Math Problem Statement

whats the right answer

Which statement explains how to determine whether triangle ABC is similar to triangle XYZ?

Solution

The correct answer in the image is "The triangles are not similar because only two pairs of angles are congruent."

Explanation:

To determine if two triangles are similar, we use the AA (Angle-Angle) Similarity Postulate, which states that two triangles are similar if at least two angles in one triangle are congruent to two angles in the other triangle.

In this problem:

  1. Triangle ABC\triangle ABC has angles 36,84,36^\circ, 84^\circ, and 6060^\circ (since the sum of a triangle's angles is 180180^\circ).
  2. Triangle XYZ\triangle XYZ has angles 50,84,50^\circ, 84^\circ, and 4646^\circ.

Although BY\angle B \cong \angle Y and CX\angle C \cong \angle X, the third angle does not match (604660^\circ \neq 46^\circ). Therefore, the triangles are not similar, as not all three angles are proportionally congruent.

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Five Related Questions:

  1. What is the Angle-Angle (AA) postulate in triangle similarity?
  2. Can two triangles be similar if only one pair of angles is congruent?
  3. What are the differences between congruence and similarity in triangles?
  4. How do you calculate the missing angle in a triangle given the other two angles?
  5. Why is it important to ensure proportionality in sides for similarity beyond angles?

Tip:

Always sum the given angles to verify the third angle when dealing with triangle problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity
Angle Properties

Formulas

Sum of angles in a triangle = 180°

Theorems

AA (Angle-Angle) Similarity Postulate

Suitable Grade Level

Grades 7-9