Question

use a two-column proof and the figure on the right to prove the following.
given: pr = 2np,
pq = 2mp
prove: $\triangle mnp \sim \triangle qrp$

statements

1. pr = 2np
2. $\frac{pr}{np} = \square$

reasons

1. given
2. dividing both sides by np

Explanation:

Step1: Divide by NP

$\frac{PR}{NP} = \frac{2NP}{NP} = 2$

Step2: Process second given side

$\frac{PQ}{MP} = \frac{2MP}{MP} = 2$

Step3: Identify vertical angles

$\angle MPN = \angle QPR$ (vertical angles are congruent)

Step4: Apply SAS similarity

Since $\frac{PR}{NP} = \frac{PQ}{MP} = 2$ and included angles are congruent, $\Delta MNP \sim \Delta QRP$

Answer:

For the blank in Step 2: $2$
Completed two-column proof (full):

Statements	Reasons
2. $\frac{PR}{NP} = 2$	2. Dividing both sides by $NP$
3. $PQ = 2MP$	3. Given
4. $\frac{PQ}{MP} = 2$	4. Dividing both sides by $MP$
5. $\angle MPN = \angle QPR$	5. Vertical angles are congruent
6. $\Delta MNP \sim \Delta QRP$	6. SAS Similarity Criterion

Final proof confirms $\Delta MNP \sim \Delta QRP$, and the value for the blank is $2$.