Question

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.

Answer 1

The triangles are similar is 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.Therefore , 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 is correct.

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

Detailed explanation:

Reflexive Property of Congruence: Any angle is congruent to itself.

Corresponding Angles Postulate: If two parallel lines are intersected by a transversal, then the corresponding angles are congruent.

Angle-Angle Similarity Postulate (AA Similarity): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Proof:

Given: ∠L≅∠L, OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯

To prove: △LOP∼△LMN

Proof:

By the Reflexive Property of Congruence, ∠L≅∠L.

Since OP¯¯¯¯¯∥MN¯¯¯¯¯¯¯, ∠LOP≅∠LMN by the Corresponding Angles Postulate.

Therefore, by the Angle-Angle Similarity Postulate (AA Similarity), △LOP∼△LMN.

Finding OP and MN:

Since the triangles are similar, we can write the following proportion:

OP/MN = LOP/LMN

Substituting the given values, we get:

OP/MN = 90/62

Solving for OP, we get:

OP = MN * (90/62)

OP = 6 * (90/62)

OP = 2

Therefore, OP = 2.

Answer 2

The correct explanation is option B;

B. ∠L ≅ ∠L By Reflexive Prop. ≅.

Since
`\overline {OP}` ║
`\overline {MN}`, ∠LOP ≅ ∠LMN by the Corr. ∠s Post. Therefore, ΔLOP ~ ΔLMN by AA ~. OP = 2 and MN = 6

Explanation:

The given parameters are;

`\overline {LP}` = 5

`\overline {NP}` = 10

`\overline {LN}` =
`\overline {LP}` +
`\overline {NP}` = 15

`\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 {LN}` = 5/15 = 1/3

∴ (x - 3)/(x + 1) = 1/3

3·(x - 3) = 1·(x + 1)

3·x - 9 = x + 1

3·x - x = 1 + 9 = 10

2·x = 10

x = 10/2 = 5

x = 5

`\overline {OP}` = (x - 3) = 5 - 3 = 2

`\overline {OP}` = 2

`\overline {MN}` = x + 1 = 5 + 1 = 6

`\overline {MN}` = 6

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 = 2 and MN = 6.