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

Test your understanding of Class IX Mathematics (Code 041) Unit IV: Geometry. This quiz covers the topic “Angles Opposite Equal Sides (Prove)”, focusing on the Isosceles property, proofs, and application of theorems. Complete the 10 MCQs below, click Submit to check your score, and download the solution PDF for revision.

Understanding Angles Opposite Equal Sides

In Class 9 Geometry, one of the most fundamental theorems relates to isosceles triangles. The theorem states: “Angles opposite to equal sides of an isosceles triangle are equal.” This property is crucial for solving various geometric proofs and problems involving triangle congruence.

Conversely, the property holds true in reverse: “The sides opposite to equal angles of a triangle are equal.” This helps us identify isosceles triangles simply by looking at their angle measures.

Key Concepts & Proof Logic

  • Isosceles Triangle: A triangle with at least two equal sides.
  • The Theorem: If AB = AC in triangle ABC, then Angle B = Angle C.
  • Proof Strategy: To prove this theorem, we typically draw an angle bisector from vertex A to the base BC (point D). Then, we use the SAS (Side-Angle-Side) congruence rule to prove triangle ABD is congruent to triangle ACD.
  • Corollaries: An equilateral triangle has three equal sides; therefore, it must have three equal angles (each 60 degrees).

Quick Revision Table

Triangle Type Side Property Angle Property
Scalene No sides equal No angles equal
Isosceles Two sides equal Angles opposite equal sides are equal
Equilateral All sides equal All angles equal (60 degrees)

Extra Practice Questions

Try solving these additional problems to strengthen your grasp of the topic:

  1. In triangle PQR, PQ = PR and Angle Q = 45 degrees. Calculate the measure of Angle P.
  2. Prove that the bisector of the vertical angle of an isosceles triangle bisects the base at right angles.
  3. If the base angles of an isosceles triangle are 4x and 4x, and the vertical angle is 2x, find the value of x.
  4. ABC is an isosceles triangle with AB = AC. Draw AP perpendicular to BC. Show that Angle B = Angle C using RHS congruence.
  5. Can a triangle have two obtuse angles? Use the angle sum property to explain why or why not.