Math / Geometry

Question23. Find the measure of $\angle C Y L$ . $\angle C Y L=$ $\qquad$ (C)Maneuvering the Middle LLC, 2017

Studdy Solution

STEP 1

1. The sum of the interior angles of a triangle is $180^\circ$ .
2. An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

STEP 2

1. Set up the equation using the exterior angle theorem.
2. Solve for $x$ .
3. Find the measure of $\angle CYL$ .

STEP 3

According to the exterior angle theorem, the exterior angle $(7x + 7)^\circ$ is equal to the sum of the two non-adjacent interior angles $\angle C$ and $\angle CYL$ . Therefore, we can write the equation:
$(7x + 7) = (4x - 26) + (5x + 3)$

STEP 4

Simplify the equation from Step 1:
$7x + 7 = 4x - 26 + 5x + 3$
Combine like terms:
$7x + 7 = 9x - 23$

STEP 5

Rearrange the equation to solve for $x$ :
Subtract $9x$ from both sides:
$7x - 9x + 7 = -23$
This simplifies to:
$-2x + 7 = -23$
Subtract 7 from both sides:
$-2x = -30$
Divide by $-2$ :
$x = 15$

STEP 6

Substitute $x = 15$ back into the expression for $\angle CYL$ :
$\angle CYL = (5x + 3)$
$\angle CYL = 5(15) + 3$
$\angle CYL = 75 + 3$
$\angle CYL = 78^\circ$
The measure of $\angle C Y L$ is:
$\boxed{78^\circ}$

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