Identify the correct explanation for why the triangles are similar. Then find OP and MN. Options:

A. ∠M≅∠N. Since OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯, ∠LOP≅∠LMN by the Alt. Int. ∠s Thm. Therefore, △LOP∼△LMNby AA∼. OP=8 and MN=4.

B. ∠L≅∠L by the Reflex Prop. of ≅. Since OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯, ∠LOP≅∠LMN by the Corr. ∠s Post. Therefore, △LOP∼△LMNby AA∼. OP=2 and MN=6.

C. ∠L≅∠L by the Reflex Prop. of ≅. Since OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯, ∠LOP≅∠LMN by the Corr. ∠s Post. Therefore, △LOP∼△LMNby AA∼. OP=4 and MN=8.

D. ∠M≅∠N. Since OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯, ∠LOP≅∠LMN by the Alt. Int. ∠s Thm. Therefore, △LOP∼△LMNby AA∼. OP=2 and MN=6.

Therefore, the correct explanation is;

C. ∠L ≅ ∠L  By Reflexive Prop. ≅.

Since ║ , ∠LOP ≅ ∠LMN by the Corr. ∠s Post. Therefore, ΔLOP ~ ΔLMN by AA ~. OP = 4 and MN = 8

Step-by-step explanation:

The given parameters are;

= 5

= 10

= x - 3

= x + 1

A two column proof is presented as follows;

Statement          Reason

∠L ≅ ∠L            By Reflexive property of congruency

║         Given

∠LOP ≅ ∠LMN  By the Corresponding angles Postulate

Therefore

ΔLOP ~ ΔLMN  By AA similarity Postulate

Where we have that ΔLOP and ΔLMN, we get;

/  = / = 5/10 = 1/2

∴ (x - 3)/(x + 1) = 1/2

2·(x - 3) = 1·(x + 1)

2·x - 6 = x + 1

2·x - x = 1 + 6 = 7

x = 7

= (x - 3) = 7 - 3 = 4

= 4

= x + 1 = 7 + 1 = 8

=  8

Therefore, the correct option is ∠L ≅ ∠L  By Reflexive Prop. ≅.

Since ║ , ∠LOP ≅ ∠LMN by the Corr. ∠s Post. Therefore, ΔLOP ~ ΔLMN by AA ~. OP = 4 and MN = 8.