Question

y is the midpoint of uw and vx. complete the proof that ∠w ≅ ∠u.
1 y is the midpoint of uw
2 y is the midpoint of vx
3 uy ≅ wy
4 vy ≅ xy
5 ∠uyx ≅ ∠vyw
6 △uxy ≅ △wvy
7 ∠w ≅ ∠u

Explanation:

Step1: Definition of mid - point

If \(Y\) is the mid - point of \(\overline{UW}\), then \(UY = WY\) (by the definition of a mid - point: a mid - point of a line segment divides the segment into two equal parts).

Step2: Definition of mid - point

If \(Y\) is the mid - point of \(\overline{VX}\), then \(VY=XY\) (by the definition of a mid - point: a mid - point of a line segment divides the segment into two equal parts).

Step3: Vertical angles

\(\angle UYX\) and \(\angle VYW\) are vertical angles. By the vertical angles theorem, \(\angle UYX\cong\angle VYW\) (vertical angles are congruent).

Step4: SAS (Side - Angle - Side) congruence criterion

In \(\triangle UXY\) and \(\triangle WYY\), we have \(UY = WY\) (from Step 1), \(\angle UYX=\angle VYW\) (from Step 3), and \(VY = XY\) (from Step 2). So, \(\triangle UXY\cong\triangle WYY\) by the SAS (Side - Angle - Side) congruence criterion.

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle UXY\cong\triangle WYY\), then \(\angle W\cong\angle U\) (by CPCTC: if two triangles are congruent, then their corresponding angles are congruent).

Answer:

1. Given (definition of mid - point)
2. Given (definition of mid - point)
3. Vertical angles theorem
4. SAS (Side - Angle - Side)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)