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 [ ], the second angle is [ ], and the third angle is [ ].
1. **State the problem:** We have an isosceles triangle with two equal angles and a third angle that is 30° more than twice either of the equal angles. We need to find the measures of all three angles.
2. **Define variables:** Let the measure of each of the two equal angles be $x$ degrees.
3. **Express the third angle:** The third angle is $30$ degrees more than twice $x$, so it is $2x + 30$ degrees.
4. **Use the triangle angle sum rule:** The sum of the angles in any triangle is $180$ degrees. So,
$$x + x + (2x + 30) = 180$$
5. **Simplify the equation:**
$$2x + 2x + 30 = 180$$
$$4x + 30 = 180$$
6. **Isolate $x$:**
$$4x = 180 - 30$$
$$4x = 150$$
7. **Solve for $x$:**
$$x = \frac{150}{4}$$
$$x = 37.5$$
8. **Find the third angle:**
$$2x + 30 = 2(37.5) + 30 = 75 + 30 = 105$$
9. **Answer:**
The two equal angles are each $37.5$ degrees, and the third angle is $105$ degrees.