Question

given: m∠acd = m∠bdc, m∠1 = m∠3 prove: m∠2 = m∠4 statements a. m∠acd = m∠bdc, m∠1 = m∠3 b. m<acd =m<1 +m<2 m<bcd = m<3 + m< 4 c. d. m<1+m<2 = m<3 + m<4 m<1+m<2 = m<1 + m<4 e. m<2 = m<4 f. reasons a. 18 b. 19 c. 20 d. 21 e. 22 f. 23

Explanation:

Step1: Given information

Given \(m\angle ACD = m\angle BDC\), \(m\angle1 = m\angle3\)

Step2: Angle - addition postulate

By the angle - addition postulate, \(m\angle ACD=m\angle1 + m\angle2\) and \(m\angle BDC=m\angle3 + m\angle4\)

Step3: Substitute equal - angle measures

Since \(m\angle ACD = m\angle BDC\), we substitute to get \(m\angle1 + m\angle2=m\angle3 + m\angle4\)

Step4: Substitute \(m\angle1 = m\angle3\)

Substitute \(m\angle1\) for \(m\angle3\) in the equation \(m\angle1 + m\angle2=m\angle3 + m\angle4\), we have \(m\angle1 + m\angle2=m\angle1 + m\angle4\)

Step5: Subtraction property of equality

Subtract \(m\angle1\) from both sides of the equation \(m\angle1 + m\angle2=m\angle1 + m\angle4\) to get \(m\angle2 = m\angle4\)

Step6: Conclusion

We have proven \(m\angle2 = m\angle4\)

Answer:

a. Given
b. Angle - addition postulate
c. \(m\angle ACD = m\angle BDC\)
d. Substitution property of equality
e. Substitution property of equality (using \(m\angle1 = m\angle3\))
f. Subtraction property of equality (subtracting \(m\angle1\) from both sides)