Question

given: mp ≅ np, mn ⊥ op
prove: mo ≅ on
statements

1. (overline{mp} cong overline{np}), (mn perp op)
2. (∠mop = 90) and (∠nop = 90)

3.

1. (∠omp cong ∠onp)
2. (△ cong △)

6.
reasons

1. given
2. perpendicular lines
3. all right angles are congruent
4. vertical angle theorem
5. aas

6.

Explanation:

Step1: Congruent right angles

Since \(MN\perp OP\), \(\angle MOP = 90^{\circ}\) and \(\angle NOP=90^{\circ}\), so \(\angle MOP\cong\angle NOP\)

Step2: Congruent triangles

We know \(MP\cong NP\) (given), \(\angle MOP\cong\angle NOP\) (from step 1), \(\angle OMP\cong\angle ONP\) (given as part of the problem setup related to vertical - like angle congruence in the triangle context). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle MOP\cong\triangle NOP\)

Step3: Corresponding parts of congruent triangles

If \(\triangle MOP\cong\triangle NOP\), then their corresponding sides \(MO\) and \(NO\) are congruent. That is, if two triangles are congruent, then their corresponding parts (CPCTC - Corresponding Parts of Congruent Triangles are Congruent) are congruent.

Answer:

1. Statement 3: \(\angle MOP\cong\angle NOP\)
2. Statement 5: \(\triangle MOP\cong\triangle NOP\)
3. Statement 6: \(MO\cong NO\) (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent)