Question

constructed response #1: using the image below, answer each of the following questions: a. if ∠1 is congruent to ∠9, explain why lines x and y are parallel. b. if ∠18 is congruent to ∠21, explain if you can prove any lines are parallel. c. if ∠8 is supplementary to ∠19, explain why lines x and z are parallel. constructed response #2: using the image below, answer each of the following questions: a. determine the angle relationships between ∠2 and ∠15. b. if a||b, m∠16 = 12x - 10 and m∠13 = 8x + 30, solve for m∠16.

Explanation:

Constructed Response #1:

a.

When two lines are cut by a transversal, if corresponding angles are congruent, the lines are parallel. $\angle1$ and $\angle9$ are corresponding angles. Since $\angle1\cong\angle9$, by the corresponding - angles postulate, lines $x$ and $y$ are parallel.

b.

$\angle18$ and $\angle21$ are vertical angles. Vertical - angle congruence is a property that holds for any two intersecting lines and does not provide information about parallel lines. So, we cannot prove any lines are parallel based on the congruence of $\angle18$ and $\angle21$.

c.

$\angle8$ and $\angle19$ are same - side interior angles. When two lines are cut by a transversal, if same - side interior angles are supplementary, the lines are parallel. Since $\angle8$ is supplementary to $\angle19$, by the same - side interior angles postulate, lines $x$ and $z$ are parallel.

Constructed Response #2:

a.

$\angle2$ and $\angle15$ are neither corresponding, alternate interior, alternate exterior, nor same - side interior angles. They have no special angle relationship with respect to the parallel lines formed by the transversals in the figure.

b.

If $a\parallel b$, then $\angle16$ and $\angle13$ are same - side interior angles. Same - side interior angles are supplementary when two parallel lines are cut by a transversal. So, $m\angle16 + m\angle13=180$.

Step1: Set up the equation

$(12x - 10)+(8x + 30)=180$

Step2: Combine like terms

$12x+8x-10 + 30=180$
$20x+20 = 180$

Step3: Solve for $x$

$20x=180 - 20$
$20x=160$
$x = 8$

Step4: Find $m\angle16$

Substitute $x = 8$ into the expression for $m\angle16$:
$m\angle16=12x-10=12\times8 - 10=96 - 10=86$

Answer:

Constructed Response #1:

a. $\angle1$ and $\angle9$ are corresponding angles. By the corresponding - angles postulate, $x\parallel y$.

b. No, because $\angle18$ and $\angle21$ are vertical angles and vertical - angle congruence does not imply parallel lines.

c. $\angle8$ and $\angle19$ are same - side interior angles. By the same - side interior angles postulate, $x\parallel z$.

Constructed Response #2:

a. No special angle relationship.

b. $m\angle16 = 86$