Question

y is the midpoint of \\(\overline{vx}\\) and \\(\overline{uw}\\). complete the proof that \\(\triangle vwy \cong \triangle xuy\\).

Explanation:

Step1: Vertical Angles

Vertical angles are equal. So, \(\angle VYW=\angle UYX\)

Step2: Definition of Midpoint

Since \(Y\) is the midpoint of \(VX\) and \(UW\), by the definition of midpoint, \(VY = XY\) and \(UY=WY\)

Step3: SAS (Side - Angle - Side) Congruence

In \(\triangle VWY\) and \(\triangle XUY\), we have \(VY = XY\) (from mid - point), \(\angle VYW=\angle UYX\) (vertical angles), \(WY = UY\) (from mid - point). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle VWY\cong\triangle XUY\)

Answer:

1. Given; 2. Given; 3. Vertical Angles Theorem; 4. Definition of midpoint; 5. SAS (Side - Angle - Side) Congruence Postulate; 6. \(\triangle VWY\cong\triangle XUY\) (by SAS)