Complete the paragraph proof.
Given: <1 and 2 are supplementary, and <2 and <3 are supplementary.
Prove: 1 =~ 3
Statement 1: <1 and <2 are supplementary Given
Statement 2: <2 and <3 are supplementary Given
Statement 3: <1 + <2 = 180° Definition of supplementary angles
Statement 4: <2 + <3 = 180° Definition of supplementary angles
Statement 5: (<1 + <2) + <3 = 180° Transitive property of equality (Statement 3 and Statement 4)
Statement 6: <1 + (<2 + <3) = 180° Associative property of addition (Statement 5)
Statement 7: <1 + 180° = 180° Substitution (Statement 6)
Statement 8: <1 = 0° Addition property of equality (Statement 7)
Statement 9: <3 = 180° - <2 Subtraction property of equality (Statement 4)
Statement 10: <3 + <2 = (180° - <2) + <2 Addition property of equality (Statement 9)
Statement 11: <3 + <2 = 180° Subtraction property of equality (Statement 10)
Statement 12: 3 =~ 1 Definition of congruent angles (Statement 8 and Statement 11)
To prove that angle 1 is congruent to angle 3, we can use the Transitive Property of Equality. This property states that if two angles are congruent to a third angle, then they are congruent to each other.
Here's the step-by-step proof:
1. Given: angle 1 and angle 2 are supplementary (1 + 2 = 180°)
2. Given: angle 2 and angle 3 are supplementary (2 + 3 = 180°)
3. We know that supplementary angles add up to 180 degrees.
4. Substituting the values from step 1 and step 2, we have (1 + 2) + (2 + 3) = 180°.
5. Simplifying the equation, we get 3 + 5 = 180°.
6. Combining like terms, we have 8 = 180°.
7. By subtracting 5 from both sides of the equation, we get 3 = 175°.
8. Since angle 3 is equal to 175°, and angle 1 plus angle 2 equals 180°, angle 1 must be equal to 5°.
9. Therefore, angle 1 is congruent to angle 3 (1 =~ 3).
This completes the paragraph proof that angle 1 is congruent to angle 3.