Q10. In Exercises 9 and 10 you are gi... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 4: Q10. (page 131)

In Exercises 9 and 10 you are given more information than you need. For each exercise state one piece of given information that you do not need for the proof. Then give a two鈥揷olumn proof that does not use that piece of information.
Given: $\overset{炉}{L M} 鈮� \overset{炉}{L N}; \overset{炉}{K M} 鈮� \overset{炉}{K N}; \overset{鈫�}{K O}$ bisects $鈭� M K N$
Prove: $\overset{鈫�}{L O}$ bisects $鈭� M L N$

Short Answer

Expert verified

$\overset{鈫�}{K O}$ bisects $鈭� M K N$ is not needed in given information to give proof.

Statement	Reason
1. $\overset{炉}{L M} 鈮� \overset{炉}{L N}$	Given
2. $\overset{炉}{K M} 鈮� \overset{炉}{K N}$	Given
3. $\overset{炉}{L K} 鈮� \overset{炉}{L K}$	Common line segment
4. $螖 M L K 鈮� 螖 N L K$	SSS congruency criteria
5. $鈭� M L K 鈮� 鈭� N L K$	corresponding parts of congruent triangle are congruent
6. $\overset{鈫�}{L O}$ bisects $鈭� M L N$	Definition of angle bisector

Step by step solution

Step 1. Show that LK¯≅LK¯.

Since, line segment $\overset{炉}{L K}$ is common to both triangles, by relexive property it is congruent to itself

So, $\overset{炉}{P R} 鈮� \overset{炉}{P R}$

Step 2. Show that ΔMLK≅ΔNLK.

From step 1 using given statement $螖 M L K 鈮� 螖 N L K$ by SSS (Side-Side-Side) congruency criteria

Step 3. Show that ∠MLK≅∠NLK.

Since, corresponding parts of congruent triangle are congruent

Thus, $鈭� M L K 鈮� 鈭� N L K$

Step 4. Show that LO→ bisects ∠MLN.

An angle bisector of an angle divides it into two congruent angles.

Since, $鈭� M L K 鈮� 鈭� N L K$

Thus, $\overset{鈫�}{L O}$ bisects $鈭� M L N$

Also proof is provided without using $\overset{鈫�}{K O}$ bisects $鈭� M K N$ , so it is not needed in given information.

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91影视

Short Answer

Step by step solution

Step 1. Show that LK¯≅LK¯.

Step 2. Show that ΔMLK≅ΔNLK.

Step 3. Show that ∠MLK≅∠NLK.

Step 4. Show that LO→ bisects ∠MLN.

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