Question

number 3: crossties l and k and parallel. both crossties are intersected by crosstie 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. 5) 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)°

Explanation:

Step1: Identify angle - pair relationship for Number 3

$\angle1$ and $\angle7$ are corresponding angles. When two parallel lines ($l$ and $k$) are cut by a transversal ($h$), corresponding angles are congruent.

Step2: Find $m\angle7$

Since $m\angle1 = 42^{\circ}$ and $\angle1\cong\angle7$, then $m\angle7=42^{\circ}$.

Step3: Identify angle - pair relationship for Number 4

$\angle3$ and $\angle4$ are alternate - interior angles. When two parallel lines (Dennis Avenue and State Road) are cut by a transversal (Newport Lane), alternate - interior angles are congruent.

Step4: Find $m\angle4$

Since $m\angle3 = 62^{\circ}$ and $\angle3\cong\angle4$, then $m\angle4 = 62^{\circ}$.

Step5: Identify angle - pair relationship for 5a

$\angle3$ and $\angle6$ are alternate - interior angles. When two parallel lines ($j$ and $k$) are cut by a transversal, alternate - interior angles are congruent. So, $m\angle3=m\angle6$.

Step6: Solve for $x$

Set up the equation $4x + 7=5x-13$.
Subtract $4x$ from both sides: $7=x - 13$.
Add 13 to both sides: $x=20$.

Step7: Identify angle - pair relationship for 5b

$\angle8$ and $\angle3$ are corresponding angles. When two parallel lines ($j$ and $k$) are cut by a transversal, corresponding angles are congruent. So, $m\angle8=m\angle3$.

Step8: Solve for $y$

Set up the equation $3y-2 = 68$.
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: $m\angle4 = 62^{\circ}$
Number 5a: $x = 20$
Number 5b: $y=\frac{70}{3}$