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W Prove: YW Statements Given: M is the midpoint of WX M is the midpoint of YZ XZ M YM MZ WM MX M is the midpoint of WX M is the midpoint of YZ ZYMW ZXMZ X AYMW AZMX YW XZ N Reasons 5 Definition of midpoint Vertical angle theorem 6 7

W Prove: YW Statements Given: M is the midpoint of WX M is the midpoint of YZ XZ M YM MZ WM MX M is the midpoint of WX M is the midpoint of YZ ZYMW ZXMZ X AYMW AZMX YW XZ N Reasons 5 Definition of midpoint Vertical angle theorem 6 7

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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HOW DO I FIND THE REASONS FOR 5, 6, AND 7?

**Title: Proving Congruent Segments in Intersecting Lines** **Introduction:** In this exercise, we are given a geometric diagram with intersecting lines and specific midpoints. The goal is to prove that certain segments are congruent using the properties of midpoints and vertical angles. **Diagram Explanation:** The diagram consists of two intersecting lines, forming a shape with vertices labeled Y, W, X, and Z. The point M is located at the intersection, serving as the midpoint of segments WX and YZ. **Given:** - M is the midpoint of segment WX. - M is the midpoint of segment YZ. **To Prove:** - Segment YW is congruent to segment XZ. **Proof Structure:** | Statements | Reasons | |---------------------------------------------|----------------------------------| | M is the midpoint of WX | Given | | M is the midpoint of YZ | Given | | YM ≅ MZ, WM ≅ MX | Definition of midpoint | | ∠YMW ≅ ∠XMZ | Vertical angle theorem | | △YMW ≅ △ZMX | (Triangular congruence criteria) | | YW ≅ XZ | (Derived from congruent triangles) | **Explanation:** 1. **Midpoint Definition:** - As M is the midpoint of WX, segments WM and MX are congruent. - Similarly, since M is the midpoint of YZ, segments YM and MZ are congruent. 2. **Vertical Angles:** - Angles ∠YMW and ∠XMZ are vertical angles and therefore congruent. 3. **Triangle Congruence:** - Using the properties of the triangles and their corresponding parts, △YMW is congruent to △ZMX. 4. **Segment Congruence:** - Given the congruency of the triangles, YW is congruent to XZ. **Conclusion:** Utilizing the definition of midpoints and the vertical angle theorem, we have successfully proven that segment YW is congruent to segment XZ. This exercise highlights the application of geometric properties to establish congruency in segments created by intersecting lines.
Transcribed Image Text:**Title: Proving Congruent Segments in Intersecting Lines** **Introduction:** In this exercise, we are given a geometric diagram with intersecting lines and specific midpoints. The goal is to prove that certain segments are congruent using the properties of midpoints and vertical angles. **Diagram Explanation:** The diagram consists of two intersecting lines, forming a shape with vertices labeled Y, W, X, and Z. The point M is located at the intersection, serving as the midpoint of segments WX and YZ. **Given:** - M is the midpoint of segment WX. - M is the midpoint of segment YZ. **To Prove:** - Segment YW is congruent to segment XZ. **Proof Structure:** | Statements | Reasons | |---------------------------------------------|----------------------------------| | M is the midpoint of WX | Given | | M is the midpoint of YZ | Given | | YM ≅ MZ, WM ≅ MX | Definition of midpoint | | ∠YMW ≅ ∠XMZ | Vertical angle theorem | | △YMW ≅ △ZMX | (Triangular congruence criteria) | | YW ≅ XZ | (Derived from congruent triangles) | **Explanation:** 1. **Midpoint Definition:** - As M is the midpoint of WX, segments WM and MX are congruent. - Similarly, since M is the midpoint of YZ, segments YM and MZ are congruent. 2. **Vertical Angles:** - Angles ∠YMW and ∠XMZ are vertical angles and therefore congruent. 3. **Triangle Congruence:** - Using the properties of the triangles and their corresponding parts, △YMW is congruent to △ZMX. 4. **Segment Congruence:** - Given the congruency of the triangles, YW is congruent to XZ. **Conclusion:** Utilizing the definition of midpoints and the vertical angle theorem, we have successfully proven that segment YW is congruent to segment XZ. This exercise highlights the application of geometric properties to establish congruency in segments created by intersecting lines.
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