Question

if two isosceles triangles have non - base angles of 40°, are they similar? justify your answer
choose the correct answer below.
a. no. nothing is known about any of the sides of these isosceles triangles, so none of the triangle similarity theorems can be used
b. yes. since the triangles are isosceles, it means that they must have congruent base angles of 70°. the triangles are similar by aa similarity
c. yes. since the triangles are isosceles, it means that they must have congruent non - base sides. the triangles are similar by sas similarity
d. no. only one angle is known in each triangle, which is not enough to show that the triangles are similar using any of the similarity theorems.

Explanation:

• For an isosceles triangle, the sum of interior angles is \(180^{\circ}\). If a non - base angle is \(40^{\circ}\), then the base angles can be calculated.
• Let the base angles be \(x\). Using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\) for \(\triangle ABC\)), if the non - base angle \(A = 40^{\circ}\), and since the triangle is isosceles (\(B = C\)), we have \(40^{\circ}+x + x=180^{\circ}\).
• Solving \(2x=180^{\circ}- 40^{\circ}=140^{\circ}\), we get \(x = 70^{\circ}\).
• So, both isosceles triangles have angles \(40^{\circ},70^{\circ},70^{\circ}\).
• By the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), the two isosceles triangles are similar.

Answer:

B. Yes. Since the triangles are isosceles, it means that they must have congruent base angles of \(70^{\circ}\). The triangles are similar by AA similarity.