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Math
GeometryElementary Geometry for College Students6th Edition

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Chapter 1.5, Problem 28E
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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 27 to 30, fill in the missing reasons for each geometric proof.

Given: E is the midpoint of D F ↔

Prove: D E = 1 2 ( D F )

Chapter 1.5, Problem 28E, In Exercises 27 to 30, fill in the missing reasons for each geometric proof. Given: E is the

Exercises 27, 28

PROOF
Statements Reasons
1. E is the midpoint of D F ↔ 1. ?
2. D E = E F 2. ?
3. D E + E F = D F 3. ?
4. D E + D E = D F 4. ?
5. 2 ( D E ) = D F 5. ?
6. D E = 1 2 ( D F ) 6. ?

To determine

To find:

The missing reasons for the geometric proof.

Explanation

Given:

The given table is shown below.

PROOF
Statements Reasons
1. E is the midpoint of DF↔ 1. ?
2. DE=EF 2. ?
3. DE+EF=DF 3. ?
4. DE+DE=DF 4. ?
5. 2(DE)=DF 5. ?
6. DE=12(DF) 6. ?

Approach:

1. Midpoint on a line is defined as the point which divides the line into two segments of equal length.

2. According to Segment-Addition Postulate, if two points A, B are given on a line segment, then third point C lies on the line segment if and only if,

AC+CB=AB

3. According to Substitution Property of Equality, if A=B, then A can be substituted for B and B can be substituted for A in any equation.

4. The Multiplication Property of Equality states that if both the sides of an equation is multiplied by a same number then the sides of the equation remains equal.

Calculation:

The first statement of the proof is given below.

E is the midpoint of DF↔

Therefore, the first reason of the proof is given.

Now, go from first statement to the second statement.

E is the midpoint of DF↔

Use Definition of Midpoint as E is the midpoint of DF↔.

DE=EF

Therefore, the second reason is Definition of Midpoint.

Now, go from second statement to the third statement.

DE=EF

Use Segment-Addition Postulate as D, E and F are collinear and E is the midpoint of DF↔.

DE+EF=DF

Therefore, the third reason is Segment-Addition Postulate.

Now, go from third statement to fourth statement.

DE+EF=DF

Use Substitution Property of Equality and substitute DE for EF as DE=EF.

DE+DE=DF

Therefore, the fourth reason for the proof is Substitution Property of Equality

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