Question

1. if (mangle cdf=(3x + 14)^{circ}), (mangle fde=(5x - 2)^{circ}), and (mangle cde=(10x - 18)^{circ}), find each measure.

x =
(mangle cdf=)
(mangle fde=)
(mangle cde=)

Explanation:

Step1: Use angle - addition postulate

Since $\angle CDE=\angle CDF+\angle FDE$, we have the equation $(10x - 18)=(3x + 14)+(5x - 2)$.

Step2: Simplify the right - hand side of the equation

Combine like terms on the right - hand side: $(3x+14)+(5x - 2)=3x+5x+14 - 2=8x + 12$. So the equation becomes $10x-18 = 8x+12$.

Step3: Solve for x

Subtract $8x$ from both sides: $10x-8x-18=8x - 8x+12$, which simplifies to $2x-18 = 12$. Then add 18 to both sides: $2x-18 + 18=12 + 18$, so $2x=30$. Divide both sides by 2: $x = 15$.

Step4: Find the measure of $\angle CDF$

Substitute $x = 15$ into the expression for $\angle CDF$: $m\angle CDF=3x + 14=3\times15+14=45 + 14=59^{\circ}$.

Step5: Find the measure of $\angle FDE$

Substitute $x = 15$ into the expression for $\angle FDE$: $m\angle FDE=5x - 2=5\times15-2=75 - 2=73^{\circ}$.

Step6: Find the measure of $\angle CDE$

Substitute $x = 15$ into the expression for $\angle CDE$: $m\angle CDE=10x - 18=10\times15-18=150 - 18=132^{\circ}$.

Answer:

$x = 15$
$m\angle CDF=59^{\circ}$
$m\angle FDE=73^{\circ}$
$m\angle CDE=132^{\circ}$