Question

1. math on the spot a graphic designer is using quadrilaterals to make a design. each quadrilateral is congruent to the one shown here. use the values of the marked angles to show that the two lines $ell_1$ and $ell_2$ are parallel.
2. math on the spot use the given angle - relationship to decide whether the lines are parallel. explain your reasoning.

a. if $angle2congangle6$
b. if $mangle4=(2x - 12)^{circ}$, $mangle5=(3x + 17)^{circ}$, and $x = 35$

1. use structure massachusetts avenue is parallel to dalton street. the acute angles formed by dalton street and scotia street and dalton street and belvidere street measure $56^{circ}$. the obtuse angle formed by massachusetts avenue and st. germain street measures $124^{circ}$. which streets are parallel?

a massachusetts avenue and st. germain street are parallel.
b massachusetts avenue and scotia street are parallel.
c st. germain street and dalton street are parallel.
d belvidere street, st. germain street, and scotia street are parallel.

Explanation:

Step1: Recall angle - related parallel - line theorems

If corresponding angles are equal, or alternate interior angles are equal, or same - side interior angles are supplementary, then two lines are parallel.

Step2: Analyze question 10

The marked angles of the congruent quadrilaterals can be used to show that the corresponding angles (or alternate interior angles) formed by $\ell_1$ and $\ell_2$ are equal. For example, if we consider the angles at the intersection of a transversal with $\ell_1$ and $\ell_2$ formed by the congruent quadrilaterals, we can use the fact that the sum of adjacent angles in a quadrilateral is $180^{\circ}$ and angle - congruence properties. Since the quadrilaterals are congruent, the angles formed by the intersection of a transversal with $\ell_1$ and $\ell_2$ are equal, which implies $\ell_1\parallel\ell_2$.

Step3: Analyze question 11A

$\angle2$ and $\angle6$ are corresponding angles. If $\angle2\cong\angle6$, by the corresponding - angles postulate, lines $\ell$ and $m$ are parallel.

Step4: Analyze question 11B

First, find the measures of $\angle4$ and $\angle5$.
Substitute $x = 35$ into the expressions for $\angle4$ and $\angle5$.
$m\angle4=(2x - 12)^{\circ}=(2\times35 - 12)^{\circ}=(70 - 12)^{\circ}=58^{\circ}$
$m\angle5=(3x + 17)^{\circ}=(3\times35+17)^{\circ}=(105 + 17)^{\circ}=122^{\circ}$
Since $m\angle4+m\angle5=58^{\circ}+122^{\circ}=180^{\circ}$, and $\angle4$ and $\angle5$ are same - side interior angles, by the same - side interior angles postulate, lines $\ell$ and $m$ are parallel.

Step5: Analyze question 12

The acute angle between Dalton Street and Scotia Street and Dalton Street and Belvidere Street is $56^{\circ}$, and the obtuse angle between Massachusetts Avenue and St. Germain Street is $124^{\circ}$. Since $56^{\circ}+124^{\circ}=180^{\circ}$, and considering the relationship between the angles formed by these streets, St. Germain Street and Dalton Street are parallel because the same - side interior angles formed by a transversal (a line intersecting these two streets) are supplementary.

Answer:

1. The congruent quadrilaterals form equal corresponding (or alternate interior) angles for $\ell_1$ and $\ell_2$, so $\ell_1\parallel\ell_2$.

11A. Lines $\ell$ and $m$ are parallel because $\angle2$ and $\angle6$ are corresponding angles and $\angle2\cong\angle6$.
11B. Lines $\ell$ and $m$ are parallel because $m\angle4 + m\angle5=180^{\circ}$ (same - side interior angles are supplementary).

1. C. St. Germain Street and Dalton Street are parallel.