Question

module 06: 14-4 triangle special segments
3
use the angle measures given to determine the value of x and the value of y.
(there is a triangle - like figure with angles: (x + 7)°, (3x - 18)°, (x - 9)° in one part and (x + 21)°, (x + 17)°, y° in another part)
if x = 40 and y = 61, then go to question 5.
if x = 40 and y = 57, then go to question 11.
if x = 40 and y = 62, then go to question 7.
if x = 62 and y = 40, then go to question 15.

Explanation:

Step1: Sum of angles in a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let's first consider the left triangle with angles \((x + 7)^{\circ}\), \((3x - 18)^{\circ}\), and \((x - 9)^{\circ}\). So we set up the equation:

$$ (x + 7)+(3x - 18)+(x - 9)=180 $$

Step2: Solve for \(x\)

Simplify the left - hand side of the equation:

$$ x + 7+3x - 18+x - 9=180\\ (1 + 3+1)x+(7-18 - 9)=180\\ 5x-20 = 180 $$

Add 20 to both sides:

$$ 5x=180 + 20\\ 5x=200 $$

Divide both sides by 5:

$$ x=\frac{200}{5}=40 $$

Step3: Sum of angles in the right triangle

Now consider the right - hand triangle with angles \((x + 21)^{\circ}\), \((x + 17)^{\circ}\), and \(y^{\circ}\). We know that \(x = 40\), so the sum of the first two angles is \((40 + 21)+(40+17)=61 + 57 = 118^{\circ}\)
Since the sum of the interior angles of a triangle is \(180^{\circ}\), we have:

$$ y=180-(40 + 21)-(40 + 17) $$

$$ y=180 - 61-57 $$

$$ y=180-(61 + 57) $$

$$ y=180 - 118=61 $$

Answer:

\(x = 40\), \(y = 61\)