Question

1. proving a theorem complete the two - column proof for the congruent supplements theorem. then write a paragraph proof. example 5

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. _____________

Explanation:

Step1: Reason for Step 2

The definition of supplementary angles states that if two angles are supplementary, then 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 step 2 is "Definition of supplementary angles".

Step2: Statement for Step 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 (if \(a = c\) and \(b = c\), then \(a=b\)), since both \(m\angle1 + m\angle2\) and \(m\angle3 + m\angle4\) equal \(180^{\circ}\), we can say \(m\angle1 + m\angle2=m\angle3 + m\angle4\). So the statement for step 3 is \(m\angle1 + m\angle2\).

Step3: Statement for Step 5

From step 4, we know \(m\angle1=m\angle4\). From step 3, we have \(m\angle1 + m\angle2=m\angle3 + m\angle4\). Using the Substitution Property of Equality (replace \(m\angle1\) with \(m\angle4\) in the left - hand side or \(m\angle4\) with \(m\angle1\) in the right - hand side), we substitute \(m\angle4\) with \(m\angle1\) in \(m\angle3 + m\angle4\) or \(m\angle1\) with \(m\angle4\) in \(m\angle1 + m\angle2\). So \(m\angle1 + m\angle2=m\angle3 + m\angle1\) (after substitution). Wait, actually, from step 3: \(m\angle1 + m\angle2=m\angle3 + m\angle4\), and step 4: \(m\angle1 = m\angle4\), so substitute \(m\angle4\) with \(m\angle1\) in the right - hand side, we get \(m\angle1 + m\angle2=m\angle3 + m\angle1\). But we can also think of it as from step 2, \(m\angle3 + m\angle4 = 180^{\circ}\), and since \(m\angle1=m\angle4\), then \(m\angle3 + m\angle1=180^{\circ}\), and from step 2, \(m\angle1 + m\angle2 = 180^{\circ}\). So by substitution, \(m\angle1 + m\angle2=m\angle3 + m\angle1\) (or \(m\angle1 + m\angle2 = 180^{\circ}\) can be written as \(m\angle1 + m\angle2=m\angle3 + m\angle4\) from step 3, and then substitute \(m\angle4\) with \(m\angle1\) to get \(m\angle1 + m\angle2=m\angle3 + m\angle1\)). But the more straightforward way is that from step 3: \(m\angle1 + m\angle2=m\angle3 + m\angle4\), and step 4: \(m\angle1 = m\angle4\), so substitute \(m\angle4\) with \(m\angle1\) in the right - hand side, so \(m\angle1 + m\angle2=m\angle3 + m\angle1\). But we can also say that since \(m\angle1 + m\angle2 = 180^{\circ}\) (step 2) and \(m\angle3 + m\angle4=180^{\circ}\) (step 2) and \(m\angle1 = m\angle4\) (step 4), then \(m\angle1 + m\angle2=m\angle3 + m\angle1\) (by substituting \(m\angle4\) with \(m\angle1\) in \(m\angle3 + m\angle4\)). So the statement for step 5 is \(m\angle3 + m\angle1\) (or \(m\angle3 + m\angle4\) can be replaced with \(m\angle1 + m\angle2\) as well, but following the substitution, since \(m\angle1=m\angle4\), we substitute \(m\angle4\) in \(m\angle3 + m\angle4\) with \(m\angle1\), so \(m\angle1 + m\angle2=m\angle3 + m\angle1\)).

Step4: Reason for Step 6

From step 5, we have \(m\angle1 + m\angle2=m\angle3 + m\angle1\). If we subtract \(m\angle1\) from both sides of the equation (Subtraction Property of Equality: if \(a + b=a + c\), then \(b = c\)), we get \(m\angle2=m\angle3\). So the reason for step 6 is "Subtraction Property of Equality".

Step5: Statement and Reason for Step 7

If \(m\angle2=m\angle3\), then by the definition of congruent angles (if the measures of two angles are equal, then the angles are congruent), we can say \(\angle2\cong\angle3\). So the statement for step 7 is \(\angle2\cong\angle3\) and the reason is "Definition of congruent angles".

Filling in the blanks:

- Step 2 Reason: Definition of supplementary angles
- Step 3 Statement: \(m\angle1 + m\angle2\)
- Step 5 Statement: \(m\angle3 + m\angle1\) (or \(m\angle3 + m\angle4\) can be considered, but more accurately \(m\angle3 + m\angle1\) after substitution)
- Step 6 Reason: Subtraction Property of Equality
- Step 7 Statement: \(\angle2\cong\angle3\); Reason: Definition of congruent angles

For the two - column proof:

| STATEMENTS | REASONS |
| --- | --- |
| 1. \(\angle1\) and \(\angle2\) are supplementary. \(\angle3\) and \(\angle4\) are supplementary. \(\angle1\cong\angle4\) | 1. Given |
| 2. \(m\angle1 + m\angle2 = 180^{\circ}\), \(m\angle3 + m\angle4=180^{\circ}\) | 2. Definition of supplementary angles |
| 3. \(m\angle1 + m\angle2=m\angle3 + m\angle4\) | 3. Transitive Property of Equality |
| 4. \(m\angle1 = m\angle4\) | 4. Definition of congruent angles |
| 5. \(m\angle1 + m\angle2=m\angle3 + m\angle1\) | 5. Substitution Property of Equality |
| 6. \(m\angle2 = m\angle3\) | 6. Subtraction Property of Equality |
| 7. \(\angle2\cong\angle3\) | 7. Definition of congruent angles |

Paragraph Proof:

We are given that \(\angle1\) and \(\angle2\) are supplementary, \(\angle3\) and \(\angle4\) are supplementary, and \(\angle1\cong\angle4\). By the definition of supplementary angles, \(m\angle1 + m\angle2 = 180^{\circ}\) and \(m\angle3 + m\angle4=180^{\circ}\). Using the Transitive Property of Equality, since both \(m\angle1 + m\angle2\) and \(m\angle3 + m\angle4\) equal \(180^{\circ}\), we have \(m\angle1 + m\angle2=m\angle3 + m\angle4\). By the definition of congruent angles, \(m\angle1 = m\angle4\). Using the Substitution Property of Equality, we substitute \(m\angle4\) with \(m\angle1\) in the equation \(m\angle1 + m\angle2=m\angle3 + m\angle4\) to get \(m\angle1 + m\angle2=m\angle3 + m\angle1\). Then, by the Subtraction Property of Equality, we subtract \(m\angle1\) from both sides of the equation, which gives \(m\angle2 = m\angle3\). Finally, by the definition of congruent angles, since \(m\angle2 = m\angle3\), we conclude that \(\angle2\cong\angle3\).

Answer:

Two - column proof blanks:

• Step 2: Definition of supplementary angles
• Step 3: \(m\angle1 + m\angle2\)
• Step 5: \(m\angle3 + m\angle1\) (or equivalent after substitution)
• Step 6: Subtraction Property of Equality
• Step 7: \(\angle2\cong\angle3\); Definition of congruent angles

Paragraph Proof:

We are given that \(\angle1\) and \(\angle2\) are supplementary, \(\angle3\) and \(\angle4\) are supplementary, and \(\angle1\cong\angle4\). By the definition of supplementary angles, \(m\angle1 + m\angle2 = 180^{\circ}\) and \(m\angle3 + m\angle4=180^{\circ}\). Using the Transitive Property of Equality, \(m\angle1 + m\angle2=m\angle3 + m\angle4\). By the definition of congruent angles, \(m\angle1 = m\angle4\). Using the Substitution Property of Equality, substitute \(m\angle4\) with \(m\angle1\) to get \(m\angle1 + m\angle2=m\angle3 + m\angle1\). By the Subtraction Property of Equality, subtract \(m\angle1\) from both sides to get \(m\angle2 = m\angle3\). By the definition of congruent angles, \(\angle2\cong\angle3\).