If one angle of the triangle measures 30°, what is the measure of the largest angle(s) of the triangle (in degrees). What is the measure of the smallest angle(s) of the triangle (in degrees)?

Question

A triangle is both isosceles and acute. If one angle of the triangle measures 30°, what is the measure of the largest angle(s) of the triangle (in degrees).

What is the measure of the smallest angle(s) of the triangle (in degrees)?

Answer 1

A triangle is both isosceles and acute. If one angle of the triangle measures 30°, what is the measure of the largest angle(s) of the triangle (in degrees).

What is the measure of the smallest angle(s) of the triangle (in degrees)?

Dear Alyssa,

A key word in this problem is "acute." If we didn't have that clue, we could not solve this problem, because there are two isosceles triangles that have an angle of 30°. However, there is only one ACUTE isosceles triangle with an angle of 30°. Since the triangle in question is ACUTE, no angle is greater than 90°.

Let's just say that 30° is the smallest angle and that the other two angles are the ones that are the same measure. We subtract 30° from 180° (the measure of the three angles of EVERY triangle) to get 150°. We need to split that in half. That tells us that each of the equal angles is 75°.

Had we thought that there were two angles of 30°, then the remaining angle would be 180° minus two times 30°; that is, the remaining angle would be 120°. However, 120° is greater than 90°, so this would NOT be an ACUTE ISOSCELES triangle.

Answer 2

In an isosceles triangle 2 angles are identical.

If the two identical angles are 30 degrees, then the other angle would be equal to (180 - 2*30) = 120 which would make the triangle obtuse, so the two other equal angles must not include the 30 degree angle and are each equal to (180 - 30)/2 = 75 degrees.

Smallest angle is 30 degrees; largest two angles are 75 degrees.