Can the angles of a triangle be in the ratio 1 : 1 : 4?

Class 7The Triangle and its PropertiesTriangles

ANo — the three parts do not add up to 180°.
BYes — and the triangle is right-angled.
CYes — the angles are 30°, 30° and 120°, an obtuse isosceles triangle.
DNo — a triangle can never have two equal angles.

Step-by-Step Solution

x + x + 4 x = 180

→

6 x = 180

→

x = 30

. The angles are 30°, 30° and 120°, which sum to 180°, so the triangle exists; two equal angles make it isosceles and the 120° angle makes it obtuse. Trap: assuming a ratio with one large part must break the 180° rule, or that two equal angles are not allowed.

Answer: C) Yes — the angles are 30°, 30° and 120°, an obtuse isosceles triangle.

Key idea: Exterior angle property

Extend one side of a triangle past a vertex. The angle formed outside the triangle is the **exterior angle** at that vertex.

**Exterior angle = sum of the two opposite interior angles.**

So if the two non-adjacent interior angles are 50° and 60°, the exterior angle = 50° + 60° = 110°.

Why? The exterior angle and the adjacent interior angle form a linear pair (sum 180°), and the three interior angles also sum to 180°. Subtract: exterior = sum of the other two interior angles.

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