Classifying Triangles
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Theory
Every triangle can be classified in two ways: by its sides (equilateral, isosceles or scalene) and by its angles (acute-angled, right-angled or obtuse-angled). Give both names together, such as a right-angled isosceles triangle. This Year 7 maths page shows how to read the marks on a diagram and use them to find missing angles and sides.
Naming. A triangle is named with the symbol \(\triangle\) and its three vertices (corners) in any order: \(\triangle ABC\), \(\triangle BCA\) and \(\triangle CAB\) are the same triangle. An angle is named with three letters and the vertex in the middle: \(\angle ABC\) is the angle at \(B\).
By sides. Tick marks show sides that are the same length.
Equilateral: all three sides are equal. All three angles are equal too, so each is \(180^\circ \div 3 = 60^\circ\).
Isosceles: two sides are equal. The two base angles, opposite the equal sides, are equal.
Scalene: no two sides are equal, and no two angles are equal.
By angles. Look at the largest angle.
Acute-angled: all three angles are less than \(90^\circ\).
Right-angled: one angle is exactly \(90^\circ\) (shown by a small square).
Obtuse-angled: one angle is between \(90^\circ\) and \(180^\circ\).
The angles of a triangle add to \(180^\circ\). So a triangle can have at most one right angle or one obtuse angle, and every triangle has at least two acute angles.
Equal angles mean equal sides. The sides opposite two equal angles are equal, so a triangle with two equal angles is isosceles even with no tick marks.
Give one name for the sides and one for the angles:
| Name | What it means |
|---|---|
| equilateral | \(3\) equal sides, each angle \(60^\circ\) |
| isosceles | \(2\) equal sides, \(2\) equal base angles |
| scalene | no equal sides, no equal angles |
| acute-angled | largest angle less than \(90^\circ\) |
| right-angled | one angle exactly \(90^\circ\) |
| obtuse-angled | one angle between \(90^\circ\) and \(180^\circ\) |
How to classify a triangle
- Sides: count the sides with matching tick marks (\(3\), \(2\) or none).
- Angles: find the largest angle. Use the angle sum (\(180^\circ\)) if one angle is missing.
- Name it both ways, for example an obtuse-angled isosceles triangle.
How to use an isosceles triangle
- Find the equal sides (the ticks). The base angles are opposite them.
- Write the base angles as equal, with the reason.
- Use the angle sum of \(180^\circ\) to find the last angle.
Name: \(\triangle PQR\) (or \(\triangle QRP\), \(\triangle RPQ\), \(\ldots\)). Sides: one, two and three ticks, so no two sides match. Angles: the largest angle is \(105^\circ\).
| \(90^\circ \lt 105^\circ\) | \(\lt\) | \(180^\circ\) |
\(\triangle PQR\) is a scalene, obtuse-angled triangle.
The ticks show \(AB = AC\), so the base angles at \(B\) and \(C\) are equal: \(a = 25\) (base angles of an isosceles triangle are equal).
| \(b + 25 + 25\) | \(=\) | \(180\) |
| \(b\) | \(=\) | \(180 - 50\) |
| \(\) | \(=\) | \(130\) |
\(a = 25\) and \(b = 130\) (angle sum of a triangle is \(180^\circ\)). The triangle is obtuse-angled and isosceles.
The two angles that are not the right angle add to \(90^\circ\) (angle sum of a triangle is \(180^\circ\)).
| \(\angle PRQ + \angle QPR\) | \(=\) | \(90^\circ\) |
| \(\angle PRQ + 20^\circ\) | \(=\) | \(90^\circ\) |
| \(\angle PRQ\) | \(=\) | \(70^\circ\) |
\(\angle PRQ = 70^\circ\). The triangle is right-angled and scalene.
Let \(\angle ABC = x^\circ\). Then \(\angle ACB = x^\circ\) too (base angles of an isosceles triangle are equal).
| \(x + x + 36\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(144\) |
| \(x\) | \(=\) | \(72\) |
\(\angle ABC = 72^\circ\). Check: \(72 + 72 + 36 = 180\).
(i) an acute-angled isosceles triangle
(ii) an obtuse-angled equilateral triangle
(iii) a triangle with two obtuse angles
| \(75 + 75 + 30\) | \(=\) | \(180\) |
| \(95 + 95\) | \(=\) | \(190\) |
(i) Yes, for example \(75^\circ\), \(75^\circ\) and \(30^\circ\).
(ii) No. Every angle of an equilateral triangle is \(60^\circ\), which is acute.
(iii) No. Two obtuse angles already add to more than \(180^\circ\), so the sides never meet.
(i) \(\angle BAC\)
(ii) \(\angle CAD\)
(iii) \(\angle BAD\)
(i) Each angle of an equilateral triangle is \(60^\circ\).
(ii) \(CA = CD\), so \(\angle CAD = \angle CDA\) (base angles of an isosceles triangle are equal).
| \(2 \times \angle CAD + 90\) | \(=\) | \(180\) |
| \(\angle CAD\) | \(=\) | \(45^\circ\) |
| \(\angle BAD\) | \(=\) | \(60^\circ + 45^\circ\) |
| \(\) | \(=\) | \(105^\circ\) |
(i) \(60^\circ\)
(ii) \(45^\circ\)
(iii) \(105^\circ\)
(ii) In \(\triangle ABC\), \(\angle A = 30^\circ\) and \(\angle B = 75^\circ\). Classify \(\triangle ABC\) by its sides and its angles.
| \(22 - 8\) | \(=\) | \(14\) |
| \(14 \div 2\) | \(=\) | \(7\) |
| \(180 - 30 - 75\) | \(=\) | \(75\) |
(i) \(7\) cm
(ii) \(\angle C = 75^\circ = \angle B\), so the sides opposite them are equal (\(AC = AB\)). The largest angle is \(75^\circ\), so \(\triangle ABC\) is an acute-angled isosceles triangle.
Common pitfalls
Frequently asked questions
What are the three types of triangle by sides?
Equilateral (three equal sides), isosceles (two equal sides) and scalene (no equal sides).
What are the three types of triangle by angles?
Acute-angled (all angles less than 90°), right-angled (one angle of 90°) and obtuse-angled (one angle more than 90°).
Can a triangle be right-angled and isosceles?
Yes. The two equal angles are then \(45^\circ\) each, because \(90 + 45 + 45 = 180\).
Can a triangle be obtuse-angled and isosceles?
Yes, for example \(25^\circ\), \(25^\circ\) and \(130^\circ\).
What is a base angle?
In an isosceles triangle, the two angles opposite the equal sides are the base angles. They are equal.