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 ∠uxy ≅ ∠vyw 6 △uxy ≅ △wvy 7 ∠w ≅ ∠u

Explanation:

Step1: Definition of midpoint

By the definition of a midpoint, if \(Y\) is the midpoint of \(\overline{UW}\), then \(UY = WY\), so \(UY\cong WY\) (by the definition of congruent segments: segments that have the same length are congruent).

Step2: Vertical angles

\(\angle UYX\) and \(\angle VYW\) are vertical angles. Vertical angles are congruent, so \(\angle UYX\cong\angle VYW\).

Step3: Definition of midpoint

Since \(Y\) is the midpoint of \(\overline{VX}\), then \(VY = XY\), so \(VY\cong XY\) (by the definition of congruent segments).

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

In \(\triangle UXY\) and \(\triangle W VY\), we have \(UY\cong WY\) (from step 1), \(\angle UYX\cong\angle VYW\) (from step 2), and \(VY\cong XY\) (from step 3). By the SAS congruence criterion, \(\triangle UXY\cong\triangle WVY\).

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

Since \(\triangle UXY\cong\triangle WVY\), then \(\angle W\cong\angle U\) (by CPCTC).

Answer:

1. Given (definition of mid - point)
2. Given (definition of mid - point)
3. Definition of mid - point (\(UY = WY\Rightarrow UY\cong WY\))
4. Vertical angles theorem (\(\angle UYX\) and \(\angle VYW\) are vertical angles)
5. Definition of mid - point (\(VY = XY\Rightarrow VY\cong XY\))
6. SAS (\(\triangle UXY\) and \(\triangle WVY\): \(UY\cong WY\), \(\angle UYX\cong\angle VYW\), \(VY\cong XY\))
7. CPCTC (\(\triangle UXY\cong\triangle WVY\Rightarrow\angle W\cong\angle U\))