Question

part 1 - multiple choice: show all work

1. in the accompanying diagram, ab and cd intersect at e. angles aec and deb measure 2x - 6 and 6x - 50, respectively. find the value of x.

(1) 29.5
(2) 11
(3) 16
(4) 164

1. in the diagram below, transversal tu intersects pq and rs at v and w, respectively.

if m∠qvw = 5x - 22 and m∠vws = 3x + 10, for which value of x is pq || rs?

1. 6
2. 16
3. 24
4. 28
5. in the diagram below, line p intersects line m and line n.

if m∠1 = 7x and m∠2 = 5x + 30, lines m and n are parallel when x equals

1. 12.5
2. 15
3. 87.5
4. 105

Explanation:

Step1: Identify vertical - angles property

Vertical angles are equal. In the first problem, angles AEC and DEB are vertical angles. So, we set up the equation \(2x - 6=6x - 50\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(-6 = 4x-50\). Then add 50 to both sides: \(44 = 4x\). Divide both sides by 4, we get \(x = 11\).

Step3: Identify corresponding - angles property for the second problem

When \(PQ\parallel RS\), corresponding angles are equal. So, \(\angle QVW=\angle VWS\). Set up the equation \(5x - 22=3x + 10\).

Step4: Solve the second - equation for \(x\)

Subtract \(3x\) from both sides: \(2x-22 = 10\). Add 22 to both sides: \(2x=32\). Divide by 2, we get \(x = 16\).

Step5: Identify alternate - interior angles property for the third problem

When \(m\parallel n\), alternate - interior angles are equal. So, \(7x=5x + 30\).

Step6: Solve the third - equation for \(x\)

Subtract \(5x\) from both sides: \(2x=30\). Divide by 2, we get \(x = 15\).

Answer:

1. (2) 11
2. (2) 16
3. (2) 15