Question

geometry the measure of the obtuse angle in the isosceles triangle is two and a half times the measure of one of the acute angles. write and solve a system of linear equations to find the measure of each angle. system of linear equations: $y = \square$ $y = \square$ the measure of the acute angles is $\boxed{0}^\circ$. the measure of the obtuse angle is $\boxed{100}^\circ$.

Explanation:

Step1: Define variables

Let \( x \) be the measure of each acute angle (in degrees), and \( y \) be the measure of the obtuse angle (in degrees).

Step2: Set up first equation (angle relationship)

The obtuse angle is two and a half times an acute angle, so \( y = 2.5x \).

Step3: Set up second equation (triangle angle sum)

The sum of angles in a triangle is \( 180^\circ \). Since there are two acute angles and one obtuse angle, \( 2x + y = 180 \).

Step4: Substitute \( y \) into the sum equation

Substitute \( y = 2.5x \) into \( 2x + y = 180 \):
\( 2x + 2.5x = 180 \)
Combine like terms: \( 4.5x = 180 \)

Step5: Solve for \( x \)

Divide both sides by \( 4.5 \):
\( x = \frac{180}{4.5} = 40 \)

Step6: Find \( y \)

Substitute \( x = 40 \) into \( y = 2.5x \):
\( y = 2.5 \times 40 = 100 \)

Answer:

System of linear equations: \( y = 2.5x \)
\( y = 180 - 2x \)

The measure of the acute angles is \( \boldsymbol{40^\circ} \).
The measure of the obtuse angle is \( \boldsymbol{100^\circ} \).