Question

the flag of a country contains an isosceles triangle. (recall that an isosceles triangle contains two angles with the same measure.) if the measure of the third angle of the triangle is 30° more than twice the measure of either of the other two angles, find the measure of each angle of the triangle. (recall that the sum of the measures of the angles of a triangle is 180°.) the first angle is \\(\square\\)°, the second angle is \\(\square\\)°, and the third angle is \\(\square\\)°.

Explanation:

Step1: Define Variables

Let the measure of each of the two equal angles be \( x \) degrees. Then the measure of the third angle is \( 2x + 30 \) degrees (since it's 30° more than twice either of the other two angles).

Step2: Use Triangle Angle Sum

The sum of the interior angles of a triangle is \( 180^\circ \). So we set up the equation:
\( x + x + (2x + 30) = 180 \)

Step3: Solve the Equation

Combine like terms:
\( 4x + 30 = 180 \)
Subtract 30 from both sides:
\( 4x = 150 \)
Divide both sides by 4:
\( x = 37.5 \)

Step4: Find the Third Angle

Now find the measure of the third angle:
\( 2x + 30 = 2(37.5) + 30 = 75 + 30 = 105 \)

Answer:

The first angle is \( 37.5^\circ \), the second angle is \( 37.5^\circ \), and the third angle is \( 105^\circ \).