Question

3.) given: ( overline{sw} cong overline{yw} ), t is the midpoint of ( overline{sy} ) prove: ( angle s cong angle y )

Explanation:

Step1: Use the definition of midpoint

Since \(T\) is the mid - point of \(SY\), by the definition of a midpoint, \(ST\cong TY\).

Step2: Use the reflexive property

\(WT\cong WT\) (reflexive property: any segment is congruent to itself).

Step3: Prove triangle congruence

We know \(SW\cong YW\) (given), \(ST\cong TY\) (from step 1), and \(WT\cong WT\) (from step 2). By the SSS (Side - Side - Side) congruence criterion, \(\triangle SWT\cong\triangle YWT\).

Step4: Use the property of congruent triangles

If \(\triangle SWT\cong\triangle YWT\), then by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) property, \(\angle S\cong\angle Y\).

Answer:

A. Given; B. \(ST\cong TY\); C. \(WT\cong WT\); D. SSS (Side - Side - Side) Congruence; E. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)