Question

1. which segment is skew to $overline{gh}$? (g1f)

a. $overline{hc}$ b. $overline{eg}$ c. $overline{ad}$ d. $overline{ef}$

1. which statement about the picture is true about $angle1$? (g6a)

a. $mangle1 + mangle3=180^{circ}$ b. $angle1congangle3$ c. $angle1congangle2$ d. $angle4congangle1$

1. if $aparallel b$ and $mangle1 = 5(y + 11), mangle2=4y - 10$, then find the measure of $angle$ angle. (g6a)
2. find $mangle1$ so that $gparallel f$, if $mangle1 = 8x + 8$ and $mangle2=4x + 28$. (g6a)
3. if the measure of $angle rst$ is $134^{circ}$, find the measure of $angle qst$. (g6a)

$(3x - 1)^{circ}$
$(x - 1)^{circ}$

1. what is the converse of the conditional statement? (g4b)

if it rains, then the grass grows.

1. what is the next shape in the pattern? (g1d)

Explanation:

1. Skew lines are lines that are not parallel and do not intersect. In a 3 - D figure, $\overline{EF}$ is skew to $\overline{GH}$.
2. When two lines are intersected by a transversal, vertical angles are congruent. $\angle1$ and $\angle3$ are vertical angles.
3. When two parallel lines are cut by a transversal, same - side interior angles are supplementary. We set up the correct equation based on this property and solve for the variable and then the angle measure.
4. When two parallel lines are cut by a transversal, corresponding angles are congruent. We solve the equation for $x$ and then find the measure of $\angle1$.
5. We use the angle - addition postulate to set up an equation involving the given angles and solve for $x$ to find the measure of $\angle QST$.
6. The converse of a conditional statement "If $p$, then $q$" is "If $q$, then $p$".
7. We observe the pattern of the number of sides of the polygons to determine the next shape.

Answer:

1. d. $\overline{EF}$
2. b. $\angle1\cong\angle3$
3. First, since $a\parallel b$, $\angle1=\angle2$. So, $5(y + 11)=4y- 10$.

Step1: Expand the left - hand side

$5y+55 = 4y - 10$

Step2: Subtract $4y$ from both sides

$5y-4y+55=4y-4y - 10$
$y+55=-10$

Step3: Subtract 55 from both sides

$y=-10 - 55$
$y=-65$
Then, $\angle1 = 5(y + 11)=5(-65 + 11)=5\times(-54)=-270$ (This is wrong, there is a mistake above. Since $a\parallel b$, $\angle1+\angle2 = 180^{\circ}$). So, $5(y + 11)+4y-10=180$.

Step1: Expand and combine like terms

$5y+55+4y-10 = 180$
$9y + 45=180$

Step2: Subtract 45 from both sides

$9y=180 - 45$
$9y=135$

Step3: Divide both sides by 9

$y = 15$
$\angle1=5(y + 11)=5\times(15 + 11)=5\times26 = 130^{\circ}$

1. Since $g\parallel f$, $\angle1=\angle2$. So, $8x + 8=4x+28$.

Step1: Subtract $4x$ from both sides

$8x-4x + 8=4x-4x+28$
$4x+8=28$

Step2: Subtract 8 from both sides

$4x=28 - 8$
$4x=20$

Step3: Divide both sides by 4

$x = 5$
$m\angle1=8x + 8=8\times5+8=40 + 8=48^{\circ}$

1. Since $\angle RST=\angle RSQ+\angle QST$, and $\angle RST = 134^{\circ}$, and $\angle RSQ=(3x - 1)^{\circ}$, $\angle QST=(x - 1)^{\circ}$. Then $(3x - 1)+(x - 1)=134$.

Step1: Combine like terms

$3x-1+x - 1=134$
$4x-2=134$

Step2: Add 2 to both sides

$4x=134 + 2$
$4x=136$

Step3: Divide both sides by 4

$x = 34$
$\angle QST=x - 1=34 - 1=33^{\circ}$

1. The converse of "If it rains, then the grass grows" is "If the grass grows, then it rains".
2. C. Octagon. The number of sides of the polygons in the pattern is increasing by 1 successively (3 - sided triangle, 4 - sided square, 5 - sided pentagon, 6 - sided hexagon, so the next is 8 - sided octagon).