2. (7 pts) prove: if (a - b - c), (b - c - d), and (a - b - d), then (a - c - d). note: this can actually be done without drawing any pictures, just use the definition and algebra! 3. (5 pts) prove: if (m) is the mid - point of (overline{ab}), then (am=\frac{1}{2}ab = mb). moral of the story: we take definitions at face value; in other words, we dont read any extra information into them. in particular, the above statement is not quite the same as the definition of mid - point in the notes; indeed, the above statement has more information in it than the definition of mid - point. and even though the above statement is “obvious”, it must be proven. 4. (10 pts) the diagram below is given, where (l) is the mid - point of (overline{km}). prove that (nl=\frac{nk + nm}{2}). hint: what to do here? certainly, its going to be all about equations and algebra, but how could you possibly arrive at the strange equation you need? in your scratchwork, write down the equation you need to prove, and start manipulating it, substitute stuff, simplify things, etc.; this is the technique of “working backwards” that you should remember from intro to proofs: play with the conclusion until something occurs to you, then write the proof! just remember the actual proof starts with the given info, and your last line should be that strange equation. 5. (6 pts) prove the vertical angles theorem from the notes.

Question

Accepted Answer

### 2.
# Explanation:
## Step1: Recall the between - ness definition
If $A - B - C$, then $AB+BC = AC$; if $B - C - D$, then $BC + CD=BD$; if $A - B - D$, then $AB+BD = AD$.
## Step2: Substitute $BD=BC + CD$ into $AB + BD=AD$
We get $AB+(BC + CD)=AD$, which can be rewritten as $(AB + BC)+CD=AD$.
## Step3: Since $AB + BC = AC$
We have $AC+CD = AD$, so $A - C - D$ by the definition of between - ness.
# Answer:
Proved that if $A - B - C$, $B - C - D$, and $A - B - D$, then $A - C - D$.

### 3.
# Explanation:
## Step1: Let $M$ be the mid - point of $\overline{AB}$
By the definition of a mid - point, $AM=MB$ and $AB=AM + MB$.
## Step2: Substitute $MB = AM$ into $AB=AM + MB$
We get $AB=AM+AM = 2AM$, so $AM=\frac{1}{2}AB$. Also, since $AM = MB$, we have $AM=\frac{1}{2}AB=MB$.
# Answer:
Proved that if $M$ is the mid - point of $\overline{AB}$, then $AM=\frac{1}{2}AB = MB$.

### 4.
# Explanation:
## Step1: Since $L$ is the mid - point of $\overline{KM}$, we have $KL=LM$
And $NL=NK + KL$, $NM=NL+LM$.
## Step2: Express $LM$ in terms of other lengths
Since $LM = KL$, and $NM=NL + KL$, we can rewrite it as $KL=NM - NL$.
## Step3: Substitute $KL$ into $NL=NK + KL$
$NL=NK+(NM - NL)$.
## Step4: Rearrange the equation
Add $NL$ to both sides: $2NL=NK + NM$. Then $NL=\frac{NK + NM}{2}$.
# Answer:
Proved that $NL=\frac{NK + NM}{2}$.

### 5.
# Explanation:
## Step1: Let two lines $AB$ and $CD$ intersect at point $O$
Form two pairs of vertical angles $\angle AOC$ and $\angle BOD$, $\angle AOD$ and $\angle BOC$.
## Step2: Use the fact that $\angle AOC+\angle AOD = 180^{\circ}$ (linear - pair postulate) and $\angle BOD+\angle AOD=180^{\circ}$ (linear - pair postulate)
Since $\angle AOC+\angle AOD=\angle BOD+\angle AOD$.
## Step3: Subtract $\angle AOD$ from both sides
We get $\angle AOC=\angle BOD$. Similarly, we can prove $\angle AOD=\angle BOC$.
# Answer:
Proved the Vertical Angles Theorem.

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QUESTION IMAGE

Question

Explanation:

2.

Step1: Recall the between - ness definition

Step2: Substitute \(BD=BC + CD\) into \(AB + BD=AD\)

Step3: Since \(AB + BC = AC\)

Step1: Let \(M\) be the mid - point of \(\overline{AB}\)

Step2: Substitute \(MB = AM\) into \(AB=AM + MB\)

Step1: Since \(L\) is the mid - point of \(\overline{KM}\), we have \(KL=LM\)

Step2: Express \(LM\) in terms of other lengths

Step3: Substitute \(KL\) into \(NL=NK + KL\)

Step4: Rearrange the equation

Answer:

3.