Question

1. proving a theorem copy and complete the two - column proof for the congruent supplement theorem (theorem 2.4). 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. ____

Explanation:

Step1: Recall supplementary - angle definition

If two angles are supplementary, the sum of their measures is 180°. So, since ∠1 and ∠2 are supplementary and ∠3 and ∠4 are supplementary, we have $m\angle1 + m\angle2=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$. The reason for this is the definition of supplementary angles.

Step2: Apply 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: Use congruent - angle definition

Given that ∠1≅∠4, by the definition of congruent angles, $m\angle1 = m\angle4$.

Step4: Substitute equal measures

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

Step5: Subtract equal measures

Subtract $m\angle1$ from both sides of the equation $m\angle1 + m\angle2=m\angle3 + m\angle1$. By the subtraction property of equality, we have $m\angle2=m\angle3$.

Step6: Use congruent - angle definition

Since $m\angle2 = m\angle3$, by the definition of congruent angles, ∠2≅∠3.

Answer:

STATEMENTS	REASONS
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. ∠2≅∠3	7. Definition of congruent angles

Paragraph proof: Given that ∠1 and ∠2 are supplementary, and ∠3 and ∠4 are supplementary, we know that $m\angle1 + m\angle2 = 180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$ by the definition of supplementary angles. By the transitive property of equality, $m\angle1 + m\angle2=m\angle3 + m\angle4$. Since ∠1≅∠4, then $m\angle1 = m\angle4$ by the definition of congruent angles. Substituting $m\angle4$ for $m\angle1$ in the equation $m\angle1 + m\angle2=m\angle3 + m\angle4$ gives $m\angle1 + m\angle2=m\angle3 + m\angle1$. Subtracting $m\angle1$ from both sides using the subtraction property of equality, we find that $m\angle2=m\angle3$. Then, by the definition of congruent angles, ∠2≅∠3.