Question

name: maileny unit 2: logic & proof
date: per: 5 homework 8: segment proofs
this is a 2 - page document!
use the segment addition postulate to write three equations using the diagram below.
1.
2.
3.

complete each proof.

1. given: x is the midpoint of \\(\overline{wy}\\), \\(\overline{wx} \cong \overline{xz}\\)

prove: \\(\overline{xy} \cong \overline{xz}\\)

statements	reasons
2. \\(wx = xy\\)	2.
3. \\(\overline{wx} \cong \overline{xz}\\)	3.
4. \\(wx = xz\\)	4.
5. \\(xy = xz\\)	5.
6. \\(\overline{xy} \cong \overline{xz}\\)	6.

1. given: \\(\overline{ab} \cong \overline{cd}\\)

prove: \\(\overline{ac} \cong \overline{bd}\\)

statements	reasons
2. \\(ab = cd\\)	2.
3. \\(ac + cd = ad\\)	3.
4. \\(ab + bd = ad\\)	4.
5. \\(cd + bd = ad\\)	5.
6. \\(ac + cd = cd + bd\\)	6.
7. \\(ac = bd\\)	7.
8. \\(\overline{ac} \cong \overline{bd}\\)	8.

Explanation:

Step1: Write segment addition equations

1. $PQ + QR = PR$
2. $QR + RS = QS$
3. $PS + ST = PT$

(Other valid equations are also acceptable, e.g., $PQ + QR + RS = PS$)

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Step2: Complete Proof 4 (Reasons)

Step2.1: Midpoint definition

Reason 2: Definition of midpoint

Step2.2: State given congruence

Reason 3: Given

Step2.3: Congruent segments = length

Reason 4: Definition of congruent segments

Step2.4: Transitive property of equality

Reason 5: Transitive property of equality

Step2.5: Congruent segments from length

Reason 6: Definition of congruent segments

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Step3: Complete Proof 5 (Reasons)

Step3.1: Congruent segments = length

Reason 2: Definition of congruent segments

Step3.2: Segment addition postulate

Reason 3: Segment addition postulate

Step3.3: Segment addition postulate

Reason 4: Segment addition postulate

Step3.4: Substitute equal lengths

Reason 5: Substitution property of equality

Step3.5: Transitive property of equality

Reason 6: Transitive property of equality

Step3.6: Subtraction property of equality

Reason 7: Subtraction property of equality

Step3.7: Congruent segments from length

Reason 8: Definition of congruent segments

Answer:

1-3. Segment Addition Equations:

1. $PQ + QR = PR$
2. $QR + RS = QS$
3. $PS + ST = PT$

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4. Proof Reasons:

Statements	Reasons
2. $WX = XY$	2. Definition of midpoint
3. $\overline{WX} \cong \overline{XZ}$	3. Given
4. $WX = XZ$	4. Definition of congruent segments
5. $XY = XZ$	5. Transitive property of equality
6. $\overline{XY} \cong \overline{XZ}$	6. Definition of congruent segments

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5. Proof Reasons:

Statements	Reasons
2. $AB = CD$	2. Definition of congruent segments
3. $AC + CD = AD$	3. Segment addition postulate
4. $AB + BD = AD$	4. Segment addition postulate
5. $CD + BD = AD$	5. Substitution property of equality
6. $AC + CD = CD + BD$	6. Transitive property of equality
7. $AC = BD$	7. Subtraction property of equality
8. $\overline{AC} \cong \overline{BD}$	8. Definition of congruent segments