Question

1. proving a theorem complete the two - column proof for the congruent supplements theorem. then write a paragraph proof. (see example 5.) given: ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. ∠1≅∠4 prove: ∠2≅∠3 statements 1. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. ∠1≅∠4 2. m∠1 + m∠2 = 180°, m∠3 + m∠4 = 180° 3. ____ = m∠3 + m∠4 4. m∠1 = m∠4 5. m∠1 + m∠2 = __ 6. m∠2 = m∠3 7. __ reasons 1. given 2. ________ 3. transitive property of equality 4. definition of congruent angles 5. substitution property of equality 6. ________ 7. ________ 21. writing explain why you do not use inductive reasoning when writing a proof. 22. 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: __ statements 1. ∠1≅∠2 ∠1 and ∠2 are supplementary. 2. m∠1 = m∠2 3. m∠1 + m∠2 = 180° 4. m∠1 + m∠1 = 180° 5. 2(m∠1) = 180° 6. m∠1 = 90° 7. m∠2 = 90° 8. __ reasons 1. given 2. definition of congruent angles 3. definition of supplementary angles 4. substitution property of equality 5. simplify. 6. division property of equality 7. transitive property of equality 8. __________

Explanation:

Step1: Recall definition of supplementary angles

If two angles are supplementary, the sum of their measures is 180°. So, since $\angle1$ and $\angle2$ are supplementary and $\angle3$ and $\angle4$ are supplementary, we have $m\angle1 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$. The reason for this step is the definition of supplementary angles.

Step2: Use transitive property

Since $m\angle1 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$, by the transitive property of equality, $m\angle1 + m\angle2=m\angle3 + m\angle4$.

Step3: Apply definition of congruent angles

Given $\angle1\cong\angle4$, by the definition of congruent angles, $m\angle1 = m\angle4$.

Step4: Use substitution property

Substitute $m\angle4$ with $m\angle1$ in the equation $m\angle1 + m\angle2=m\angle3 + m\angle4$. We get $m\angle1 + m\angle2=m\angle3 + m\angle1$.

Step5: Subtract $m\angle1$ from both sides

Using the subtraction - property of equality, if $m\angle1 + m\angle2=m\angle3 + m\angle1$, then $m\angle2=m\angle3$.

Step6: Recall definition of congruent angles

Since $m\angle2=m\angle3$, by the definition of congruent angles, $\angle2\cong\angle3$.

Inductive reasoning is based on patterns and observations. Proofs require deductive reasoning, which starts with known facts, postulates, and theorems and uses logical steps to reach a conclusion. Inductive reasoning cannot guarantee the truth of a statement for all cases, while a proof must be valid for every possible case.

By looking at the given statements and the steps of the proof, we can see that the goal is to find the measure of the angles. Since we start with $\angle1\cong\angle2$ and they are supplementary, we are trying to prove that $\angle1 = 90^{\circ}$ and $\angle2 = 90^{\circ}$, i.e., that $\angle1$ and $\angle2$ are right - angles.

Answer:

1. Definition of supplementary angles
2. $m\angle1 + m\angle2$
3. $m\angle3 + m\angle1$
4. Subtraction property of equality
5. $\angle2\cong\angle3$; Definition of congruent angles

For question 21: