Question

Prove LO bisects angle MLN and identify which given statement is unnecessary.​

Answer 1

To prove that LO bisects angle MLN, we can use the Angle Bisector Theorem. According to this theorem, if a line bisects an angle, it divides the opposite side into two segments that are proportional to the adjacent sides of the angle. We can prove this by using the given information and applying the theorem.

Step-by-step explanation:

To prove that LO bisects angle MLN, we can use the Angle Bisector Theorem. According to this theorem, if a line bisects an angle, it divides the opposite side into two segments that are proportional to the adjacent sides of the angle. In this case, LO is the angle bisector of angle MLN if and only if (MN/NO) = (ML/LO). We can prove this by using the given information and applying the theorem.

Given:

• LO bisects angle MLN
• TLX TL

To prove:

• LO bisects angle MLN

Proof:

1. Since LO bisects angle MLN, TLX and TL are alternate interior angles and are therefore congruent.
2. In triangle MLN, using the Law of Sines, we can express the ratio of sides as follows: MN/sin(TLX) = ML/sin(TL)
3. Since sin(TLX) = sin(TL) (by the congruent angles), we can simplify the expression to: MN/ML = sin(TLX)/sin(TL)
4. According to the Angle Bisector Theorem, we know that TO/OL = TN/NL, which can be written as: OL/TN = LO/NL
5. Using the Sine Law in triangle OTL, we have: LO/sin(OTL) = TL/sin(LOT)
6. Since OTL is congruent to TLX (by the congruent angles), we can simplify the expression to: LO/TL = sin(TLX)/sin(TL)
7. Comparing the expressions MN/ML = sin(TLX)/sin(TL) and LO/TL = sin(TLX)/sin(TL), we can conclude that MN/ML = LO/TL
8. Therefore, LO bisects angle MLN (by the Angle Bisector Theorem)

So, we have proven that LO bisects angle MLN.

The given statement that is unnecessary for this proof is 'TLX TL' because it does not provide any additional information or requirement for proving that LO bisects angle MLN.

Answer 2

Given:

LM = LN, KM = KN, KO bisects angle MKN

What we want to prove:

LO bisects angle MLN

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Proof:

Statement 1: KM = KN

Reason 1: Given

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Statement 2: KO bisects angle MKN

Reason 2: Given

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Statement 3: Angle MKO = Angle NKO

Reason: Definition of angle bisection

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Statement 4: KO = KO

Reason 4: Reflexive property of congruence

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Statement 5: Triangle MKO = Triangle NKO

Reason 5: SAS postulate

note: this combines statements 1, 3, and 4 (for the "S", "A", and "S" in that order)

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Statement 6: MO = ON

Reason 6: CPCTC

note: CPCTC stands for "corresponding parts of congruent triangles are congruent"

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Statement 7: Angle KOM = Angle KON

Reason: CPCTC

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Statement 8: OL = OL

Reason 8: Reflexive property of congruence

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Statement 9: Triangle MOL = Triangle NOL

Reason 9: SAS postulate

note: used statements 6, 7, 8 for the "S", "A", "S" in that order

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Statement 10: Angle MLO = Angle NLO

Reason 10: CPCTC

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Statement 11: LO bisects angle MLN

Reason 11: Definition of angle bisection

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That concludes the proof. We did not use the statement "LM = LN", so it is unnecessary.