Question

3
given: (overline{ac} cong overline{de}), (b) is the midpoint of (overline{ad}) and (overline{ce})
prove: (\triangle abc cong \triangle dbe)

statements	reasons
2. (b) is the midpoint of (overline{ad}) and (overline{ce})	2.
3.	3. definition of midpoint
4.	4. definition of midpoint
5. (\triangle abc cong \triangle dbe)	5.

7
given: (su) and (rt) bisect each other, (overline{st} cong overline{ur})
prove: (angle sti cong angle uri)

statements	reasons
2. (overline{si} cong overline{ui}); (overline{ri} cong overline{ti})	2.
3.	3. given
4.	4. side - side - side (sss)
5. (angle sti cong angle uri)	5.

Explanation:

Problem 3 (Triangle Congruence Proof)

Step 1: Identify Given Information

• Statement 1: \(\overline{AC} \cong \overline{DE}\) (Reason: Given)
• Statement 2: \(B\) is the midpoint of \(\overline{AD}\) and \(\overline{CE}\) (Reason: Given)

Step 2: Use Midpoint Definition

• Statement 3: \(\overline{AB} \cong \overline{DB}\) (Reason: Definition of Midpoint, since \(B\) is midpoint of \(\overline{AD}\))
• Statement 4: \(\overline{CB} \cong \overline{EB}\) (Reason: Definition of Midpoint, since \(B\) is midpoint of \(\overline{CE}\))

Step 3: Prove Triangle Congruence

• Statement 5: \(\triangle ABC \cong \triangle DBE\) (Reason: Side - Side - Side (SSS) Congruence Postulate, as \(\overline{AC} \cong \overline{DE}\), \(\overline{AB} \cong \overline{DB}\), \(\overline{CB} \cong \overline{EB}\))

Step 1: Identify Given Information

• Statement 1: \(SU\) and \(RT\) bisect each other (Reason: Given)
• Statement 2: \(\overline{SI} \cong \overline{UI}\); \(\overline{RI} \cong \overline{TI}\) (Reason: Definition of Bisector, since \(SU\) and \(RT\) bisect each other, they divide each other into two congruent segments)

Step 2: Use Given Side Congruence

• Statement 3: \(\overline{ST} \cong \overline{UR}\) (Reason: Given)

Step 3: Prove Triangle Congruence

• Statement 4: \(\triangle STI \cong \triangle URI\) (Reason: Side - Side - Side (SSS) Congruence Postulate, as \(\overline{SI} \cong \overline{UI}\), \(\overline{RI} \cong \overline{TI}\), \(\overline{ST} \cong \overline{UR}\))

Step 4: Prove Angle Congruence

• Statement 5: \(\angle STI \cong \angle URI\) (Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Answer:

Statements	Reasons
2. \(B\) is the midpoint of \(\overline{AD}\) and \(\overline{CE}\)	2. Given
3. \(\overline{AB} \cong \overline{DB}\)	3. Definition of Midpoint
4. \(\overline{CB} \cong \overline{EB}\)	4. Definition of Midpoint
5. \(\triangle ABC \cong \triangle DBE\)	5. SSS Congruence Postulate

Problem 7 (Angle Congruence Proof)