Write a paragraph proof. Given: AC bisects <DAB and <DCB . Prove: DAC=BAC?
t7t7
To prove that DAC = BAC, we can use the given information that AC bisects angles DAB and DCB. In a paragraph proof, we need to provide a step-by-step explanation showing how we arrive at the conclusion.
Here's a paragraph proof for the given statement:
Given: AC bisects <DAB and <DCB
To Prove: DAC = BAC
Proof:
1. AC bisects <DAB and <DCB (Given)
2. m<DAC = m<DAB/2 (Definition of angle bisector)
3. m<BAC = m<DCB/2 (Definition of angle bisector)
4. m<DAC + m<BAC = m<DAB/2 + m<DCB/2 (Adding equations 2 and 3)
5. m<DAC + m<BAC = (m<DAB + m<DCB)/2 (Combining like terms)
6. 2(m<DAC + m<BAC) = m<DAB + m<DCB (Multiplying both sides by 2)
7. 2m<DAC + 2m<BAC = m<DAB + m<DCB (Distributive property)
8. m<DAC + m<DAC + m<BAC + m<BAC = m<DAB + m<DCB (Rearranging terms)
9. 2m<DAC + 2m<BAC = 2m<DAB + 2m<DCB (Equal angles have equal measures)
10. 2m<DAC = 2m<DAB (Subtracting 2m<BAC from both sides)
11. m<DAC = m<DAB (Dividing both sides by 2)
12. DAC = DAB (By angle congruence)
13. m<BAC = m<DCB/2 (Given)
14. m<BAC = m<DCB/2 (Dividing both sides by 2)
15. m<BAC = m<DCB (By substitution)
16. BAC = DCB (By angle congruence)
17. DAC = BAC (By transitive property of equality)
Thus, the proof shows that DAC = BAC using the given information and a step-by-step logical reasoning based on angles bisector definition and congruence properties.