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QuestionOne angle is 66∘66^{\circ}, and the third angle is 57∘57^{\circ} more than half the second angle. Find the second and third angles.

Studdy Solution

STEP 1

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

STEP 2

Let's denote the measure of the second angle as xx and the measure of the third angle as yy. We know that y=12x+57∘y = \frac{1}{2}x +57^{\circ}.

STEP 3

We also know that the sum of the angles in a triangle is 180∘180^{\circ}. Therefore, we can write another equation 66∘+x+y=180∘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 yy from the first equation into the second equation to solve for xx66∘+x+(12x+57∘)=180∘.66^{\circ} + x + \left(\frac{1}{2}x +57^{\circ}\right) =180^{\circ}.

STEP 6

implify the equationx+12x=180∘−66∘−57∘.x + \frac{1}{2}x =180^{\circ} -66^{\circ} -57^{\circ}.

STEP 7

Combine like terms32x=57∘.\frac{3}{2}x =57^{\circ}.

STEP 8

To solve for xx, multiply both sides of the equation by 23\frac{2}{3}x=57∘×23.x =57^{\circ} \times \frac{2}{3}.

STEP 9

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

STEP 10

Now that we have the value of xx, we can substitute it into the first equation to find the value of yyy=2×38∘+57∘.y = \frac{}{2} \times38^{\circ} +57^{\circ}.

STEP 11

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

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