Question

1. given that m∠1 = 50° and m∠2 = m∠1, find m∠2. which property allows this? write a short explanation.
2. in a figure, you know ab ≅ cd, cd ≅ ef, and ef ≅ gh. using transitive and substitution, what can you conclude about ab and gh?

Explanation:

8.

Step1: Substitute the value of ∠1

Since \(m\angle2 = m\angle1\) and \(m\angle1 = 50^{\circ}\), we substitute the value of \(m\angle1\) into the equation for \(m\angle2\).
\(m\angle2=50^{\circ}\)

Step2: Identify the property

The property that allows this is the substitution property. It states that if \(a = b\), then \(b\) can be substituted for \(a\) in any equation or expression. Here \(a = m\angle1\) and \(b = m\angle2\).

Step1: Apply the transitive property

The transitive property of congruence states that if \(a\cong b\) and \(b\cong c\), then \(a\cong c\). Given \(AB\cong CD\), \(CD\cong EF\), by the transitive - property, \(AB\cong EF\).
Since \(EF\cong GH\) and we just found \(AB\cong EF\), applying the transitive property again, we get \(AB\cong GH\).

Step2: Justify with substitution (optional)

We can also think of it in terms of substitution. If we consider the congruence statements as equal - length statements (since congruent segments have equal lengths), we can substitute the lengths of the segments in the equations. For example, if \(l(AB)=l(CD)\), \(l(CD)=l(EF)\) and \(l(EF)=l(GH)\), we can substitute and get \(l(AB)=l(GH)\) which means \(AB\cong GH\).

Answer:

\(m\angle2 = 50^{\circ}\), Substitution property

9.