chapter 3: angle pairs & perpendicular
(c)16. the difference between 3 times an angles supplement and 100 is 17. find the measure of the angle.
complete each sentence below with always, sometimes, or never.
(d)17. adjacent angles are ____ supplementary.
a. always
b. sometimes
c. never
18. the midpoint of a line segment is ____ equidistant (equally far) from each end point.
a. always
b. sometimes
c. never
19. vertical angles are ____ adjacent.
a. always
b. sometimes
c. never
answer each question below.
20. if $\overline{qx}=\overline{qy}=\overline{qz}$, and if $m\angle xqy = 4a + 8$, $m\angle xqz=8a$, and $m\angle yqz = 28$, find $m\angle xqz$.
21. if line $\ell$ bisects $\overline{ab}$ at point $m$, and if $am = 4x$ and $mb=5x - 4$, find $ab$.
tell whether the arguments below are valid or invalid deductions.
22. all the players on the volleyball team have sand in their toes. allison is not a player on the volleyball team, so she must not have sand in her toes.
23. all of the ballerinas are wearing tutus. hank is not wearing a tutu, so he must not be a ballerina.
do the proof below.
(e)24. given: $\overline{fj}\perp\overline{fg}$, $\overline{gh}\perp\overline{fg}$, and $\angle2\cong\angle4$ prove: $\angle1\cong\angle3$

Question

Accepted Answer

### 16.
# Explanation:
## Step1: Let the angle be $x$.
Its supplement is $180 - x$. The equation based on the problem is $3(180 - x)-100 = 17$.
## Step2: Expand the left - hand side.
$540-3x - 100=17$, which simplifies to $440-3x = 17$.
## Step3: Isolate the variable term.
Subtract 440 from both sides: $-3x=17 - 440=-423$.
## Step4: Solve for $x$.
Divide both sides by $- 3$: $x = 141$.
# Answer:
$141$

### 17.
# Brief Explanations:
Adjacent angles are two angles that have a common vertex and a common side. They are supplementary when they form a linear pair (sum to 180 degrees), but not all adjacent angles form a linear pair.
# Answer:
B. sometimes

### 18.
# Brief Explanations:
By the definition of a mid - point of a line segment, it is the point that divides the line segment into two equal parts, so it is equidistant from each endpoint.
# Answer:
A. always

### 19.
# Brief Explanations:
Vertical angles are formed by two intersecting lines and are opposite each other. Adjacent angles share a common side and a common vertex. Vertical angles do not share a common side, so they are never adjacent.
# Answer:
C. never

### 20.
# Explanation:
## Step1: Since $\overrightarrow{QX}-\overrightarrow{QY}-\overrightarrow{QZ}$, then $m\angle XQZ=m\angle XQY + m\angle YQZ$.
We have the equation $8a=(4a + 8)+28$.
## Step2: Simplify the right - hand side of the equation.
$8a=4a+36$.
## Step3: Isolate the variable term.
Subtract $4a$ from both sides: $8a-4a=36$, so $4a = 36$.
## Step4: Solve for $a$.
Divide both sides by 4: $a = 9$.
## Step5: Find $m\angle XQZ$.
Substitute $a = 9$ into $m\angle XQZ = 8a$, so $m\angle XQZ=72$.
# Answer:
$72$

### 21.
# Explanation:
## Step1: Since line $\ell$ bisects $\overline{AB}$ at point $M$, then $AM = MB$.
Set up the equation $4x=5x - 4$.
## Step2: Isolate the variable term.
Subtract $4x$ from both sides: $0=x - 4$.
## Step3: Solve for $x$.
Add 4 to both sides: $x = 4$.
## Step4: Find $AM$ and $MB$.
$AM = 4x=16$ and $MB = 5x - 4=16$.
## Step5: Find $AB$.
$AB=AM + MB=32$.
# Answer:
$32$

### 22.
# Brief Explanations:
The statement "All the players on the volleyball team have sand in their toes" only tells us about players on the volleyball team. Just because Allison is not a player on the volleyball team, we cannot conclude that she does not have sand in her toes. There could be other reasons for her to have sand in her toes.
# Answer:
Invalid

### 23.
# Brief Explanations:
The statement "All of the ballerinas are wearing tutus" is a conditional statement. If Hank is not wearing a tutu, by the contra - positive of the statement (if not $q$ then not $p$ where $p$ is being a ballerina and $q$ is wearing a tutu), we can conclude that he is not a ballerina.
# Answer:
Valid

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QUESTION IMAGE

Question

Explanation:

16.

Step1: Let the angle be $x$.

Step2: Expand the left - hand side.

Step3: Isolate the variable term.

Step4: Solve for $x$.

Answer:

17.