Question

in a triangle, the measure of the middle angle is triple the measure of the smallest angle, and the measure of the largest angle is 55° greater than the measure of the smallest angle. find the measures of the angles.

Explanation:

Step1: Let the smallest angle be $x$.

Set the smallest - angle as $x$. Then the middle - angle is $3x$ (since it's triple the smallest), and the largest angle is $x + 55$.

Step2: Use the angle - sum property of a triangle.

The sum of the interior angles of a triangle is $180^{\circ}$. So, $x+3x+(x + 55)=180$.

Step3: Combine like terms.

Combining like terms gives $5x+55 = 180$.

Step4: Solve for $x$.

Subtract 55 from both sides: $5x=180 - 55=125$. Then divide both sides by 5: $x=\frac{125}{5}=25$.

Step5: Find the other angles.

The middle angle is $3x = 3\times25 = 75^{\circ}$, and the largest angle is $x + 55=25+55 = 80^{\circ}$.

Answer:

The measures of the angles are $25^{\circ}$, $75^{\circ}$, and $80^{\circ}$.