Question

1. given: \\( \overline { f h } \\) bisects \\( \angle e h g, \angle f e h \cong \angle f g h \\) prove: \\( \triangle f e h \cong \triangle f g h \\) 5. given: \\( \angle r p q \cong \angle r t s, r \\) is the midpoint of \\( \overline { q s } \\) prove: \\( \triangle p q r \cong \triangle t s r \\) complete proof 6 using hypotenuse-leg (hl). 6. given: \\( \triangle a b c \\) and \\( \triangle d c b \\) are right triangles, \\( \overline { a b } \cong \overline { d c } \\) prove: \\( \triangle a b c \cong \triangle d c b \\)

Explanation:

Step1: For problem 4

• Statement 1: \(\overline{FH}\) bisects \(\angle EHG\)
• Reason 1: Given
• Statement 2: \(\angle EHF=\angle GHF\)
• Reason 2: Definition of angle - bisector
• Statement 3: \(\angle FEH\cong\angle FGH\)
• Reason 3: Given
• Statement 4: \(\overline{FH}\cong\overline{FH}\)
• Reason 4: Reflexive property of congruence
• Statement 5: \(\triangle FEH\cong\triangle FGH\)
• Reason 5: AAS (Angle - Angle - Side) congruence criterion

Step2: For problem 5

• Statement 1: \(R\) is the mid - point of \(\overline{QS}\)
• Reason 1: Given
• Statement 2: \(\overline{QR}\cong\overline{SR}\)
• Reason 2: Definition of mid - point
• Statement 3: \(\angle RPQ\cong\angle RTS\)
• Reason 3: Given
• Statement 4: \(\angle PRQ\cong\angle TRS\)
• Reason 4: Vertical angles are congruent
• Statement 5: \(\triangle PQR\cong\triangle TSR\)
• Reason 5: AAS (Angle - Angle - Side) congruence criterion

Step3: For problem 6

• Statement 1: \(\triangle ABC\) and \(\triangle DCB\) are right - triangles
• Reason 1: Given
• Statement 2: \(\overline{AB}\cong\overline{DC}\)
• Reason 2: Given
• Statement 3: \(\overline{BC}\cong\overline{CB}\)
• Reason 3: Reflexive property of congruence
• Statement 4: \(\triangle ABC\cong\triangle DCB\)
• Reason 4: HL (Hypotenuse - Leg) congruence criterion

Answer:

For problem 4:

Statements	Reasons
2. \(\angle EHF=\angle GHF\)	2. Definition of angle - bisector
3. \(\angle FEH\cong\angle FGH\)	3. Given
4. \(\overline{FH}\cong\overline{FH}\)	4. Reflexive property of congruence
5. \(\triangle FEH\cong\triangle FGH\)	5. AAS

For problem 5:

Statements	Reasons
2. \(\overline{QR}\cong\overline{SR}\)	2. Definition of mid - point
3. \(\angle RPQ\cong\angle RTS\)	3. Given
4. \(\angle PRQ\cong\angle TRS\)	4. Vertical angles are congruent
5. \(\triangle PQR\cong\triangle TSR\)	5. AAS

For problem 6:

Statements	Reasons
2. \(\overline{AB}\cong\overline{DC}\)	2. Given
3. \(\overline{BC}\cong\overline{CB}\)	3. Reflexive property of congruence
4. \(\triangle ABC\cong\triangle DCB\)	4. HL