Given: UR=VW Prove: UV=RW(Midpoint)
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To prove that UV = RW, we need to show that U and R are the midpoints of UV and RW, respectively.
Here's how we can approach the proof:
1. Given that UR = VW, we can start by assuming that U and R are midpoints of UV and RW, respectively.
2. Let's draw a diagram with line segments UV and RW intersecting at point P, which is the midpoint of both UV and RW.
3. By definition, the midpoint of a line segment divides it into two equal parts. Therefore, UP = PV and RP = PW.
4. Consider the triangles URP and VWP. Using the Side-Side-Side (SSS) congruence criterion, we can show that these triangles are congruent.
a. UR = VW (Given)
b. UP = PV (By the midpoint property)
c. RP = PW (By the midpoint property)
d. Therefore, by SSS, URP ≅ VWP.
5. By triangle congruence, corresponding angles are also congruent. Therefore, angle URP ≅ angle VWP.
6. Since angle URP is formed by UV and RW, and angle VWP is formed by WV and RP (which is RW), we can conclude that angle UVW ≅ angle WRP.
7. Since angles UVW and WRP are corresponding angles, it follows that they are equal in measure.
8. Now, let's consider the triangle UVW. Using the Angle-Side-Angle (ASA) congruence criterion, we can prove that UV ≅ RW.
a. Angle UVW ≅ angle WRP (proved in the previous step)
b. VW = VW (Reflexive property)
c. Therefore, by ASA, UVW ≅ RWP.
9. By triangle congruence, corresponding sides are also congruent. Therefore, UV ≅ RW.
10. Since we assumed that U and R are the midpoints of UV and RW, respectively, and we have shown that UV ≅ RW, we can conclude that U and R are indeed the midpoints of UV and RW.
Therefore, we have proved that if UR = VW, then UV = RW (midpoint).