Question

question 1 | 3.4-writing proofs homework
given
\overline{xz}. w is collinear with
x, y, and z. z is the
midpoint of \overline{yw}.

1. \boxed{xy = zw}
2. \boxed{\overline{xy} \cong \overline{yz} \text{ and } \overline{yz} \cong \overline{zw}}
3. \boxed{xy = yz \text{ and } yz = zw}
4. \overline{xy} \cong \overline{zw}
5. midpoint thm.
6. def. of \cong segments
7. transitive prop. of equality
8. \boxed{def of \cong segs}

Explanation:

To solve this proof, we analyze each step:

Step 1: Identify Given Information

We know \( Z \) is the midpoint of \( \overline{YW} \), so by the Midpoint Theorem, \( YZ = ZW \) (since a midpoint divides a segment into two equal parts). Also, from the diagram/context, \( XY = YZ \) (likely from collinearity and segment equality).

Step 2: Apply Transitive Property of Equality

If \( XY = YZ \) and \( YZ = ZW \), the Transitive Property of Equality tells us \( XY = ZW \).

Step 3: Use Definition of Congruent Segments

The definition of congruent segments states that if two segments have equal length, they are congruent. Since \( XY = ZW \), we conclude \( \overline{XY} \cong \overline{ZW} \).

For the missing step (step 1 on the left, corresponding to "Given" on the right), the given information is:

• \( Y \) is on \( \overline{XZ} \) (so \( X, Y, Z \) are collinear).
• \( Z \) is the midpoint of \( \overline{YW} \).

For the missing reason (step 5 on the right), the reason is the Definition of Congruent Segments (since \( XY = ZW \) implies \( \overline{XY} \cong \overline{ZW} \)).

If the question is to fill in the missing "Given" or reason, here’s the breakdown:

• Step 1 (Left): Given: \( Y \) is on \( \overline{XZ} \) (so \( X, Y, Z \) are collinear), and \( Z \) is the midpoint of \( \overline{YW} \).
• Step 5 (Right Reason): Definition of congruent segments (since \( XY = ZW \) implies \( \overline{XY} \cong \overline{ZW} \)).

If the goal is to complete the proof, the key logical flow is:

1. Given: \( Y \in \overline{XZ} \), \( Z \) is midpoint of \( \overline{YW} \).
2. Midpoint Thm: \( YZ = ZW \); (and \( XY = YZ \) from collinearity/other given).
3. Def of \( \cong \) segments: \( \overline{XY} \cong \overline{YZ} \), \( \overline{YZ} \cong \overline{ZW} \).
4. Transitive Prop of Equality: \( XY = YZ \), \( YZ = ZW \implies XY = ZW \).
5. Def of \( \cong \) segments: \( \overline{XY} \cong \overline{ZW} \) (since \( XY = ZW \)).

Final Answer (for Step 5 Reason):

The reason for step 5 is the Definition of Congruent Segments (since equal lengths imply congruent segments).

(If the question was to fill the "Given" for step 1, the answer is: \( Y \) is on \( \overline{XZ} \), and \( Z \) is the midpoint of \( \overline{YW} \).)

Answer:

To solve this proof, we analyze each step:

Step 1: Identify Given Information

We know \( Z \) is the midpoint of \( \overline{YW} \), so by the Midpoint Theorem, \( YZ = ZW \) (since a midpoint divides a segment into two equal parts). Also, from the diagram/context, \( XY = YZ \) (likely from collinearity and segment equality).

Step 2: Apply Transitive Property of Equality

If \( XY = YZ \) and \( YZ = ZW \), the Transitive Property of Equality tells us \( XY = ZW \).

Step 3: Use Definition of Congruent Segments

The definition of congruent segments states that if two segments have equal length, they are congruent. Since \( XY = ZW \), we conclude \( \overline{XY} \cong \overline{ZW} \).

For the missing step (step 1 on the left, corresponding to "Given" on the right), the given information is:

• \( Y \) is on \( \overline{XZ} \) (so \( X, Y, Z \) are collinear).
• \( Z \) is the midpoint of \( \overline{YW} \).

For the missing reason (step 5 on the right), the reason is the Definition of Congruent Segments (since \( XY = ZW \) implies \( \overline{XY} \cong \overline{ZW} \)).

If the question is to fill in the missing "Given" or reason, here’s the breakdown:

• Step 1 (Left): Given: \( Y \) is on \( \overline{XZ} \) (so \( X, Y, Z \) are collinear), and \( Z \) is the midpoint of \( \overline{YW} \).
• Step 5 (Right Reason): Definition of congruent segments (since \( XY = ZW \) implies \( \overline{XY} \cong \overline{ZW} \)).

If the goal is to complete the proof, the key logical flow is:

1. Given: \( Y \in \overline{XZ} \), \( Z \) is midpoint of \( \overline{YW} \).
2. Midpoint Thm: \( YZ = ZW \); (and \( XY = YZ \) from collinearity/other given).
3. Def of \( \cong \) segments: \( \overline{XY} \cong \overline{YZ} \), \( \overline{YZ} \cong \overline{ZW} \).
4. Transitive Prop of Equality: \( XY = YZ \), \( YZ = ZW \implies XY = ZW \).
5. Def of \( \cong \) segments: \( \overline{XY} \cong \overline{ZW} \) (since \( XY = ZW \)).

Final Answer (for Step 5 Reason):

The reason for step 5 is the Definition of Congruent Segments (since equal lengths imply congruent segments).

(If the question was to fill the "Given" for step 1, the answer is: \( Y \) is on \( \overline{XZ} \), and \( Z \) is the midpoint of \( \overline{YW} \).)