Question

△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:

1. First, use the Law of Cosines to find the vertex - angle (the smallest angle in this case):

• The Law of Cosines for a triangle with sides \(a = 12\), \(b = 12\), and \(c = 3\) is \(c^{2}=a^{2}+b^{2}-2ab\cos C\).
• Substitute \(a = 12\), \(b = 12\), and \(c = 3\) into the formula:
• \(3^{2}=12^{2}+12^{2}-2\times12\times12\times\cos D\).
• \(9 = 144 + 144-288\cos D\).
• \(9=288 - 288\cos D\).
• \(288\cos D=288 - 9=279\).
• \(\cos D=\frac{279}{288}\approx0.96875\).
• \(D=\cos^{-1}(0.96875)\approx14^{\circ}\).

1. Then, use the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)) to find the base - angles:

• Let the base - angles \(\angle E=\angle F = x\).
• We know that \(D + 2x=180^{\circ}\).
• Since \(D\approx14^{\circ}\), then \(2x = 180 - 14=166^{\circ}\).
• \(x = 83^{\circ}\).

Answer:

(1) 14
(2) 83