Question

geometry
find the value of each variable.
1)
(3x)° (8y - 102)°
(2y + 6)°
2)
(5x + 4)° (3x - 24)°
114° (2y)°
name an angle or angle pair that satisfies each condition in the diagram to the right.

1. two adjacent angles
2. two vertical angles
3. a linear pair that has vertex f
4. pool felipe uses a computer program to model the paths of pool balls. ∠gfh is a straight angle that represents the rail of the pool table. if fk bisects ∠jfl, and m∠jfl = 90°, what is m∠lfk?

Explanation:

Step1: Use vertical - angle property for 1)

Set \(3x = 2y + 6\) and \(8y-102=2y + 6\). Solve \(8y-102=2y + 6\):
\(8y-2y=6 + 102\), \(6y=108\), \(y = 18\). Substitute \(y = 18\) into \(3x=2y + 6\), \(3x=2\times18 + 6=42\), \(x = 14\).

Step2: Use vertical - angle and linear - pair properties for 2)

Set \(5x + 4=114\), \(5x=110\), \(x = 22\). Set \(2y+3x - 24=180\), substitute \(x = 22\), \(2y+3\times22-24 = 180\), \(2y+66 - 24=180\), \(2y=138\), \(y = 69\).

Step3: For 3)

Adjacent angles share a common side and a common vertex. For example, \(\angle BCF\) and \(\angle FCD\).

Step4: For 4)

Vertical angles are opposite each other. For example, \(\angle BCF\) and \(\angle DFH\).

Step5: For 5)

A linear - pair at vertex \(F\): \(\angle CFG\) and \(\angle GFD\).

Step6: For 6)

If \(\overrightarrow{FK}\) bisects \(\angle JFL\) and \(m\angle JFL = 90^{\circ}\), then \(m\angle LFK=\frac{90^{\circ}}{2}=45^{\circ}\).

Answer:

1. \(x = 14\), \(y = 18\)
2. \(x = 22\), \(y = 69\)
3. \(\angle BCF\) and \(\angle FCD\)
4. \(\angle BCF\) and \(\angle DFH\)
5. \(\angle CFG\) and \(\angle GFD\)
6. \(45^{\circ}\)