Question

1. determine if each statement must be true, could possibly be true, or definitely can’t be true. explain or show your reasoning.

a. two equilateral triangles are similar.
all equilateral triangles have all angles = 60° so they are always similar. must be true
b. an equilateral triangle and a square are similar.
definitely cant be true

Explanation:

Part a:

Step1: Recall similarity of triangles

Similar triangles have corresponding angles equal and sides in proportion. For equilateral triangles, each angle is \(60^\circ\).

Step2: Check angles of equilateral triangles

Any equilateral triangle has all angles \(60^\circ\). So, for two equilateral triangles, corresponding angles are equal (all \(60^\circ\)), and sides will be in proportion (since all sides of an equilateral triangle are equal, the ratio of sides between two equilateral triangles will be constant). Thus, by AA (Angle - Angle) similarity criterion (since all angles are equal), two equilateral triangles must be similar.

Part b:

Step1: Recall the definition of similar figures

Similar figures must have the same shape, which means the number of sides must be the same (for polygons) and corresponding angles equal, sides in proportion.

Step2: Compare equilateral triangle and square

An equilateral triangle has 3 sides and internal angles of \(60^\circ\) each. A square has 4 sides and internal angles of \(90^\circ\) each. The number of sides is different, and the angle measures are different. So, they can never be similar (since the basic shape is different - one is a 3 - sided polygon, the other is a 4 - sided polygon, and angle measures don't match). So, an equilateral triangle and a square definitely can't be similar.

Answer:

a. Two equilateral triangles are similar: Must be true (because all equilateral triangles have all angles equal to \(60^\circ\), so by AA similarity, they are similar).
b. An equilateral triangle and a square are similar: Definitely can't be true (because they have a different number of sides and different angle measures, so they can't have the same shape required for similarity).