Question

Find the value of x.

Answer 1

`\large\boxed{\textsf{x=3}}`

Explanation:

`\textsf{We are asked to find the value of x.}`

`\textsf{Notice that we have Intersecting Chords inside of the circle.}`

`\large\underline{\textsf{What are Chords?}}`

`\textsf{Chords are line segments that have endpoints on the circumference of a circle.}`

`\textsf{Note that some Chords can be Diameters, but not for this problem. There's no Center.}`

`\textsf{These Chords don't create Perpendicular Lines, which means the angles aren't equal.}`

`\textsf{We can identify the unknown angles with a postulate.}`

`\large\underline{\textsf{What is a Postulate?}}`

`\textsf{A Postulate is a statement that can never be false. No matter what, it's always}`

`\textsf{true.}`

`\textsf{We can use the Linear Pair Postulate to find the needed angle.}`

`\large\underline{\textsf{What is the Linear Pair Postulate?}}`

`\textsf{Linear Pair Postulate is a Postulate that proves 2 angles add up to 180}^(\circ).`

`\textsf{A Linear Pair is 2 Adjacent Angles that form a Straight Angle.}`

`\large\underline{\textsf{For our Problem;}}`

`\tt \angle HLJ \ and \ \angle ILJ \ are \ Linear \ Pairs.`

`\large\underline{\textsf{Find} \ \angle \textsf{HLJ;}}`

`\tt \angle HLJ + \angle ILJ = 180^(\circ).`

`\tt \angle HLJ + 85.5^(\circ) = 180^(\circ).`

`\underline{\textsf{Subtract 85.5 from both sides of the equation;}}`

`\tt \angle HLJ = 94.5^(\circ)`

`\textsf{Now that we have} \tt \ \angle HLJ, \textsf{we can find x.}`

`\textsf{We are given 2 arc measures and we have 2 Intersecting Chords.}`

`\textsf{We should use} \tt \ \angle HLJ \ \textsf{to find x.}`

`\large\underline{\textsf{How?}}`

`\textsf{The Angle is equal to half the sum of the 2 arcs.}`

`\underline{\textsf{For our Problem;}}`

`\tt \angle HJL = (1)/(2) (42x-6 + 15x+24)`

`\tt 94.5^(\circ) = (1)/(2) (42x-6 + 15x+24)`

`\large\underline{\textsf{Solve;}}`

`\textsf{Multiply each side by 2 first.}`

`\tt 189^(\circ) = 42x-6 + 15x+24`

`\underline{\textsf{Combine Like Terms;}}`

`\tt 189^(\circ) = 57x+18`

`\underline{\textsf{Subtract 18 from both sides;}}`

`\tt 171^(\circ) = 57x`

`\underline{\textsf{Divide each side by 57;}}`

`\large\boxed{\textsf{x=3}}`