What is the angle sum property of a triangle?

The angle sum property states that the sum of all three interior angles of a triangle = 180°. Proof uses parallel lines: draw a line through the apex parallel to the base; the base angles equal the alternate interior angles at the apex; since the three angles at the apex form a straight line (180°), the triangle angles also sum to 180°.

What are corresponding angles and alternate angles?

When a transversal cuts two parallel lines: Corresponding angles are in the same position at each intersection and are equal (e.g., both above-left). Alternate interior angles are between the two lines on opposite sides of the transversal and are equal. Co-interior (same-side interior) angles are supplementary (sum = 180°).

What is the exterior angle theorem for triangles?

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent (remote) interior angles. Example: If a triangle has angles 60°, 70°, 50°, the exterior angle at the vertex with 50° = 60° + 70° = 130°. This theorem is also a corollary of the angle sum property.

Angle Sum Property and Parallel Lines

Lines and Angles • Class 9 Mathematics • NCERT • CBSE

The angle sum property of a triangle states that the sum of three interior angles of any triangle is 180°. For parallel lines cut by a transversal, alternate interior angles are equal, corresponding angles are equal, and co-interior angles are supplementary (sum = 180°).

Key Formulas

Angle sum property: ∠A + ∠B + ∠C = 180° (in any triangle)
Exterior angle = sum of two non-adjacent interior angles
Vertically opposite angles are equal
Linear pair: two adjacent angles sum to 180°
Co-interior angles (same side of transversal): sum = 180°

Frequently Asked Questions

What is the angle sum property of a triangle?: The angle sum property states that the sum of all three interior angles of a triangle = 180°. Proof uses parallel lines: draw a line through the apex parallel to the base; the base angles equal the alternate interior angles at the apex; since the three angles at the apex form a straight line (180°), the triangle angles also sum to 180°.
What are corresponding angles and alternate angles?: When a transversal cuts two parallel lines: Corresponding angles are in the same position at each intersection and are equal (e.g., both above-left). Alternate interior angles are between the two lines on opposite sides of the transversal and are equal. Co-interior (same-side interior) angles are supplementary (sum = 180°).
What is the exterior angle theorem for triangles?: The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent (remote) interior angles. Example: If a triangle has angles 60°, 70°, 50°, the exterior angle at the vertex with 50° = 60° + 70° = 130°. This theorem is also a corollary of the angle sum property.

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