Question

10 given: mn || po, mp || no. prove: mp ≅ on statements reasons 1. mn || po 1. given 2. mp || no 2. given 3. 3. 4. 4. 5. 5. 6. 6. 7. 7.

Explanation:

Step1: Alternate interior angles

Since \(MN\parallel PO\), \(\angle NMP=\angle OPM\) (Alternate - interior angles of parallel lines \(MN\) and \(PO\) with transversal \(MP\)).

Step2: Alternate interior angles

Since \(MP\parallel NO\), \(\angle PMO = \angle NOM\) (Alternate - interior angles of parallel lines \(MP\) and \(NO\) with transversal \(MO\)).

Step3: Common side

\(MO = MO\) (Reflexive property of equality).

Step4: ASA (Angle - Side - Angle)

\(\triangle MPO\cong\triangle ONM\) (By ASA congruence criterion: \(\angle NMP=\angle OPM\), \(MO = MO\), \(\angle PMO=\angle NOM\)).

Step5: Corresponding parts of congruent triangles

\(MP = ON\) (Corresponding parts of congruent triangles \(\triangle MPO\) and \(\triangle ONM\) are equal).

Answer:

1. \(\angle NMP=\angle OPM\); Alternate - interior angles of parallel lines \(MN\) and \(PO\) with transversal \(MP\)
2. \(\angle PMO=\angle NOM\); Alternate - interior angles of parallel lines \(MP\) and \(NO\) with transversal \(MO\)
3. \(MO = MO\); Reflexive property of equality
4. \(\triangle MPO\cong\triangle ONM\); ASA congruence criterion
5. \(MP = ON\); Corresponding parts of congruent triangles