Question

triangles 1 and 2 are both isosceles. they each have a 30° angle. explain why these triangles do not have to be similar to each other. type your response in the space below.

Explanation:

Step1: Consider the cases for the isosceles triangle with a \(30^{\circ}\) angle

In an isosceles triangle, if the \(30^{\circ}\) angle is the vertex angle, then the base - angles are \(\frac{180 - 30}{2}=75^{\circ}\) (using the angle - sum property of a triangle \(A + B + C=180^{\circ}\) for \(\triangle ABC\) with \(A = 30^{\circ}\) and \(B = C\)).

Step2: Consider another case for the isosceles triangle with a \(30^{\circ}\) angle

If the \(30^{\circ}\) angle is a base - angle, then the vertex angle is \(180-(30 + 30)=120^{\circ}\) (using the angle - sum property of a triangle \(A + B + C = 180^{\circ}\) for \(\triangle ABC\) with \(A=B = 30^{\circ}\)).

Answer:

Triangle 1 could have angles \(30^{\circ},75^{\circ},75^{\circ}\) (if the \(30^{\circ}\) angle is the vertex angle) and triangle 2 could have angles \(30^{\circ},30^{\circ},120^{\circ}\) (if the \(30^{\circ}\) angle is a base - angle). Since the corresponding angles of the two triangles are not equal (one has angles \(30,75,75\) and the other has \(30,30,120\)), by the AA (angle - angle) similarity criterion (which states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), the two isosceles triangles with a \(30^{\circ}\) angle are not similar.