Question

number 3: cross - ties l and k are parallel. both cross - ties are intersected by cross - tie h. if m∠1 = 42°, find m∠7
number 4: dennis avenue and state road are parallel streets that intersect newport lane along the south side of oak creek park. if m∠3 = 62°, find m∠4.
number 5 - 7: use the figure to find the value of the indicated variable. justify your reasoning.

1. a. if m∠3=(4x + 7)° and m∠6=(5x - 13)°, find x.

b. find y if m∠8 = 68° and m∠3=(3y - 2)°
6)
7)

Explanation:

Step1: Identify angle - relationships for Number 3

Since \(l\) and \(k\) are parallel and intersected by \(h\), \(\angle1\) and \(\angle7\) are corresponding angles. Corresponding angles are congruent when two parallel lines are cut by a transversal.
So \(m\angle7=m\angle1\)

Step2: Calculate \(m\angle7\)

Given \(m\angle1 = 42^{\circ}\), then \(m\angle7=42^{\circ}\)

Step3: Identify angle - relationships for Number 4

Since Dennis Avenue and State Road are parallel and intersected by Newport Lane, \(\angle3\) and \(\angle4\) are alternate - interior angles. Alternate - interior angles are congruent when two parallel lines are cut by a transversal.
So \(m\angle4=m\angle3\)

Step4: Calculate \(m\angle4\)

Given \(m\angle3 = 62^{\circ}\), then \(m\angle4=62^{\circ}\)

Step5: Identify angle - relationships for 5a

\(\angle3\) and \(\angle6\) are alternate - interior angles. Since \(j\) and \(l\) are parallel, \(m\angle3=m\angle6\).
Set up the equation \(4x + 7=5x-13\)

Step6: Solve the equation for \(x\) in 5a

Subtract \(4x\) from both sides: \(7=x - 13\)
Add 13 to both sides: \(x=20\)

Step7: Identify angle - relationships for 5b

\(\angle8\) and \(\angle3\) are vertical angles. Vertical angles are congruent. So \(m\angle8=m\angle3\)
Set up the equation \(3y-2 = 68\)

Step8: Solve the equation for \(y\) in 5b

Add 2 to both sides: \(3y=70\)
Divide both sides by 3: \(y=\frac{70}{3}\)

Answer:

Number 3: \(m\angle7 = 42^{\circ}\)
Number 4: \(m\angle4 = 62^{\circ}\)
5a: \(x = 20\)
5b: \(y=\frac{70}{3}\)