Question

name: nayeliz dejesus
date: 9/10
bell:
unit 2: logic & proof
homework 7: segment proofs
this is a 2 - page document!
use the segment addition postulate to write three equations using the diagram below.
1.
2.
3.
diagram: p---q---r---s---t
complete the proofs below by filling in the missing statements and reasons

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

prove: (overline{xy} cong overline{xz})
diagram: w---x---y, z above x
statements | reasons

1. x is the midpoint of (overline{wy}) | 1.
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.
7. given: (overline{ab} cong overline{cd})

prove: (overline{ac} cong overline{bd})
diagram: a---b---c---d
statements | reasons

1. (overline{ab} cong overline{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. (overline{ac} cong overline{bd}) | 8.

© gina wilson (all things algebra), 2014

Explanation:

Problem 1: Using Segment Addition Postulate

The diagram has points \( P, Q, R, S, T \) in order. The segment addition postulate states that if three points are collinear, the sum of the lengths of the two smaller segments equals the length of the larger segment.

Step 1: Equation with \( P, Q, R \)

If \( Q \) is between \( P \) and \( R \), then \( PQ + QR = PR \).

Step 2: Equation with \( Q, R, S \)

If \( R \) is between \( Q \) and \( S \), then \( QR + RS = QS \).

Step 3: Equation with \( R, S, T \)

If \( S \) is between \( R \) and \( T \), then \( RS + ST = RT \).

Problem 4: Proof (Midpoint and Congruence)

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. Definition of midpoint (a midpoint divides a segment into two equal parts)
3. \( \overline{WX} \cong \overline{XZ} \)	3. Given
4. \( WX = XZ \)	4. Definition of congruent segments (congruent segments have equal length)
5. \( XY = XZ \)	5. Substitution Property (replace \( WX \) with \( XY \) from Step 2)
6. \( \overline{XY} \cong \overline{XZ} \)	6. Definition of congruent segments (if lengths are equal, segments are congruent)

Problem 5: Proof (Congruent Segments)

Given: \( \overline{AB} \cong \overline{CD} \)
Prove: \( \overline{AC} \cong \overline{BD} \)

Statements	Reasons
2. \( AB = CD \)	2. Definition of congruent segments
3. \( AC + CD = AD \)	3. Segment Addition Postulate ( \( C \) is between \( A \) and \( D \), so \( AC + CD = AD \))
4. \( AB + BD = AD \)	4. Segment Addition Postulate ( \( B \) is between \( A \) and \( D \), so \( AB + BD = AD \))
5. \( CD + BD = AD \)	5. Substitution Property (replace \( AB \) with \( CD \) from Step 2)
6. \( AC + CD = CD + BD \)	6. Transitive Property (both equal \( AD \), so they equal each other)
7. \( AC = BD \)	7. Subtraction Property (subtract \( CD \) from both sides)
8. \( \overline{AC} \cong \overline{BD} \)	8. Definition of congruent segments

Final Answers (Key Equations/Proof Steps)

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

(For Proofs 4 and 5, the completed tables above show the full reasoning.)

Answer:

Problem 1: Using Segment Addition Postulate

The diagram has points \( P, Q, R, S, T \) in order. The segment addition postulate states that if three points are collinear, the sum of the lengths of the two smaller segments equals the length of the larger segment.

Step 1: Equation with \( P, Q, R \)

If \( Q \) is between \( P \) and \( R \), then \( PQ + QR = PR \).

Step 2: Equation with \( Q, R, S \)

If \( R \) is between \( Q \) and \( S \), then \( QR + RS = QS \).

Step 3: Equation with \( R, S, T \)

If \( S \) is between \( R \) and \( T \), then \( RS + ST = RT \).

Problem 4: Proof (Midpoint and Congruence)

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. Definition of midpoint (a midpoint divides a segment into two equal parts)
3. \( \overline{WX} \cong \overline{XZ} \)	3. Given
4. \( WX = XZ \)	4. Definition of congruent segments (congruent segments have equal length)
5. \( XY = XZ \)	5. Substitution Property (replace \( WX \) with \( XY \) from Step 2)
6. \( \overline{XY} \cong \overline{XZ} \)	6. Definition of congruent segments (if lengths are equal, segments are congruent)

Problem 5: Proof (Congruent Segments)

Given: \( \overline{AB} \cong \overline{CD} \)
Prove: \( \overline{AC} \cong \overline{BD} \)

Statements	Reasons
2. \( AB = CD \)	2. Definition of congruent segments
3. \( AC + CD = AD \)	3. Segment Addition Postulate ( \( C \) is between \( A \) and \( D \), so \( AC + CD = AD \))
4. \( AB + BD = AD \)	4. Segment Addition Postulate ( \( B \) is between \( A \) and \( D \), so \( AB + BD = AD \))
5. \( CD + BD = AD \)	5. Substitution Property (replace \( AB \) with \( CD \) from Step 2)
6. \( AC + CD = CD + BD \)	6. Transitive Property (both equal \( AD \), so they equal each other)
7. \( AC = BD \)	7. Subtraction Property (subtract \( CD \) from both sides)
8. \( \overline{AC} \cong \overline{BD} \)	8. Definition of congruent segments

Final Answers (Key Equations/Proof Steps)

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

(For Proofs 4 and 5, the completed tables above show the full reasoning.)