Question

1. 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 middle angle. find the measures of the angles.

Explanation:

Step1: Set up variables

Let the measure of the smallest angle be \(x\). Then the measure of the middle angle is \(3x\), and the measure of the largest angle is \(x - 55\).

Step2: Use the triangle - angle sum formula

Since the sum of the interior angles of a triangle is \(180^{\circ}\), we have the equation \(x+3x+(x - 55)=180\).
Simplify the left - hand side: \(x+3x+x-55 = 5x-55\).
So, \(5x-55 = 180\).

Step3: Solve for \(x\)

Add \(55\) to both sides of the equation: \(5x-55 + 55=180 + 55\), which gives \(5x=235\).
Divide both sides by \(5\): \(x=\frac{235}{5}=47\).

Step4: Find the measures of the other angles

The middle angle: \(3x = 3\times47 = 141\).
The largest angle: \(x - 55=47-55=- 8\). But this is wrong. Let's re - check the variable assignment.

Let the measure of the middle angle be \(x\). Then the measure of the largest angle is \(3x\), and the measure of the smallest angle is \(x + 55\).
Using the triangle - angle sum formula \(x+3x+(x + 55)=180\).
Simplify the left - hand side: \(x+3x+x + 55=5x + 55\).
Set up the equation \(5x+55 = 180\).
Subtract \(55\) from both sides: \(5x+55-55 = 180-55\), so \(5x=125\).
Divide both sides by \(5\): \(x = 25\).

The middle angle \(x = 25^{\circ}\).
The largest angle \(3x=3\times25 = 75^{\circ}\).
The smallest angle \(x + 55=25 + 55=80^{\circ}\).

Answer:

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