Question

9.) select which goes in the corresponding space reason (4)
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 congruence
transitive property of equality
subtraction property of equality
definition of supplementary angles
substitution property of equality

Explanation:

Step1: Analyze the statements

We know that \(m\angle1 + m\angle2=180\), \(m\angle3 + m\angle4 = 180\) and \(m\angle1=m\angle4\). In statement 4, we replace \(180\) in the first - two equations with each other since they are both equal to \(180\), getting \(m\angle1 + m\angle2=m\angle3 + m\angle4\).

Step2: Recall the property

The substitution property of equality states that if \(a = b\), then \(a\) can be substituted for \(b\) in any equation or expression. Here, since \(m\angle1 + m\angle2\) and \(m\angle3 + m\angle4\) are both equal to \(180\), and \(m\angle1=m\angle4\), we use the substitution property to get \(m\angle1 + m\angle2=m\angle3 + m\angle4\).

Answer:

Substitution Property of Equality