Question

which one of the following sets of data determines a unique triangle?
choose the correct answer below.
a. a = 7, b = 24, c = 25
b. a = 3, b = 5, c = 20
c. a = 50°, b = 50°, c = 80°
d. a = 40°, b = 20°, c = 30°

Explanation:

Step1: Check the triangle - inequality theorem

The triangle - inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\).
For option B: \(a = 3\), \(b = 5\), \(c = 20\). Then \(a + b=3 + 5=8<20\), so a triangle with side lengths \(3\), \(5\), and \(20\) does not exist.
For option D: \(A = 40^{\circ}\), \(B = 20^{\circ}\), \(C = 30^{\circ}\). Since \(A + B + C=40^{\circ}+20^{\circ}+30^{\circ}=90^{\circ}
eq180^{\circ}\), a triangle with these angle measures does not exist.

Step2: Check the uniqueness of triangles

For option C: \(A = 50^{\circ}\), \(B = 50^{\circ}\), \(C = 80^{\circ}\). There are infinitely many similar triangles with these angle measures (they can have different side - length ratios).
For option A: \(a = 7\), \(b = 24\), \(c = 25\). First, check the triangle - inequality: \(7+24 = 31>25\), \(7 + 25=32>24\), \(24+25 = 49>7\). Also, by the SSS (side - side - side) congruence criterion, a triangle with side lengths \(a = 7\), \(b = 24\), and \(c = 25\) is unique.

Answer:

A. \(a = 7\), \(b = 24\), \(c = 25\)