1. given: o is the mid - point of mn, om = ow. prove: ow = on.
2. given: ab = cd. prove: ac = bd.
3. given: m∠1 = 90°. prove: m∠2 = 90°.
4. given: ∠1 and ∠2 are complementary, ∠3 and ∠2 are complementary. prove: m∠1 = m∠3.
5. given: m∠1 = m∠3. prove: m∠jol = m∠kom.
6. given: m∠1 = 90°. prove: m∠2+90 = 180.
7. given: pr≅ln, q is the mid - point of pr, m is the mid - point of ln. prove: pq = lm.
8. given: ef⊥eg, d is in the interior of ∠feg. prove: ∠fed and ∠deg are complementary.
9. given: ab≅cd. prove: ac≅bd.
10. given: ∠1 and ∠2 are supplementary, ∠1≅∠2. prove: ∠1 and ∠2 are right angles.
11. given: ∠1≅∠2. prove: ∠1 and ∠2 are right angles.
12. given: ∠1 and ∠2 are complementary. prove: ∠2 and ∠3 are complementary.

Question

Accepted Answer

1.
# Explanation:
## Step1: Definition of mid - point
Since O is the mid - point of $\overline{MN}$, by the definition of mid - point, $OM = ON$.
## Step2: Given condition
Given $OM = OW$.
## Step3: Transitive property
By the transitive property of equality (if $a = b$ and $b = c$, then $a = c$), since $OM = ON$ and $OM = OW$, we have $OW=ON$.
# Answer:
The proof is completed as above.

2.
# Explanation:
## Step1: Segment addition postulate
We know that $AC=AB + BC$ and $BD=CD + BC$.
## Step2: Given condition
Given $AB = CD$.
## Step3: Substitution
Substitute $AB$ for $CD$ in the equation for $BD$. So $BD=AB + BC$. Since $AC=AB + BC$ and $BD=AB + BC$, by the transitive property of equality, $AC = BD$.
# Answer:
The proof is completed as above.

3.
# Explanation:
## Step1: Vertical angles
$\angle1$ and $\angle2$ are vertical angles.
## Step2: Vertical - angle theorem
By the vertical - angle theorem, vertical angles are congruent. Since $m\angle1 = 90^{\circ}$ and $\angle1\cong\angle2$, then $m\angle2=90^{\circ}$.
# Answer:
The proof is completed as above.

4.
# Explanation:
## Step1: Complementary - angle definition
If $\angle1$ and $\angle2$ are complementary, then $m\angle1 + m\angle2=90^{\circ}$. If $\angle3$ and $\angle2$ are complementary, then $m\angle3 + m\angle2 = 90^{\circ}$.
## Step2: Solve for $m\angle1$ and $m\angle3$
From $m\angle1 + m\angle2=90^{\circ}$, we get $m\angle1=90^{\circ}-m\angle2$. From $m\angle3 + m\angle2 = 90^{\circ}$, we get $m\angle3=90^{\circ}-m\angle2$.
## Step3: Transitive property
By the transitive property of equality, since $m\angle1=90^{\circ}-m\angle2$ and $m\angle3=90^{\circ}-m\angle2$, then $m\angle1 = m\angle3$.
# Answer:
The proof is completed as above.

5.
# Explanation:
## Step1: Angle addition postulate
$m\angle JOL=m\angle1 + m\angle KOL$ and $m\angle KOM=m\angle3 + m\angle KOL$.
## Step2: Given condition
Given $m\angle1 = m\angle3$.
## Step3: Substitution
Substitute $m\angle1$ for $m\angle3$ in the equation for $m\angle KOM$. So $m\angle KOM=m\angle1 + m\angle KOL$. Since $m\angle JOL=m\angle1 + m\angle KOL$ and $m\angle KOM=m\angle1 + m\angle KOL$, by the transitive property of equality, $m\angle JOL=m\angle KOM$.
# Answer:
The proof is completed as above.

6.
# Explanation:
## Step1: Given condition
Given $m\angle1 = 90^{\circ}$.
## Step2: Angle - addition postulate
$\angle1$ and $\angle2$ are adjacent angles that form a straight - line. By the angle - addition postulate, $m\angle1+m\angle2 = 180^{\circ}$.
## Step3: Substitution
Substitute $m\angle1 = 90^{\circ}$ into $m\angle1+m\angle2 = 180^{\circ}$, we get $90^{\circ}+m\angle2 = 180^{\circ}$.
# Answer:
The proof is completed as above.

7.
# Explanation:
## Step1: Definition of mid - point
Since Q is the mid - point of $\overline{PR}$, then $PQ=\frac{1}{2}PR$. Since M is the mid - point of $\overline{LN}$, then $LM=\frac{1}{2}LN$.
## Step2: Given condition
Given $\overline{PR}\cong\overline{LN}$, so $PR = LN$.
## Step3: Substitution
Substitute $PR = LN$ into the equations for $PQ$ and $LM$. We have $PQ=\frac{1}{2}PR$ and $LM=\frac{1}{2}LN=\frac{1}{2}PR$. So $PQ = LM$.
# Answer:
The proof is completed as above.

8.
# Explanation:
## Step1: Definition of perpendicular lines
Since $\overline{EF}\perp\overline{EG}$, then $\angle FEG = 90^{\circ}$.
## Step2: Angle - addition postulate
$\angle FEG=\angle FED+\angle DEG$.
## Step3: Complementary - angle definition
Since $\angle FEG = 90^{\circ}$ and $\angle FEG=\angle FED+\angle DEG$, by the definition of complementary angles (two angles whose sum is $90^{\circ}$), $\angle FED$ and $\angle DEG$ are complementary.
# Answer:
The proof is completed as above.

9.
# Explanation:
## Step1: Segment addition postulate
$AC=AB + BC$ and $BD=CD + BC$.
## Step2: Given condition
Given $\overline{AB}\cong\overline{CD}$, so $AB = CD$.
## Step3: Substitution
Substitute $AB$ for $CD$ in the equation for $BD$. So $BD=AB + BC$. Since $AC=AB + BC$ and $BD=AB + BC$, by the transitive property of equality, $\overline{AC}\cong\overline{BD}$.
# Answer:
The proof is completed as above.

10.
# Explanation:
## Step1: Supplementary - angle definition
Since $\angle1$ and $\angle2$ are supplementary, $m\angle1 + m\angle2=180^{\circ}$.
## Step2: Given condition
Given $\angle1\cong\angle2$, so $m\angle1 = m\angle2$.
## Step3: Substitution
Substitute $m\angle1$ for $m\angle2$ in $m\angle1 + m\angle2=180^{\circ}$, we get $m\angle1+m\angle1 = 180^{\circ}$, or $2m\angle1 = 180^{\circ}$, then $m\angle1 = 90^{\circ}$. Since $m\angle1 = m\angle2$, $m\angle2 = 90^{\circ}$. So $\angle1$ and $\angle2$ are right angles.
# Answer:
The proof is completed as above.

11.
# Explanation:
## Step1: Given condition
Given $\angle1\cong\angle2$.
## Step2: Angle - addition postulate
$\angle1$ and $\angle2$ are adjacent angles that form a straight - line, so $m\angle1 + m\angle2=180^{\circ}$.
## Step3: Substitution
Substitute $m\angle1$ for $m\angle2$ in $m\angle1 + m\angle2=180^{\circ}$, we get $m\angle1+m\angle1 = 180^{\circ}$, or $2m\angle1 = 180^{\circ}$, then $m\angle1 = 90^{\circ}$. Since $m\angle1 = m\angle2$, $m\angle2 = 90^{\circ}$. So $\angle1$ and $\angle2$ are right angles.
# Answer:
The proof is completed as above.

12.
# Explanation:
There is not enough information given to prove that $\angle2$ and $\angle3$ are complementary. We only know that $\angle1$ and $\angle2$ are complementary, but there is no relation given between $\angle2$ and $\angle3$.
# Answer:
The proof cannot be completed with the given information.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Definition of mid - point

Step2: Given condition

Step3: Transitive property

Step1: Segment addition postulate

Step2: Given condition

Step3: Substitution

Step1: Vertical angles

Step2: Vertical - angle theorem

Answer: