Question

the measures of the angles of a triangle are shown in the figure below. find the measure of the smallest angle. 105° (5x + 14)° (6x + 17)°

Explanation:

Step1: Use the triangle - angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(105+(6x + 17)+(5x+14)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(105 + 6x+17+5x + 14=(6x + 5x)+(105 + 17+14)=11x+136\).
The equation becomes \(11x+136 = 180\).

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 angles

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

Answer:

The measure of the smallest angle is \(34^{\circ}\).