Question

given: <1 & <2, <3 & <4 are supplements. <1 ~= <4
prove: <2 ~= <3
statements reasons

1. <1 & <2, <3 & <4 are supplements and <1 ~= <4 1) given
2. m<1+m<2=180 m<3+m<4=180 2) definition of supplementary angles
3. m<1=m<4 3) definition of congruence
4. m<1+m<2=m<3+m<4 4)

5)m<1+m<2=m<3+m<1 5)
6)m<2=m<3 6)
7)<2~=<3 7) definition of congruency
transitive property of equality
definition of supplementary angles
subtraction property of equality
substitution property of equality

Explanation:

Step1: Analyze step 4

We know from step 2 that \(m\angle1 + m\angle2=180\) and \(m\angle3 + m\angle4 = 180\), and from step 3 \(m\angle1=m\angle4\). So, by the substitution property of equality, we can substitute \(m\angle4\) for \(m\angle1\) in the second - equality related to supplementary angles, getting \(m\angle1 + m\angle2=m\angle3 + m\angle4\). So, the reason for step 4 is the Substitution Property of Equality.

Step2: Analyze step 5

Since \(m\angle1=m\angle4\) (from step 3), we substitute \(m\angle1\) for \(m\angle4\) in the equation \(m\angle1 + m\angle2=m\angle3 + m\angle4\) to get \(m\angle1 + m\angle2=m\angle3 + m\angle1\). The reason for step 5 is the Substitution Property of Equality.

Step3: Analyze step 6

We start with the equation \(m\angle1 + m\angle2=m\angle3 + m\angle1\). By the subtraction property of equality, if we subtract \(m\angle1\) from both sides of the equation (\(m\angle1 + m\angle2−m\angle1=m\angle3 + m\angle1−m\angle1\)), we get \(m\angle2=m\angle3\).

Answer:

Subtraction Property of Equality