Question

given: ( overline{mp} cong overline{lp} ) ( overline{ln} cong overline{mo} ) prove: ( pn = po ) statements reasons ( overline{mp} cong overline{lp} ) & ( overline{ln} cong overline{mo} ) definition of ( cong ) segments ( mo = mp + po ) & ( ln = lp + pn ) ( lp + pn = mp + po ) ( lp + pn = lp + po ) subtraction property of equality

Explanation:

Step1: Given

$$\overline{MP}\cong\overline{LP},\overline{LN}\cong\overline{MO}$$
By the definition of congruent segments, \(MP = LP\) and \(LN=MO\).

Step2: Segment Addition Postulate

By the segment addition postulate, \(MO=MP + PO\) and \(LN=LP + PN\).
Since \(LN = MO\), we substitute to get \(LP + PN=MP + PO\).

Step3: Substitute \(MP\) with \(LP\)

Because \(MP = LP\) (from congruent segments), substitute \(MP\) in \(LP + PN=MP + PO\) to get \(LP + PN=LP + PO\).

Step4: Subtraction Property of Equality

Subtract \(LP\) from both sides of the equation \(LP + PN=LP + PO\).
$$(LP + PN)-LP=(LP + PO)-LP$$
$$PN = PO$$

Answer:

\(PN = PO\)