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Q7.

Expert-verified

Geometry

Found in: Page 211

Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

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Short Answer

For each statement in Ex 5-10 copy and complete a table like the one shown below.
If $A M = M B$ , then M is the midpoint of $\bar{A B}$ .

Statement	If ?, then ?	True/false
1. Given	If $A M = M B$ , then M is the midpoint of $\bar{A B}$ .	False
2. Contrapositive	If M is not a midpoint of $\bar{A B}$ then $A M \neq M B$ .	False
3. Converse	If M is the midpoint of $\bar{A B}$ then role="math" localid="1638260520254" $A M = M B$ .	True
4. Inverse	If $A M \neq M B$ , then M is not the midpoint of $\bar{A B}$ .	False

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Define contrapositive, inverse, and converse for the given statement

If the given statement is ‘’If p then q’’, the contrapositive is ‘’If not q then not p’’.

If the given statement is ‘’If p then q’’, the converse is ‘’If q then p’’.

If the given statement is ‘’If p then q’’, the inverse is ‘’If not p then not q’’.

Step 2. Contrapositive of the given statement

The given statement is If $A M = M B$ , then M is the midpoint of $\bar{A B}$ .

If A, M, and B are non-collinear for example vertex of an equilateral triangle.

Therefore, the given statement: If $A M = M B$ , then M is the midpoint of $\bar{A B}$

is false.

Suppose A, M, and B are the vertex of an equilateral triangle.

The contrapositive of the given statement is If M is not a midpoint of $\bar{A B}$ then $A M \neq M B$ .

Thus, it is also false.

Step 3. Check inverse and converse of the given statement

Since the midpoint is equidistant from both the endpoints of the line segment.

The converse of the given statement is If M is the midpoint of $\bar{A B}$ then $A M = M B$ .

Thus, it is true.

The inverse of the given statement is If $A M \neq M B$ , then M is not the midpoint of the line segment $\bar{A B}$ .

Thus, it is false.

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