Math Problem Statement

The diagram shows a triangle with two equal sides. In addition, the sum of the two equal angles in this triangle is less than 120°. Based on this information, which of the following must be true?

Solution

The question discusses a triangle with two equal sides (an isosceles triangle) where the sum of the two equal angles is less than 120°. Let's analyze the options:

Key Observations:

  1. Two equal sides: This automatically makes the triangle isosceles.
  2. Sum of the two equal angles < 120°: Since the sum of all angles in a triangle is 180°, the third angle must be greater than 60° (because 180°sum of equal angles>180°120°=60°180° - \text{sum of equal angles} > 180° - 120° = 60°).

Analysis of Options:

  • (A) Incorrect. The triangle is not scalene, as it has two equal sides and angles.
  • (B) Incorrect. The statement suggests that both equal angles sum up to exactly 120° or more. The problem states that the sum is less than 120°.
  • (C) Incorrect. While the triangle has two equal sides, this does not imply it is a right triangle. Nothing in the information given guarantees a 90° angle.
  • (D) Correct. The triangle is isosceles, but since the sum of the two equal angles is less than 120°, it cannot be equilateral.

Final Answer:

D. The triangle is isosceles but could also be equilateral depending on the angles.

Do you want further clarification or a breakdown of the reasoning? Here are 5 follow-up questions:

  1. How does the property of equal sides in an isosceles triangle influence its angles?
  2. Can an isosceles triangle have its equal angles summing to more than 120°?
  3. Why is it impossible for the triangle to be scalene in this situation?
  4. How do we use the angle sum property of a triangle in this problem?
  5. Can a right triangle also be isosceles?

Tip: Always remember the sum of angles in a triangle is 180180^\circ; this principle helps verify constraints on possible angle values.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Isosceles Triangles
Angle Sum Property

Formulas

Sum of angles in a triangle = 180°

Theorems

Isosceles triangle theorem

Suitable Grade Level

Grades 7-9