Write a flow proof:if angle 1 and angle 2 are vertical angles, and angle 2 and angle 3 are complementary, then angle 1 and angle 3 are complementary.
Sure! Here's a flow proof for the given statement:
Statement | Reason
--------------------------------------------------------|-------------------------------------------------
1. Angle 1 and Angle 2 are vertical angles | Given
2. Angle 2 and Angle 3 are complementary | Given
3. Angle 1 and Angle 2 share a common ray | Definition of Vertical Angles
4. Angle 3 and Angle 2 share a common ray | Definition of Complementary Angles
5. Angle 1, Angle 2, and Angle 3 form a straight line | Linear Pair Postulate
6. Angle 1 and Angle 2 are supplementary | Definition of a Straight Line
7. Angle 2 and Angle 3 are supplementary | Definition of Complementary Angles
8. Angle 1 and Angle 2 have a sum of 180 degrees | Definition of Supplementary Angles
9. Angle 2 and Angle 3 have a sum of 90 degrees | Definition of Complementary Angles
10. Angle 1 and Angle 3 have a sum of 90 degrees | Transitive Property of Equality
11. Angle 1 and Angle 3 are complementary | Definition of Complementary Angles
In this flow proof, we start by stating that Angle 1 and Angle 2 are vertical angles (given). Then we state that Angle 2 and Angle 3 are complementary (given). From the definition of vertical angles, we can conclude that Angle 1 and Angle 2 share a common ray (Statement 3). Similarly, from the definition of complementary angles, we can conclude that Angle 3 and Angle 2 share a common ray (Statement 4).
Next, we use the fact that Angle 1, Angle 2, and Angle 3 form a straight line (Linear Pair Postulate) to conclude that Angle 1 and Angle 2 are supplementary (Definition of a Straight Line). Similarly, Angle 2 and Angle 3 are supplementary (Definition of Complementary Angles) (Statements 6 and 7).
Using the definition of supplementary angles, we state that Angle 1 and Angle 2 have a sum of 180 degrees (Definition of Supplementary Angles). Similarly, Angle 2 and Angle 3 have a sum of 90 degrees (Definition of Complementary Angles) (Statements 8 and 9).
Finally, using the transitive property of equality, we can conclude that Angle 1 and Angle 3 have a sum of 90 degrees (Statement 10). Hence, we can say that Angle 1 and Angle 3 are complementary (Definition of Complementary Angles) (Statement 11).
This completes the flow proof showing that if Angle 1 and Angle 2 are vertical angles and Angle 2 and Angle 3 are complementary, then Angle 1 and Angle 3 are complementary.