecting Lines · Lesson 9 of 10
Types of Triangles
“Classify triangles by their sides and angles, then explore how the two descriptions can be combined.”
• Distinguish equilateral, isosceles, and scalene triangles by their sides. • Distinguish acute, right, and obtuse triangles by their angles. • Explain why one acute angle is not enough to identify an acute triangle. • Give both a side-based and an angle-based description when enough information is available. • Explore triangle types by construction without assuming a picture proves its measurements.
Types of Triangles
During our constructions, we have met triangles with equal sides, unequal sides, a right angle, and an obtuse angle. We can organise them using two different questions: which sides are equal, and how large are the angles? The answers describe different features, so one triangle can have a description from each scheme.
For example, “isosceles” tells us something about side lengths. “Right-angled” tells us about an angle. Neither description replaces the other. A triangle can be both isosceles and right-angled if two of its sides are equal and one of its angles is 90°.
Classification by Sides
Compare all three lengths. If all are equal, the triangle is equilateral. If two are equal, it is isosceles. If all three differ, it is scalene. In the side-classification table below we list the all-three-equal case separately, so the isosceles row describes exactly two equal sides.
| Type | Side-length feature | Valid example |
|---|---|---|
| Equilateral | All three sides equal | 4 cm, 4 cm, 4 cm |
| Isosceles | Two equal sides; all-three-equal cases listed separately here | 5 cm, 5 cm, 6 cm |
| Scalene | All three side lengths different | 3 cm, 4 cm, 5 cm |
A triangle whose three side lengths are all different.
A triangle’s orientation does not determine its type. An isosceles triangle need not have its equal sides pointing upward, and a scalene triangle may appear quite balanced. Use given or accurately measured lengths. Also check that the lengths can form a triangle before classifying it: a list such as 2, 2, 5 does not describe a triangle at all.
Problem
Classify side sets 4, 4, 4; 5, 5, 6; and 3, 4, 5, all in centimetres.
- 1.Each set passes the triangle inequality: 4 < 4 + 4, 6 < 5 + 5, and 5 < 3 + 4.
- 2.For 4, 4, 4, all lengths are equal, so the triangle is equilateral.
- 3.For 5, 5, 6, two are equal, so it is isosceles.
- 4.For 3, 4, 5, all differ, so it is scalene.
Classification by Angles
An acute angle is smaller than 90°, a right angle equals 90°, and an obtuse angle is between 90° and 180°. A triangle is acute-angled only when all three angles are acute. A triangle with one right angle is right-angled, and one with an obtuse angle is obtuse-angled.
| Type | Interior-angle feature | Valid example |
|---|---|---|
| Acute-angled | All three angles are less than 90° | 50°, 60°, 70° |
| Right-angled | One angle equals 90° | 30°, 60°, 90° |
| Obtuse-angled | One angle exceeds 90° | 20°, 40°, 120° |
Why not define an acute triangle as one with a single acute angle? Because right and obtuse triangles also contain acute angles. For example, 20°, 40°, 120° contains two acute angles, yet it is obtuse-angled. We need to inspect all three angles before saying a triangle is acute.
A triangle cannot have two right angles: those would already total 180°, leaving no positive third angle. Two obtuse angles would total more than 180°, which is also impossible. In fact every triangle has at least two acute angles; otherwise two angles would each be at least 90° and their sum would leave no room for the third.
Problem
Classify triangles with angles 36°, 72°, 72°; 90°, 30°, 60°; and 120°, 30°, 30°.
- 1.First check each total: all three sets add to 180°, and every angle is positive.
- 2.36°, 72°, 72° are all smaller than 90°, so the triangle is acute-angled.
- 3.90°, 30°, 60° has a right angle, so it is right-angled.
- 4.120°, 30°, 30° has an obtuse angle, so it is obtuse-angled, even though the other two angles are acute.
Combining the Two Descriptions
To construct an isosceles right triangle, draw two equal sides meeting at 90°, then join their free endpoints. To construct an isosceles obtuse triangle, draw two equal sides meeting at an obtuse angle such as 120°, then join the endpoints. Both constructions have two equal sides, but their angle classifications differ.
Problem
Construct AB = AC = 4 cm with ∠A = 90°. Describe its types.
- 1.Draw AB = 4 cm. Construct a perpendicular ray at A.
- 2.Mark C 4 cm from A on that ray and join BC.
- 3.AB and AC are equal, so the triangle is isosceles.
- 4.∠A is 90°, so it is also right-angled. The remaining angles together total 90°.
Explore the equilateral case by constructing it carefully. Its three equal lengths give a symmetric shape: folding along the line from a vertex to the midpoint of the opposite side matches the two sides and swaps the other two vertices. Repeating this at another vertex shows that the three corner angles match. Since equal angles share a total of 180°, each is 60°. Thus an equilateral triangle is acute, never right or obtuse.
This chapter uses constructions to explore how side types and angle types combine. A later study will develop the general relationships between equal sides and equal angles. Do not assume a complete side-angle rule from one rough picture.
How Much Information Is Enough?
Some measurements allow many different triangles. If a right triangle has one 90° angle and the opposite side AC is 5 cm, the other angles are not yet fixed. They only need to be positive and add to 90°. For each choice, use AC as a base and construct the two endpoint angles; the third angle will be 90°.
Problem
Construct right triangles with ∠B = 90° and AC = 5 cm. Is there just one?
- 1.Draw AC = 5 cm. Choose endpoint angles ∠A = 30° and ∠C = 60°; their rays meet at B.
- 2.The angle sum makes ∠B = 180° − 30° − 60° = 90°.
- 3.Repeat with endpoint angles 45° and 45°, or 20° and 70°. Each gives a different triangle with the same specified right angle and opposite side.
- 4.There are infinitely many choices of positive endpoint angles adding to 90°, so the original information does not specify just one triangle.
An acute angle does not by itself make an acute triangle. Equal-looking sides do not establish equal lengths, and labels such as “isosceles” or “right” are not substitutes for checking that the given measurements are possible.
Quiz
Which side set describes a scalene triangle?
Which statement defines an acute-angled triangle?
A triangle has angles 20°, 40°, and 120°. What is its angle type?
Can a triangle be both isosceles and right-angled?
Why can a triangle not have two obtuse angles?
A right triangle has opposite side AC = 5 cm but no other measurement is fixed. How many different triangles are possible?
Practice Problems
- Classify the valid side sets 6, 6, 6; 4, 4, 6; and 4, 6, 8. Check existence before naming each type.
- Classify angle sets 50°, 60°, 70°; 90°, 45°, 45°; and 100°, 35°, 45°. Explain why each is valid.
- Construct an isosceles triangle with two 4 cm sides and included angle 90°, then another with included angle 120°. Give both classifications.
- Construct an equilateral triangle and explore by folding whether its corner angles match. Use the angle sum to explain why it cannot be right or obtuse.
- Construct two different triangles with ∠B = 90° and AC = 5 cm. State the chosen endpoint angles in each.
- Explain why defining an acute triangle as “one with an acute angle” would also include right and obtuse triangles.
- Explain why two right angles or two obtuse angles leave no room for a third positive angle.
Key Takeaways
• Side lengths distinguish equilateral, isosceles, and scalene triangles. • Angle measures distinguish acute, right, and obtuse triangles. • An acute triangle must have all three angles acute. • One triangle can have a side classification and an angle classification. • Some incomplete sets of measurements allow many triangles rather than one.