Question

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Provide reasons for the proof.

Given: angle 2=4
and 2 and 3 are supplementary
Prove: angle 1=3

Statement: Reason
1. 2=4 1. Given
2. m2=m4 2. Angle congruence postulate
3. angles 2 and 3 are supplementary 3. Given
4. m2+m3=180 4. Definition of supplementary angles
5. angles 1 and 4 are supplementary 5. Fill in blank
6. m1+m4=180 6. Definition of supplementary angles
7. m1+m4=m2+m3 7. Fill in blank
8. m1+m4=m4+m3 8. Fill in blank
9. m1=m3 9. Fill in blank
10. 1=3 10. Fill in blank

Answer 1

Statement 1: ∠2=∠4 .. Given

Statement 2: Measure of angle 2 = Measure of angle 4 = Alternate angles

Statement 3: ∠2=∠3 , Given Supplementary Angles

Statement 4 : ∠2+∠3=180 , Sum of supplementary angles is equal to 180 degree.

Statement 5: ∠1 and ∠4 are supplementary angles because angle of a straight line is equal to 180 degree.

Statement 6: Measure of angle 1 + measure of angle 4 = 180, sum of angles of supplementary angles is 180 degree.

Statement 7:  ∠1+∠4= ∠2+∠3 ... Both sums are 180 degrees.

Statement 8: ∠1+∠4=∠4+∠3 .. ∠2 and ∠4 are equal (alternate angles)

Statement 9: ∠1= ∠3 , because ∠4 is common on both sides.

Statement 10: ∠1 = ∠3 .. hence, proved

Answer 2

Definitions, postulates, and theorems are defined relationships, between the angles, that are used to determine the values of the angles

The two column proof is presented as follows;

Statement ${}$                                      Reason

1. 2 = 4   ${}$                                           1. Given

2. m∠2 = m∠4 ${}$                                 2. Angle congruence postulate

3. ∠2 and ∠ 3 are supp ∠s ${}$             3. Given

4. m∠2 + m∠3 = 180°  ${}$                     4. Definition Supplementary angles

5. ∠1 and 4 are supplementary   ${}$    5. Linear pair angles theorem

6. m∠1 + m∠4 = 180°  ${}$                      6. Definition of supplementary angles

7. m∠1 + m∠4 = m∠2 + m∠3   ${}$         7. Transitive property of equality

8. m∠1 + m∠4 = m∠4 + m∠3   ${}$         8. Transitive property of equality

9. m∠1 = m∠3                 ${}$                   9. Addition property of equality

10. m∠1 = m∠3                 ${}$                 10. Definition of equality

Reasons:

• Linear pair angles theorem states that together form a straight line are supplementary (they sum up to 180°)

• Transitive property of equality; If two variables are each equal to a third variable, then the two variables are equal to each other

• Addition property; Where two variables are equal then the values obtained by adding the same quantities to both variables are also equal

• Definition of equality; Two quantities are equal when their measures are equal

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