Question

given ln ≅ on, ap ≅ ap, cb ≅ eb, defn. of midpoint, df || hg, vertical ∠’s, ∠dfe ≅ ∠hgf, δsvt ≅ δuvr; angle - side - angle, ab ≅ dh, jk ≅ lm, reflexive property, pq || rs, alt. interior ∠’s, ∠qpr ≅ ∠srp, ∠zxv ≅ ∠yxv, δjkl ≅ δlmj.

1. given: pq ≅ rs, ∠pqr ≅ ∠ksp (maybe typo, should be ∠rsp?), prove: δpqr ≅ δrsp.

statements: 1. 2. 3. ∠pqr ≅ ∠rsp 4. overline{rp} ≅ overline{pr} 5. δpqr ≅ δrsp
reasons: 1. given 2. alternate interior angles 3. 4. 5.

1. given: n is the midpoint of lq, ∠nlm ≅ ∠nqp (maybe typo, original is ∠nlm ≅ ∠nqp?), prove: δmln ≅ δpnq.

statements: 1. n is the midpoint of lq 2. 3. ∠lnm ≅ ∠qnp 4. ∠nlm ≅ ∠nqp 5. δmln ≅ δpnq
reasons: 1. given 2. definition of midpoint 3. 4. 5.

Explanation:

Step1: Analyze Statement 2

The reason is "Definition of Midpoint", so the statement should be \( LN = QN \) (since \( N \) is the midpoint of \( LQ \), midpoint divides a segment into two equal parts).

Step2: Analyze Statement 3

The statement is \( \angle LNM \cong \angle QNP \), these are vertical angles, so the reason is "Vertical Angles Theorem" (vertical angles are congruent).

Step3: Analyze Statement 4

The statement is \( \angle NLM \cong \angle NQP \), this is given in the problem, so the reason is "Given".

Step4: Analyze Statement 5

To prove \( \triangle MLN \cong \triangle PQN \), we have \( LN = QN \) (from step 2), \( \angle LNM = \angle QNP \) (from step 3), \( \angle NLM = \angle NQP \) (from step 4). So by Angle - Angle - Side (AAS) congruence criterion, the triangles are congruent. The reason is "AAS (Angle - Angle - Side) Congruence Theorem".

Answer:

1. Given
2. \( LN = QN \)
3. Vertical Angles Theorem
4. Given
5. AAS (Angle - Angle - Side) Congruence Theorem