Question

question the measures of the angles of a triangle are shown in the figure below. find the measure of the smallest angle.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, we have the equation \(79+(3x + 15)+(4x-19)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(79+3x + 15+4x-19=(3x + 4x)+(79 + 15-19)=7x + 75\). So the equation becomes \(7x+75 = 180\).

Step3: Solve for x

Subtract 75 from both sides: \(7x=180 - 75\), so \(7x=105\). Then divide both sides by 7: \(x=\frac{105}{7}=15\).

Step4: Find the measures of the non - given angles

For the angle \((3x + 15)^{\circ}\), substitute \(x = 15\): \(3\times15+15=45 + 15=60^{\circ}\).
For the angle \((4x-19)^{\circ}\), substitute \(x = 15\): \(4\times15-19=60-19 = 41^{\circ}\).

Step5: Identify the smallest angle

Compare the angles: \(41^{\circ}<60^{\circ}<79^{\circ}\).

Answer:

\(41^{\circ}\)