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, \(105+(6x + 17)+(5x+14)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(105 + 6x+17 + 5x+14=180\), which gives \((6x + 5x)+(105 + 17+14)=180\), then \(11x+136 = 180\).

Step3: Solve for \(x\)

Subtract 136 from both sides: \(11x=180 - 136\), so \(11x = 44\). Divide both sides by 11: \(x = 4\).

Step4: Find the measures of the non - given angles

For the angle \((6x + 17)^{\circ}\), substitute \(x = 4\): \(6\times4+17=24 + 17=41^{\circ}\).
For the angle \((5x + 14)^{\circ}\), substitute \(x = 4\): \(5\times4+14=20 + 14=34^{\circ}\).

Step5: Compare the non - given angles with the given angle

We have angles \(105^{\circ}\), \(41^{\circ}\), and \(34^{\circ}\).

Answer:

\(34^{\circ}\)