Question
Rachel has been working on a proof of the Base Angles Theorem. She marked up a triangle to help her with the proof. Help her to construct a viable argument. Fill in the missing statement and the missing reason in her proof.
Given: Isosceles △EFG
with EG¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯
Prove: ∠G≅∠F
Statements Reasons
1. Isosceles △EFG
with EG¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯
1. Given
2. K
, the midpoint of GF¯¯¯¯¯¯¯¯
EK¯¯¯¯¯¯¯¯
through points E
and K
2. Construction
3. GK¯¯¯¯¯¯¯¯≅KF¯¯¯¯¯¯¯¯
3. Definition of midpoint
4. 4. Reflexive property of congruence
5. △EKG≅△EKF
5. SSS Congruence Theorem
6. ∠G≅∠F
6.
(1 point)
Responses
Statement 4 should say "△EFG≅△EFG
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " triangle upper E upper F upper G congruent to triangle upper E upper F upper G ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "∠GEK≅∠FEK
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " angle upper G upper E upper K congruent to angle upper F upper E upper K ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " Modifying above upper E upper K with bar congruent to Modifying above upper E upper K with bar ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯
," and Reason 6 should say “Base Angles Theorem.”
Given: Isosceles △EFG
with EG¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯
Prove: ∠G≅∠F
Statements Reasons
1. Isosceles △EFG
with EG¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯
1. Given
2. K
, the midpoint of GF¯¯¯¯¯¯¯¯
EK¯¯¯¯¯¯¯¯
through points E
and K
2. Construction
3. GK¯¯¯¯¯¯¯¯≅KF¯¯¯¯¯¯¯¯
3. Definition of midpoint
4. 4. Reflexive property of congruence
5. △EKG≅△EKF
5. SSS Congruence Theorem
6. ∠G≅∠F
6.
(1 point)
Responses
Statement 4 should say "△EFG≅△EFG
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " triangle upper E upper F upper G congruent to triangle upper E upper F upper G ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "∠GEK≅∠FEK
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " angle upper G upper E upper K congruent to angle upper F upper E upper K ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯
," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say " Modifying above upper E upper K with bar congruent to Modifying above upper E upper K with bar ," and Reason 6 should say “CPCTC Theorem.”
Statement 4 should say "EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯
," and Reason 6 should say “Base Angles Theorem.”
Answers
GPT-4o mini
In order to complete Rachel's proof of the Base Angles Theorem, we need to look at Statement 4 and Reason 6, which relate to the angles in the isosceles triangle.
Here's how we should fill in the missing parts:
### Completed Proof
- **Statement 4**: ∠GEK ≅ ∠FEK
- **Reason 4**: Reflexive property of congruence
- **Statement 6**: ∠G ≅ ∠F
- **Reason 6**: CPCTC (Corresponding Parts of Congruent Triangles are Congruent Theorem)
The reason for Statement 6 being "CPCTC Theorem" is appropriate because once we prove that triangles EKG and EKF are congruent, we can deduce that their corresponding parts (the base angles ∠G and ∠F) are congruent as well.
Thus, the correct option would be:
**Statement 4 should say "∠GEK≅∠FEK," and Reason 6 should say "CPCTC Theorem."**
Here's how we should fill in the missing parts:
### Completed Proof
- **Statement 4**: ∠GEK ≅ ∠FEK
- **Reason 4**: Reflexive property of congruence
- **Statement 6**: ∠G ≅ ∠F
- **Reason 6**: CPCTC (Corresponding Parts of Congruent Triangles are Congruent Theorem)
The reason for Statement 6 being "CPCTC Theorem" is appropriate because once we prove that triangles EKG and EKF are congruent, we can deduce that their corresponding parts (the base angles ∠G and ∠F) are congruent as well.
Thus, the correct option would be:
**Statement 4 should say "∠GEK≅∠FEK," and Reason 6 should say "CPCTC Theorem."**