Properties of triangles

1Summary 重點摘要

Any triangle's three interior angles sum to 180°; isosceles base angles are equal, equilateral angles are 60°, and an exterior angle equals the two opposite interior angles.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a triangle?A triangle is a polygon formed by three sides, with three sides, three vertices and three angles — the polygon with the fewest sides. Where two neighbouring sides meet there is a vertex, and each vertex forms an interior angle. Triangles are everywhere in daily life: set squares, roof supports and warning road signs. Because the three sides brace each other into a rigid structure, triangles reinforce building frames, and later every polygon can be cut into triangles for study.
2. 2. Angle sumThe three interior angles of a triangle always add up to 180° — the most important property of all. For an equilateral triangle the three angles are equal, so each is 180° ÷ 3 = 60°; check: 60° + 60° + 60° = 180°. Whatever the shape — long and thin or flat and wide — as long as it is a triangle, the angle sum never changes. This rule is the key tool for finding angles in every type below.
3. 3. Equilateral triangleAn equilateral triangle has all three sides equal, say 5 cm each. When the three sides are equal, the three angles must also be equal, each 180° ÷ 3 = 60° — so no measuring is needed at all. Conversely, if all three angles of a triangle are 60°, it must be equilateral. Equal sides giving equal angles is the basic rule for telling triangle types apart and for checking your drawings.
4. 4. Isosceles triangleAn isosceles triangle has two equal sides, called the legs — say 5 cm each — while the third side is the base, say 6 cm. Its key property is that the two base angles are equal: sides that are equal face equal angles. In practice it is combined with the angle sum: the apex angle gives the two base angles, and the base angles give the apex, as the worked example below demonstrates.
5. 5. Right-angled triangleA right-angled triangle has one angle of exactly 90°, and the longest side facing the right angle is the hypotenuse. Since the angle sum is 180°, the two acute angles must add up to 180° − 90° = 90°. For example, acute angles of 30° and 60°: 30° + 60° = 90°, and with the right angle the total is exactly 180°. This relation is very practical: know one acute angle, subtract it from 90°, and the other follows — no need for 180°.
6. 6. Scalene triangleA scalene triangle has three sides of different lengths, such as 3 cm, 4 cm and 5 cm; when no two sides are equal, no two angles are equal either. To identify one, measure the sides — if no two match, it is scalene. As a bonus, the 3-4-5 lengths hide a special fact: 3² + 4² = 9 + 16 = 25 = 5², so it is actually right-angled. This rule is formally learned in senior maths; for now, remember the example.
7. 7. Find unknown angleTo find an unknown angle, start from the angle-sum equation. If the angles are 50°, 60° and an unknown: 50° + 60° + unknown = 180°; adding the known pair gives 110°, and 180° − 110° = 70°, so the third angle is 70°. Check: 50° + 60° + 70° = 180°. For an isosceles triangle add the equal-base-angles rule: with apex 40°, each base angle = (180° − 40°) ÷ 2 = 70°, matching the worked example.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Triangle: 3 sides, 3 angles
• Angle sum = 180°
• Equilateral: all sides equal, all angles equal (60° each)
• Isosceles: two sides equal, two base angles equal
• Right-angled: one angle = 90°