Question

unit 4 triangle properties test
given \\( \triangle a b c \\) below, order the sides from shortest to longest, with the shortest side going first and the longest side going last.
\\( \overline{a b} \\)
\\( \overline{b c} \\)
\\( \overline{a c} \\)

Explanation:

Step1: Find the value of \(x\)

The sum of angles in a triangle is \(180^{\circ}\). So, \((3x + 12)+(x + 39)+(2x + 15)=180\).
Combine like - terms: \(3x+x + 2x+12 + 39+15=180\), \(6x+66 = 180\).
Subtract 66 from both sides: \(6x=180 - 66=114\).
Divide by 6: \(x=\frac{114}{6}=19\).

Step2: Calculate the measures of the angles

• \(\angle A=(3x + 12)^{\circ}\). Substitute \(x = 19\): \(\angle A=(3\times19+12)^{\circ}=(57 + 12)^{\circ}=69^{\circ}\).
• \(\angle B=(x + 39)^{\circ}\). Substitute \(x = 19\): \(\angle B=(19+39)^{\circ}=58^{\circ}\).
• \(\angle C=(2x + 15)^{\circ}\). Substitute \(x = 19\): \(\angle C=(2\times19+15)^{\circ}=(38+15)^{\circ}=53^{\circ}\).

Step3: Use the angle - side relationship

In a triangle, the side opposite the smaller angle is shorter.
Since \(\angle C<\angle B<\angle A\), the side opposite \(\angle C\) is \(\overline{AB}\), the side opposite \(\angle B\) is \(\overline{AC}\), and the side opposite \(\angle A\) is \(\overline{BC}\).

Answer:

\(\overline{AB},\overline{AC},\overline{BC}\)