Question

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

Answer 1

Answer:

Therefore, the correct explanation is;

C. ∠L ≅ ∠L  By Reflexive Prop. ≅.

Since $\overline {OP}$ ║ $\overline {MN}$, ∠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;

$\overline {LP}$ = 5

$\overline {NP}$ = 10

$\overline {OP}$ = x - 3

$\overline {MN}$ = x + 1

A two column proof is presented as follows;

Statement    ${}$      Reason

∠L ≅ ∠L     ${}$        By Reflexive property of congruency

$\overline {OP}$ ║ $\overline {MN}$     ${}$     Given

∠LOP ≅ ∠LMN ${}$  By the Corresponding angles Postulate

Therefore

ΔLOP ~ ΔLMN ${}$  By AA similarity Postulate

Where we have that ΔLOP and ΔLMN, we get;

$\overline {OP}$/$\overline {MN}$  = $\overline {LP}$/$\overline {NP}$ = 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

$\overline {OP}$ = (x - 3) = 7 - 3 = 4

$\overline {OP}$ = 4

$\overline {MN}$ = x + 1 = 7 + 1 = 8

$\overline {MN}$ =  8

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

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

Answer 2

Answer:

∠L≅∠L by the Reflex Prop. of ≅.

Since OP⎯⎯⎯⎯⎯∥MN⎯⎯⎯⎯⎯⎯,∠LOP≅∠LMN by the

Corr. ∠s Post. Therefore, △LOP∼△LMN

by AA∼. OP=2 and MN=6.

Step-by-step explanation: