Sides Opposite Equal Angles (Motivate) MCQs Quiz | Class 9

This quiz targets Class IX Mathematics (Code 041), Unit IV: Geometry, specifically addressing the converse isosceles property where sides opposite to equal angles are equal. Please complete the 10 multiple-choice questions below, click submit to view your detailed results, and download the solution PDF for your records.

Overview: Sides Opposite Equal Angles

In Class 9 Geometry, we explore the properties of triangles. One fundamental theorem is the Converse of the Isosceles Triangle Property. This theorem motivates the understanding that geometric equality works both ways: just as equal sides imply equal angles, equal angles imply equal sides.

The Theorem Statement

“The sides opposite to equal angles of a triangle are equal.”

This means if you have a triangle ABC where angle B is equal to angle C, then the side opposite angle B (which is AC) is equal to the side opposite angle C (which is AB). Thus, AB equals AC.

Key Concepts & Proof Motivation

  • Motivation: To prove this, we typically construct an angle bisector from vertex A to side BC and use triangle congruence criteria (like AAS or ASA).
  • Isosceles Triangle: A triangle with two equal angles is always an isosceles triangle.
  • Equilateral Triangle: If all three angles are equal (60 degrees each), then all three sides are equal, making it equilateral.

Comparison Table

Given Condition Conclusion Property Name
AB = AC (Sides are equal) Angle C = Angle B Isosceles Triangle Property
Angle B = Angle C (Angles are equal) AC = AB Converse Property

Extra Practice Questions

  1. In triangle PQR, if angle P = 45 degrees and angle Q = 45 degrees, which two sides are equal? (Answer: PR and QR)
  2. Can a triangle have two equal angles of 90 degrees? (Answer: No, sum would be 180 without the third angle)
  3. If one angle of a right-angled triangle is 45 degrees, is the triangle isosceles? (Answer: Yes)
  4. In an equilateral triangle, are sides opposite equal angles equal? (Answer: Yes, all sides are equal)
  5. If angle A = 70 degrees and angle B = 70 degrees in triangle ABC, find AC if BC = 10cm. (Answer: AC = 10cm)