Question

1. (l27) select the pair of lines in the diagram below that must be parallel if \\( \angle 6 \cong \angle 8 \\). if there are no lines that must be parallel, select

one.\
a. \\( k \\| j \\)
b. \\( d \\| e \\)
c. \\( k \\| i \\)
d. \\( d \\| f \\)
e. none

1. (l18) find \\( x \\) and \\( y \\).

a. \\( x = 15 ; y = 45 \\)
b. \\( x = 20 ; y = 60 \\)
c. \\( x = 18 ; y = 72 \\)
d. \\( x = 21 ; y = 85 \\)
e. none of the above

1. (l31) find \\( y \\).

a. \\( y = 95 \\)
b. \\( y = 121 \\)
c. \\( y = 180 \\)
d. \\( y = 139 \\)
e. none of the above

1. (l30) find \\( x \\) and \\( y \\).

a. \\( x = 48 ; y = 100 \\)
b. \\( x = 56 ; y = 110 \\)
c. \\( x = 42 ; y = 85 \\)
d. \\( x = 45 ; y = 90 \\)
e. none of the above

1. (l31) find \\( x \\) and \\( y \\).

a. \\( x = 34 ; y = 90 \\)
b. \\( x = 42 ; y = 84 \\)
c. \\( x = 25 ; y = 45 \\)
d. \\( x = 26 ; y = 92 \\)
e. none of the above

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. In the diagram for problem 21, \((4x - 16)^{\circ}\) and \((3x + 10)^{\circ}\) are vertical angles. So, \(4x-16=3x + 10\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides of the equation \(4x-16=3x + 10\). We get \(4x-3x-16=3x-3x + 10\), which simplifies to \(x-16=10\). Then add 16 to both sides: \(x=10 + 16=26\).

Step3: Use the linear - pair property

\(y^{\circ}\) and \((4x - 16)^{\circ}\) form a linear pair. Since \(x = 26\), then \(4x-16=4\times26-16=104 - 16=88\). And \(y+88 = 180\) (because the sum of angles in a linear pair is \(180^{\circ}\)).

Step4: Solve for \(y\)

Subtract 88 from both sides of the equation \(y + 88=180\). So, \(y=180-88 = 92\).

Answer:

D. \(x = 26;y = 92\)