20. proving a theorem complete the two - column proof for the congruent supplements theorem. then write a paragraph proof. example 3
given ∠1 and ∠2 are supplementary.
∠3 and ∠4 are supplementary.
∠1 ≅ ∠4
prove ∠2 ≅ ∠3

statements | reasons
1. ∠1 and ∠2 are supplementary, ∠3 and ∠4 are supplementary, ∠1 ≅ ∠4 | 1. given
2. m∠1 + m∠2 = 180°, m∠3 + m∠4 = 180° | 2. ______
3. ______ = m∠3 + m∠4 | 3. transitive property of equality
4. m∠1 = m∠4 | 4. definition of congruent angles
5. m∠1 + m∠2 = ______ | 5. substitution property of equality
6. m∠2 = m∠3 | 6. ______
7. ______ | 7. ______

21. writing explain why you do not use inductive reasoning when writing a proof.

. how do you see it?
consider the two - column proof. what is the writer trying to prove?
given ∠1 ≅ ∠2
∠1 and ∠2 are supplementary.
prove ________

Question

20. proving a theorem complete the two - column proof for the congruent supplements theorem. then write a paragraph proof. example 3
given ∠1 and ∠2 are supplementary.
∠3 and ∠4 are supplementary.
∠1 ≅ ∠4
prove ∠2 ≅ ∠3

21. writing explain why you do not use inductive reasoning when writing a proof.

. how do you see it?
consider the two - column proof. what is the writer trying to prove?
given ∠1 ≅ ∠2
∠1 and ∠2 are supplementary.
prove ________

Accepted Answer

### Problem 20 (Two - column proof for Congruent Supplements Theorem)

#### Step 1: Reason for Statement 2
The definition of supplementary angles states that if two angles are supplementary, the sum of their measures is $180^{\circ}$. So for $\angle1$ and $\angle2$ being supplementary, $m\angle1 + m\angle2=180^{\circ}$, and for $\angle3$ and $\angle4$ being supplementary, $m\angle3 + m\angle4 = 180^{\circ}$. So the reason for statement 2 is "Definition of supplementary angles".

#### Step 2: Statement 3
From step 2, we have $m\angle1 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$. By the Transitive Property of Equality (since both equal $180^{\circ}$), we can say $m\angle1 + m\angle2=m\angle3 + m\angle4$.

#### Step 3: Statement 5
We know from step 2 that $m\angle3 + m\angle4 = 180^{\circ}$, and from the Substitution Property of Equality (since $m\angle1 + m\angle2=m\angle3 + m\angle4$), we substitute $m\angle3 + m\angle4$ with $180^{\circ}$ in the equation $m\angle1 + m\angle2=m\angle3 + m\angle4$ to get $m\angle1 + m\angle2 = 180^{\circ}$ (wait, actually, we can also use the fact that $m\angle1=m\angle4$ from step 4. Let's correct this. Since $m\angle1 + m\angle2=180^{\circ}$ and $m\angle1=m\angle4$, by substitution, we can also look at the equation $m\angle1 + m\angle2=m\angle3 + m\angle4$. Substitute $m\angle1$ with $m\angle4$ in $m\angle1 + m\angle2$, we get $m\angle4 + m\angle2$. But also, $m\angle3 + m\angle4=180^{\circ}$ and $m\angle1 + m\angle2 = 180^{\circ}$. Since $m\angle1=m\angle4$, then $m\angle1 + m\angle2=m\angle4 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$. So $m\angle4 + m\angle2=m\angle3 + m\angle4$. Then, by Subtraction Property of Equality (subtract $m\angle4$ from both sides), we get $m\angle2=m\angle3$. But let's follow the two - column proof:

- Statement 5: Since $m\angle1 + m\angle2 = 180^{\circ}$ (from step 2) and $m\angle3 + m\angle4=180^{\circ}$ (from step 2) and $m\angle1=m\angle4$ (from step 4), by substitution, $m\angle1 + m\angle2=m\angle3 + m\angle1$ (substituting $m\angle4$ with $m\angle1$ in $m\angle3 + m\angle4$). Wait, no, the correct substitution is: from $m\angle1 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$ and $m\angle1=m\angle4$, so $m\angle1 + m\angle2=m\angle3 + m\angle1$ (because $m\angle4 = m\angle1$). Then, by Subtraction Property of Equality (subtract $m\angle1$ from both sides), we get $m\angle2=m\angle3$. But in the given two - column proof, statement 5 is $m\angle1 + m\angle2=\underline{180^{\circ}}$ (wait, no, let's re - examine the table:

Looking at the table:

- Statement 2: $m\angle1 + m\angle2 = 180^{\circ}$, $m\angle3 + m\angle4=180^{\circ}$
- Statement 3: $\underline{m\angle1 + m\angle2}=m\angle3 + m\angle4$ (by Transitive Property, since both equal $180^{\circ}$)
- Statement 4: $m\angle1=m\angle4$ (Definition of congruent angles, since $\angle1\cong\angle4$)
- Statement 5: $m\angle1 + m\angle2=\underline{m\angle3 + m\angle1}$ (by Substitution Property, substituting $m\angle4$ with $m\angle1$ in $m\angle3 + m\angle4$)
- Statement 6: $m\angle2=m\angle3$ (by Subtraction Property of Equality, subtract $m\angle1$ from both sides of $m\angle1 + m\angle2=m\angle3 + m\angle1$)
- Statement 7: $\angle2\cong\angle3$ (by Definition of congruent angles, since $m\angle2 = m\angle3$)

And the reasons:

- Reason 2: Definition of supplementary angles
- Reason 6: Subtraction Property of Equality
- Reason 7: Definition of congruent angles

### Problem 21: Why no inductive reasoning in proof?

# Brief Explanations:
Inductive reasoning is based on observing patterns in specific cases (e.g., looking at several examples and generalizing). However, a mathematical proof requires a general, deductive argument that shows a statement is true for all cases (not just observed ones). For example, if we use inductive reasoning to prove "all triangles have sum of interior angles $180^{\circ}$", we might measure a few triangles, but this doesn't prove it for every triangle. A deductive proof (using postulates, theorems) shows it for any triangle. So in writing a proof, we need a logical, deductive chain (using definitions, postulates, theorems) that is valid for all instances, not just observed ones.
# Answer:
Inductive reasoning generalizes from specific cases (patterns in examples) but doesn't guarantee truth for all cases. Proofs need deductive reasoning (using definitions, postulates, theorems) to show a statement is true for all instances, not just observed ones.

### "How Do You See It?" Problem

Given $\angle1\cong\angle2$ and $\angle1$ and $\angle2$ are supplementary. Let's find what to prove.

If $\angle1\cong\angle2$, then $m\angle1 = m\angle2$ (Definition of congruent angles). If $\angle1$ and $\angle2$ are supplementary, then $m\angle1 + m\angle2=180^{\circ}$ (Definition of supplementary angles). Substitute $m\angle2$ with $m\angle1$ (since $m\angle1 = m\angle2$): $m\angle1+m\angle1 = 180^{\circ}$, $2m\angle1 = 180^{\circ}$, $m\angle1 = 90^{\circ}$. So $m\angle2=90^{\circ}$ too. So $\angle1$ and $\angle2$ are right angles. So we need to prove that $\angle1$ and $\angle2$ are right angles (or $\angle1\cong90^{\circ}$, $\angle2\cong90^{\circ}$).

# Brief Explanations:
1. From $\angle1\cong\angle2$, $m\angle1 = m\angle2$ (def. of congruent angles).
2. From $\angle1$ and $\angle2$ supplementary, $m\angle1 + m\angle2 = 180^{\circ}$ (def. of supplementary angles).
3. Substitute $m\angle2$ with $m\angle1$: $2m\angle1=180^{\circ}\Rightarrow m\angle1 = 90^{\circ}$, so $m\angle2 = 90^{\circ}$. Thus, $\angle1$ and $\angle2$ are right angles.
# Answer:
$\angle1$ and $\angle2$ are right angles (or $\angle1\cong90^{\circ}$, $\angle2\cong90^{\circ}$)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 20 (Two - column proof for Congruent Supplements Theorem)

Step 1: Reason for Statement 2

Step 2: Statement 3

Step 3: Statement 5

Answer:

"How Do You See It?" Problem