Question

extra practice level up your skills! mixed proofs proof write the correct statements and reasons to complete the two - column proof. given: \\( \overline { m p } \cong \overline { l p } \\) \\( \overline { ln } \cong \overline { m o } \\) prove: \\( p n = p o \\) proof: 1. \\( \overline { m p } \cong \overline { l p } \\) \\( \overline { ln } \cong \overline { m o } \\) 2. 3. \\( m o = m p + p o \\) \\( l n = l p + p n \\) 4. \\( l p + p n = m p + p o \\) 5. \\( l p + p n = l p + p o \\) 6. basket of choices substitution property segment addition postulate \\( p n = p o \\) \\( m p = l p, l n = m o \\) given substitution property

Explanation:

Step1: Reasons for Statement 1

The first statement is given in the problem. So the reason for statement 1 is "Given".

Step2: Statements for Reason 2

Since the reason is "Definition of ≅ segments", the statement should be \(MP = LP\) and \(LN=MO\).

Step3: Reasons for Statement 3

The Segment Addition Postulate states that if \(A - B - C\), then \(AC=AB + BC\). So the reason for statement 3 is "Segment Addition Postulate".

Step4: Reasons for Statement 4

Since \(LN = MP+PO\) and \(LN = LP + PN\) (from statements 1 - 3), by substitution (if \(a=b\) and \(a = c\), then \(b = c\)), the reason for statement 4 is "Substitution Property".

Step5: Reasons for Statement 5

Since \(MP = LP\) (from statement 2), we substitute \(MP\) with \(LP\) in \(MP+PO\). So the reason for statement 5 is "Substitution Property".

Step6: Statements for Reason 6

Starting from \(LP + PN=LP + PO\), using the Subtraction Property of Equality (\(a + b=a + c\) implies \(b = c\) when we subtract \(a\) from both sides), we get \(PN=PO\).

Answer:

Statements	Reasons
2. \(MP = LP\), \(LN = MO\)	2. Definition of \(\cong\) segments
3. \(MO=MP + PO\), \(LN=LP + PN\)	3. Segment Addition Postulate
4. \(LP + PN=MP + PO\)	4. Substitution Property
5. \(LP + PN=LP + PO\)	5. Substitution Property
6. \(PN = PO\)	6. Subtraction Property of Equality