Question

which best explains whether or not all isosceles triangles are similar?
all isosceles triangles are similar. two angles within each triangle are always congruent.
all isosceles triangles are similar. the triangle sum theorem states that the sum of the angles in a triangle is 180°. therefore, the third angle can always be determined.
all isosceles triangles are not similar. the pair of congruent angles within one triangle is not necessarily congruent to the pair of congruent angles within the other triangle.
all isosceles triangles are not similar. given only the vertex angle of an isosceles triangle, there is not enough information to determine the measures of the base angles. therefore, it is not possible to determine if the base angles of one isosceles triangle are congruent to the base angles of another.

Explanation:

To determine if all isosceles triangles are similar, we use the definition of similar triangles (AA similarity: if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar). An isosceles triangle has two congruent angles (the base angles). However, consider two isosceles triangles: one with a vertex angle of \(30^\circ\) (so base angles are \(\frac{180 - 30}{2}=75^\circ\)) and another with a vertex angle of \(60^\circ\) (so base angles are \(\frac{180 - 60}{2}=60^\circ\)). The angle pairs (\(30^\circ,75^\circ,75^\circ\) vs \(60^\circ,60^\circ,60^\circ\)) are not congruent. So the pair of congruent angles in one isosceles triangle isn't necessarily congruent to the pair in another.

• First option: Wrong, as shown by the example, not all isosceles triangles have corresponding angles congruent.
• Second option: Wrong, the triangle sum theorem alone doesn't ensure angle congruence between different isosceles triangles.
• Third option: Correct, because the congruent angle pairs (base angles) can differ in measure between different isosceles triangles, so AA similarity isn't guaranteed.
• Fourth option: Wrong, we can determine base angles from the vertex angle (\(base\ angle=\frac{180 - vertex\ angle}{2}\)), but the issue is the angles may not match between triangles, not the inability to calculate them.

Answer:

C. All isosceles triangles are not similar. The pair of congruent angles within one triangle is not necessarily congruent to the pair of congruent angles within the other triangle.