Question

three triangular shapes were measured by different students. analyze the following triangular measurements to determine similarity relationships.

short response questions (6 questions)

question 6
4 points
true or false: if two triangles have the same three angle measures, they must be congruent. explain your reasoning using examples from the given triangles.

Explanation:

Step1: Recall the triangle congruence criteria

Triangle congruence requires that all corresponding sides and angles are equal. Having the same three angle measures (AAA) only guarantees similarity, not congruence.

Step2: Provide an example

For instance, consider triangle \(ABC\) with angles \(50^{\circ}, 60^{\circ}, 70^{\circ}\) and side lengths \(3,4,5\) and triangle \(XYZ\) with angles \(50^{\circ}, 60^{\circ}, 70^{\circ}\) but side lengths \(6,8,10\). They have the same angle measures (similar by AAA) but are not congruent as side lengths differ.

Answer:

False. Two triangles with the same three - angle measures are similar (by the AA (Angle - Angle) similarity criterion, since if two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal as the sum of angles in a triangle is \(180^{\circ}\)), but not necessarily congruent. Congruence requires all corresponding sides to be equal in length as well. For example, an equilateral triangle with side length \(2\) and another equilateral triangle with side length \(3\) have all angles equal to \(60^{\circ}\) but are not congruent.