11. given: $om + mp = op$. prove: $o - m - p$. a. transitive property b. betweenness of points c. addition property d. betweenness of rays e. definition of a segment bisector. 12. given: $angle1$ and $angle2$ are supplements of $angle3$. prove: $angle1congangle2$. a. transitive property b. if two angles are supplementary to the same angle or congruent angles, then they are congruent. c. definition of an angle bisector d. to every real number from 0 to 180, there corresponds exactly one ray. e. if two angles are complementary to the same angle or congruent angles, then they are congruent. 13. in the figure below, $angle sot$ and $angle tou$ are complementary. if $mangle sot = 2x + 4$ and $mangle tou = x - 16$, find $x$. 14. $angle abc$ and $angle def$ are supplementary. if $mangle abc = 2y$ and $mangle def = 3y - 5$, find $y$. use betweenness of points or rays to answer each question below. 15. if points $q$, $r$, and $s$ are collinear and if $rs = 15.7$, $rq = 9.1$, and $qs = 6.6$, which point is between the other two? a. $q$ is between $r$ and $s$. b. $r$ is between $q$ and $s$. c. $s$ is between $q$ and $r$. 16. if point $k$ lies in the interior of $angle jml$ and if $mangle kml = 24$, $mangle jml = 71$, and $mangle jmk = 47$, which of the following rays is between the other two: $overrightarrow{mk}$, $overrightarrow{mj}$, or $overrightarrow{ml}$? a. $overrightarrow{mk}$ is between $overrightarrow{mj}$ and $overrightarrow{ml}$. b. $overrightarrow{mj}$ is between $overrightarrow{mk}$ and $overrightarrow{ml}$. c. $overrightarrow{ml}$ is between $overrightarrow{mj}$ and $overrightarrow{mk}$. use the definition of a segment or angle bisector to answer each question below. 17. if $overrightarrow{pq}$ bisects $overrightarrow{cx}$ at point $j$, and if $jx = 4y$ and $cj = y + 9$, find $y$. 18. if $overrightarrow{ol}$ bisects $angle hop$, and if $mangle hol = 2x - 5$ and $mangle lop = x + 11$, find $mangle hop$. is $angle hop$ acute, right, or obtuse? a. $mangle hop = 27$; acute b. $mangle hop = 54$; acute c. $mangle hop = 27$; right d. $mangle hop = 54$; obtuse e. $mangle hop = 16$; acute select the set of conditional statements that represents each venn diagram below. 19. espresso drinkers. latte drinkers. linda. a. if a person is an espresso drinker, then he/she is a latte drinker. if linda is an espresso drinker, then she is a latte drinker. b. if a person is a latte drinker, then he/she is an espresso drinker. if linda is an espresso drinker, then she is a latte drinker.

Question

Accepted Answer

### 11.
# Explanation:
## Step1: Recall betweenness of points
If $OM + MP=OP$, by the definition of betweenness of points, point $M$ lies between $O$ and $P$, i.e., $O - M - P$.
# Answer:
B. Betweenness of Points

### 12.
# Explanation:
## Step1: Use angle - supplement property
If $\angle1$ and $\angle2$ are supplements of $\angle3$, then by the property "If two angles are supplementary to the same angle or congruent angles, then they are congruent", $\angle1\cong\angle2$.
# Answer:
B. If two angles are supplementary to the same angle or congruent angles, then they are congruent.

### 13.
# Explanation:
## Step1: Use complementary - angle property
Since $\angle SOT$ and $\angle TOU$ are complementary, $m\angle SOT + m\angle TOU=90^{\circ}$. So, $(2x + 4)+(x - 16)=90$.
## Step2: Simplify the equation
Combine like - terms: $2x+x+4 - 16 = 90$, which gives $3x-12 = 90$.
## Step3: Solve for $x$
Add 12 to both sides: $3x=90 + 12=102$. Then divide by 3: $x=\frac{102}{3}=34$.
# Answer:
$x = 34$

### 14.
# Explanation:
## Step1: Use supplementary - angle property
Since $\angle ABC$ and $\angle DEF$ are supplementary, $m\angle ABC+m\angle DEF = 180^{\circ}$. So, $2y+(3y - 5)=180$.
## Step2: Simplify the equation
Combine like - terms: $2y+3y-5 = 180$, which gives $5y-5 = 180$.
## Step3: Solve for $y$
Add 5 to both sides: $5y=180 + 5=185$. Then divide by 5: $y=\frac{185}{5}=37$.
# Answer:
$y = 37$

### 15.
# Explanation:
## Step1: Check the sum of lengths
We know that if $Q$, $R$, and $S$ are collinear, and $RQ+QS=RS$ (since $9.1 + 6.6=15.7$), then $Q$ is between $R$ and $S$.
# Answer:
A. $Q$ is between $R$ and $S$

### 16.
# Explanation:
## Step1: Check angle measures
Since $m\angle KML=24$, $m\angle JML = 71$, and $m\angle JMK=47$, and $m\angle JMK+m\angle KML=m\angle JML$ ($47 + 24=71$), $\overrightarrow{MK}$ is between $\overrightarrow{MJ}$ and $\overrightarrow{ML}$.
# Answer:
A. $\overrightarrow{MK}$ is between $\overrightarrow{MJ}$ and $\overrightarrow{ML}$

### 17.
# Explanation:
## Step1: Use the definition of a segment bisector
If $\overline{PQ}$ bisects $\overline{CX}$ at point $J$, then $CJ=JX$. So, $y + 9=4y$.
## Step2: Solve for $y$
Subtract $y$ from both sides: $9=4y - y=3y$. Then divide by 3: $y = 3$.
# Answer:
$y = 3$

### 18.
# Explanation:
## Step1: Use the definition of an angle bisector
If $\overrightarrow{OL}$ bisects $\angle HOP$, then $m\angle HOL=m\angle LOP$. So, $2x-5=x + 11$.
## Step2: Solve for $x$
Subtract $x$ from both sides: $2x-x-5=x-x + 11$, which gives $x-5 = 11$. Add 5 to both sides: $x=11 + 5=16$.
## Step3: Find $m\angle HOP$
$m\angle HOP=m\angle HOL+m\angle LOP=(2x - 5)+(x + 11)$. Substitute $x = 16$: $m\angle HOP=(2\times16-5)+(16 + 11)=(32-5)+27=27+27 = 54^{\circ}$. Since $0^{\circ}<54^{\circ}<90^{\circ}$, $\angle HOP$ is acute.
# Answer:
B. $m\angle HOP = 54$; acute

### 19.
# Explanation:
## Step1: Analyze the Venn - diagram relationship
All latte drinkers are espresso drinkers, and Linda is a latte drinker. So, the correct conditional statements are: If a person is a latte drinker, then he/she is an espresso drinker. If Linda is a latte drinker, then she is an espresso drinker.
# Answer:
B. If a person is a latte drinker, then he/she is an espresso drinker. If Linda is a latte drinker, then she is an espresso drinker.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

11.

Step1: Recall betweenness of points

Step1: Use angle - supplement property

Step1: Use complementary - angle property

Step2: Simplify the equation

Step3: Solve for \(x\)

Answer:

12.