Question

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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 wy, wx ≅ xz

prove: xy ≅ xz
statements reasons

1. x is the midpoint of wy 1.
2. wx = xy 2.
3. wx ≅ xz 3.
4. wx = xz 4.
5. xy = xz 5.
6. xy ≅ xz 6.
7. given: ab ≅ cd

prove: ac ≅ bd
statements reasons

1. ab ≅ cd 1.
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. ac ≅ bd 8.

Explanation:

Step1: Write segment - addition equations

For the points \(P,Q,R,S,T\) on a line, by the segment - addition postulate:

1. \(PQ + QR=PR\)
2. \(QR+RS = QS\)
3. \(RS + ST=RT\)

Step2: Complete proof for \(XY\cong XZ\)

1. Given
2. Definition of mid - point (If \(X\) is the mid - point of \(\overline{WY}\), then \(WX = XY\))
3. Given
4. Definition of congruent segments (\(\overline{WX}\cong\overline{XZ}\) implies \(WX = XZ\))
5. Transitive property of equality (\(WX = XY\) and \(WX = XZ\) implies \(XY = XZ\))
6. Definition of congruent segments (\(XY = XZ\) implies \(\overline{XY}\cong\overline{XZ}\))

Step3: Complete proof for \(AC\cong BD\)

1. Given
2. Definition of congruent segments (\(\overline{AB}\cong\overline{CD}\) implies \(AB = CD\))
3. Segment - addition postulate (For points \(A,C,D\) on a line, \(AC + CD=AD\))
4. Segment - addition postulate (For points \(A,B,D\) on a line, \(AB + BD=AD\))
5. Substitution property of equality (Since \(AB = CD\), substitute \(AB\) with \(CD\) in \(AB + BD=AD\))
6. Transitive property of equality (\(AC + CD=AD\) and \(CD + BD=AD\) implies \(AC + CD=CD + BD\))
7. Subtraction property of equality (Subtract \(CD\) from both sides of \(AC + CD=CD + BD\))
8. Definition of congruent segments (\(AC = BD\) implies \(\overline{AC}\cong\overline{BD}\))

Answer:

1. \(PQ + QR=PR\)
2. \(QR+RS = QS\)
3. \(RS + ST=RT\)

4.

Statements	Reasons
2. \(WX = XY\)	2. Definition of mid - point
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

5.

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