In the diagram,<LMN=<ONM=90.P is the midpoint of MN.MN=2ML and MN=NO.prove that <OPN=<LNO

Question

In the diagram,<LMN=<ONM=90.P is the midpoint of MN.MN=2ML and MN=NO.prove that <OPN=<LNO

Answer 1

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Answer 2

To prove that <OPN = <LNO, we can use the fact that P is the midpoint of MN and MN = NO.

Given:
∠LMN = ∠ONM = 90°
P is the midpoint of MN
MN = 2ML
MN = NO

To prove:
∠OPN = ∠LNO

Proof:
1. Triangles LNO and PON are right triangles (since ∠LMN = ∠ONM = 90°).
2. By the definition of a right triangle, ∠NPO = ∠ONL and ∠NOP = ∠LON.
3. Since P is the midpoint of MN, NP = MP.
4. From the given information, MN = NO.
5. By the definition of a midpoint, MN = 2ML.
6. Combining the above equations, 2ML = NO.
7. Since NP = MP and NO = 2ML, triangle NOP is congruent to triangle LNP by the Side-Angle-Side (SAS) congruence criterion.
8. By the congruence of triangles NOP and LNP, ∠OPN = ∠LNO.

Therefore, <OPN = <LNO.

Answer 3

To prove that ∠OPN = ∠LNO, we can use the properties of isosceles triangles and parallel lines.

Here are the steps to prove this statement:

Step 1: Given information
- <LMN = <ONM = 90° (Right angles)
- P is the midpoint of MN (PN = PM)
- MN = 2ML (Two times the length of ML)
- MN = NO (The sides MN and NO are equal)

Step 2: Drawing additional lines
- Extend ML to point Q such that LQ = LP. Now, we have two congruent segments PQ and PL.
- Draw line OQ.

Step 3: Identifying congruent triangles
- In triangle MPL and triangle NPL, we have MP = NP (P is the midpoint of MN) and PL = PL (common side). Additionally, angle MPL = angle NPL = 90° (given).
- Therefore, triangle MPL and triangle NPL are congruent by the hypotenuse-leg congruence theorem.

Step 4: Proving parallel lines
- Since triangle MPL and triangle NPL are congruent, the corresponding angles are congruent as well.
- Therefore, angle LNO = angle LNP (corresponding angles of congruent triangles).

Step 5: Proving isosceles triangles
- In triangle LQN, we have LQ = LP (constructed line) and angle LNO = angle LNP (proved in the previous step).
- Therefore, triangle LNO is an isosceles triangle.

Step 6: Proving congruent base angles
- In an isosceles triangle, the base angles are congruent.
- Hence, angle OPN = angle LNO (congruent base angles of isosceles triangle LNO).

Therefore, we have proved that ∠OPN = ∠LNO.

Answer 4

When you sketch out the diagram... two triangles. Since MN=2ML, and MN=N0, then NO=2ML and it also provides that angle LMN is 90 degrees and since it is an isosceles triange, angle LMN = angle LNM and they are both 45 degrees. Triangle LMN has a similar triange sitting on top of it (you can't see it without the diagram), but it contains angle PNL as it's end point. Since angle M = Angle P and they are 90 degrees and since the angle MLN equals the angle ONL from Parallel lines Alternate angles, and the angle zPN (that is in the small similar triangle, that is sitting on top of triangle LMN), is equal to MLN, we thus know that angle OPN=angle LNO and they are 45 degrees. QED : )