Question

unit 4 triangle properties test
given \\( \triangle a b c \\) below, order the angles from smallest to largest, with the smallest angle going first and the largest angle going last.
\\( \equiv \angle a \\)
\\( \equiv \angle b \\)
\\( \equiv \angle c \\)

Explanation:

Step1: Use the triangle angle - sum property

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

Step2: Calculate the measure of each angle

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

Step3: Order the angles

Since \(53^{\circ}<58^{\circ}<69^{\circ}\).

Answer:

\(\angle C,\angle B,\angle A\)