Question

find the values of x and y; state which theorem(s) you used.
find the value of x that makes m || n.

Explanation:

Step1: Identify angle - relationship for problem 1

Since the two angles are corresponding angles and corresponding angles are equal when lines are parallel, we have $x = 61$. And $y$ is also equal to $61$ as it is a corresponding angle to the $61^{\circ}$ angle.

Step2: Identify angle - relationship for problem 2

The two angles are vertical angles. Vertical angles are equal. So $13y+19 = 90$. Solve the equation for $y$:

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Step3: Identify angle - relationship for problem 3

The two angles are alternate - interior angles. Alternate - interior angles are equal when lines are parallel. So $6y-9 = 61$, then $6y=61 + 9=70$, and $y=\frac{70}{6}=\frac{35}{3}\approx11.67$.

Step4: Identify angle - relationship for problem 4

For $m\parallel n$, the two angles are same - side interior angles and they are supplementary. So $x+97 = 180$, then $x=180 - 97=83$.

Step5: Identify angle - relationship for problem 5

The two angles are corresponding angles. For $m\parallel n$, we set $4x + 24=84$. Solve for $x$:

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Step6: Identify angle - relationship for problem 6

The two angles are corresponding angles. For $m\parallel n$, we set $11x+32=6x - 6$.

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Answer:

Problem 1: $x = 61,y = 61$
Problem 2: $y=\frac{71}{13}$
Problem 3: $y=\frac{35}{3}$
Problem 4: $x = 83$
Problem 5: $x = 15$
Problem 6: $x=-\frac{38}{5}$