Geometry (Classes VI–VIII) • Topic 3 of 6

Triangles & Their Properties

The single most-tested fact in the whole chapter is the angle sum of a triangle = 180°. From it flow the exterior-angle theorem (an exterior angle equals the sum of the two opposite interior angles) and the angle-side relationships in scalene, isosceles and equilateral triangles (in an isosceles triangle the angles opposite the equal sides are equal; an equilateral triangle has three 60° angles). The triangle inequality — the sum of any two sides is greater than the third — is a CTET favourite for 'can these be the sides of a triangle?' items. Pedagogically, the powerful classroom move is the torn-corners activity: a child tears the three corners of any paper triangle and fits them along a straight line to 'discover' that they make 180°. That is van Hiele Level 2 (Informal Deduction) emerging from Level 1 (Analysis). The deep misconception is believing the angle sum changes for a bigger triangle — children think a large triangle 'has more angle'. CTET tests the 180° fact through find-the-missing-angle computation, the exterior-angle theorem directly, classification by sides/angles, and the torn-corners activity as pedagogy.

✅ Solved examples

1. Two angles of a triangle are 48° and 67°. Find the third angle.
The angle sum of a triangle is 180°, so the third angle = 180° − (48° + 67°) = 180° − 115° = 65°.
2. An exterior angle of a triangle is 120° and one of the two opposite interior angles is 45°. Find the other opposite interior angle.
By the exterior-angle theorem, the exterior angle equals the sum of the two opposite interior angles: 120° = 45° + x, so x = 75°.
3. Can a triangle have sides of length 4 cm, 5 cm and 10 cm?
No. By the triangle inequality, the sum of any two sides must exceed the third. Here 4 + 5 = 9, which is less than 10, so such a triangle cannot exist.
4. A teacher has each child tear the three corners off a paper triangle and arrange them along a ruler edge. What is this activity designed to let children discover?
That the three interior angles of a triangle together make a straight angle, i.e. their sum is 180°. It is a concrete, discovery route to the angle-sum property (van Hiele Level 2 reasoning growing out of hands-on analysis).

✏️ Practice — try these, take hints as needed

1. Each angle of an equilateral triangle measures:
Three equal angles summing to 180°.
180 ÷ 3.
60°
2. A right triangle has one angle of 90° and another of 35°. Its third angle is:
Angles add to 180°.
180 − 90 − 35.
55°
3. In an isosceles triangle the two equal sides meet at the apex. The base angles are always:
Angles opposite equal sides.
They match each other.
Equal to each other
4. An exterior angle of a triangle measures 105°. The sum of its two opposite interior angles is:
Exterior-angle theorem.
It equals the exterior angle.
105°

📝 Topic test — 8 questions

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