Triangles
ICSE Class 7 · Mathematics · Chapter 15
What this chapter covers
- A triangle and its parts
- Classification of triangles
- Angle sum property
- Exterior angle property
- Construction of simple triangles
Key points
- A triangle has three sides, three angles and three vertices; these six sides and angles are its elements.
- By sides, a triangle is equilateral (all sides equal), isosceles (at least two sides equal) or scalene (no sides equal).
- By angles, a triangle is acute-angled, right-angled or obtuse-angled.
- The angles opposite the equal sides of an isosceles triangle are equal; each angle of an equilateral triangle is 60°.
- The angle sum property: the three angles of any triangle add up to 180°.
- A triangle has at most one right angle or one obtuse angle; the two acute angles of a right-angled triangle add up to 90°.
- An exterior angle and the interior angle next to it form a linear pair and add up to 180°.
- The exterior angle property: an exterior angle equals the sum of its two interior opposite angles.
- A triangle can be constructed when three suitable elements are known: SSS, SAS, ASA or RHS.
- Three lengths form a triangle only if the two shorter ones together are longer than the longest.
Questions answered
How many elements does a triangle have? Name them for △ XYZ.
Six elements: the sides XY, YZ and ZX, and the angles ∠ X, ∠ Y and ∠ Z.
Classify a triangle with sides 8 cm, 8 cm and 5 cm by its sides.
Two sides are equal, so it is an isosceles triangle.
Can a triangle have angles of 100° and 95°? Give a reason.
No. These two angles already add up to 195°, which is more than the 180° allowed for all three angles.
Two angles of a triangle are 35° and 85°. Find the third angle and classify the triangle by its angles.
180° - 35° - 85° = 60°. All three angles are less than 90°, so it is acute-angled.
One acute angle of a right-angled triangle is 28°. Find the other acute angle.
The two acute angles add up to 90°, so the other is 90° - 28° = 62°.
The angles of a triangle are x, 2x and 3x. Find them.
6x = 180°, so x = 30°. The angles are 30°, 60° and 90°.
An exterior angle of a triangle is 132° and its interior opposite angles are equal. Find them.
Each is 132° ÷ 2 = 66°.
In △ ABC, BC is extended to D. ∠ ACD = 105° and ∠ B = 40°. Find ∠ A and ∠ ACB.
∠ A = 105° - 40° = 65°, and ∠ ACB = 180° - 105° = 75°.