Question

△uvx ≅ △wtx. complete the proof that △vwx ≅ △tux.
(image of a quadrilateral with vertices u, v, w, t and diagonals intersecting at x)
statement | reason
1 △uvx ≅ △wtx | dropdown
2 (overline{tx} cong overline{vx}) | dropdown
3 (overline{ux} cong overline{wx}) | dropdown
4 (angle txu cong angle vxw) | dropdown
5 △vwx ≅ △tux | dropdown

Explanation:

Step1: Reason for Statement 1

Given that \(\triangle UVX \cong \triangle WTX\), so the reason is "Given".

Step2: Reason for Statement 2

From \(\triangle UVX \cong \triangle WTX\), corresponding parts are congruent. So \(TX \cong VX\) (corresponding sides of congruent triangles are congruent, CPCTC).

Step3: Reason for Statement 3

From \(\triangle UVX \cong \triangle WTX\), \(UX \cong WX\) (CPCTC).

Step4: Reason for Statement 4

\(\angle TXU\) and \(\angle VXW\) are vertical angles, and vertical angles are congruent. So the reason is "Vertical angles are congruent".

Step5: Reason for Statement 5

We have \(TX \cong VX\), \(UX \cong WX\), and \(\angle TXU \cong \angle VXW\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle VWX \cong \triangle TUX\). So the reason is "SAS (Side - Angle - Side) congruence criterion".

Answer:

1. Reason: Given
2. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
3. Reason: CPCTC
4. Reason: Vertical angles are congruent
5. Reason: SAS (Side - Angle - Side) congruence criterion