Math Snap

STEP 1

Assumptions1. One angle of the triangle measures $66^{\circ}$.
. The measure of the third angle is $57^{\circ}$ more than $\frac{1}{}$ the measure of the second angle.
3. The sum of the angle measures of a triangle is $180^{\circ}$.

STEP 2

Let's denote the measure of the second angle as $x$ and the measure of the third angle as $y$. We know that $y = \frac{1}{2}x +57^{\circ}$.

STEP 3

We also know that the sum of the angles in a triangle is $180^{\circ}$. Therefore, we can write another equation $66^{\circ} + x + y =180^{\circ}$.

STEP 4

Now we have a system of two equations\begin{align} y &= \frac{1}{2}x +57^{\circ},\\66^{\circ} + x + y &=180^{\circ}. \end{align}

STEP 5

We can substitute $y$ from the first equation into the second equation to solve for $x$$$66^{\circ} + x + \left(\frac{1}{2}x +57^{\circ}\right) =180^{\circ}.$$

STEP 6

implify the equation$$x + \frac{1}{2}x =180^{\circ} -66^{\circ} -57^{\circ}.$$

STEP 7

Combine like terms$$\frac{3}{2}x =57^{\circ}.$$

STEP 8

To solve for $x$, multiply both sides of the equation by $\frac{2}{3}$$$x =57^{\circ} \times \frac{2}{3}.$$

STEP 9

Calculate the value of $x$$$x =38^{\circ}.$$

STEP 10

Now that we have the value of $x$, we can substitute it into the first equation to find the value of $y$$$y = \frac{}{2} \times38^{\circ} +57^{\circ}.$$

Calculate the value of $y$$$y =19^{\circ} +57^{\circ} =76^{\circ}.$$ The measure of the second angle is $38^{\circ}$ and the measure of the third angle is $76^{\circ}$.