Question

finding the angle measures of a triangle
△def is an isosceles triangle with base angles e and f.
what is the angle measure of the smallest angle in the triangle? approximate to the nearest degree.
□ degrees
what are the measures of the two congruent base angles?
□ degrees

Explanation:

Step1: Find the smallest angle

Since the triangle is isosceles with \(DF = DE=12\) and \(FE = 3\), the smallest angle is \(\angle D\).
We use the Law of Cosines: \(\cos D=\frac{DF^{2}+DE^{2}-FE^{2}}{2\cdot DF\cdot DE}\)
Substitute \(DF = DE = 12\) and \(FE = 3\) into the formula:

$$
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Then \(D=\cos^{-1}(0.96875)\approx14^{\circ}\)

Step2: Find the base angles

We know that the sum of angles in a triangle is \(180^{\circ}\). Let the base angles \(\angle E=\angle F=x\)
We have \(D + 2x=180^{\circ}\)
Since \(D\approx14^{\circ}\), then \(2x=180 - 14=166^{\circ}\)
So \(x = 83^{\circ}\)

Answer:

14 degrees
83 degrees