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

By the angle - addition postulate, \(m\angle CDF+m\angle FDE = m\angle CDE\).
Substitute the given expressions: \((3x + 14)+(5x-2)=10x - 18\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(3x+14 + 5x-2=(3x + 5x)+(14 - 2)=8x+12\).
So the equation becomes \(8x + 12=10x-18\).

Step3: Solve for \(x\)

Subtract \(8x\) from both sides: \(8x+12-8x=10x - 18-8x\), which gives \(12 = 2x-18\).
Add \(18\) to both sides: \(12 + 18=2x-18+18\), so \(30 = 2x\).
Divide both sides by \(2\): \(x=\frac{30}{2}=15\).

Step4: Find \(m\angle CDF\)

Substitute \(x = 15\) into \(m\angle CDF=(3x + 14)^{\circ}\).
\(m\angle CDF=(3\times15 + 14)^{\circ}=(45+14)^{\circ}=59^{\circ}\).

Step5: Find \(m\angle FDE\)

Substitute \(x = 15\) into \(m\angle FDE=(5x - 2)^{\circ}\).
\(m\angle FDE=(5\times15-2)^{\circ}=(75 - 2)^{\circ}=73^{\circ}\).

Step6: Find \(m\angle CDE\)

Substitute \(x = 15\) into \(m\angle CDE=(10x - 18)^{\circ}\).
\(m\angle CDE=(10\times15-18)^{\circ}=(150-18)^{\circ}=132^{\circ}\).

Answer:

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