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Question

An isosceles triangle with a downward orientation is marked counter-clockwise from the bottom vertex as upper E upper F upper G. Sides upper E upper F and upper E upper G are both marked with a single congruent tick mark. A line bisects the triangle.

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 "EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯
," and Reason 6 should say “Base Angles 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 “Base Angles Theorem.”

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.”
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Answer 1

To complete Rachel's proof of the Base Angles Theorem, the missing statements and reasons can be filled in as follows:

For Statement 4, it should state: EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯ This is correct because it's referring to the segments that were shown to be congruent as part of the construction.

For Reason 6, it should state: CPCTC Theorem. This means "Corresponding Parts of Congruent Triangles are Congruent," which is valid because triangles EKG and EKF are shown to be congruent, allowing us to conclude that their corresponding angles are also congruent.

So the completed proof would look like this:

| Statements | Reasons | |-------------------------------------------------|--------------------------------| | 1. Isosceles △EFG with EG¯¯¯¯¯¯¯¯≅EF¯¯¯¯¯¯¯¯ | 1. Given | | 2. K, the midpoint of GF¯¯¯¯¯¯¯¯ | 2. Construction | | 3. GK¯¯¯¯¯¯¯¯≅KF¯¯¯¯¯¯¯¯¯ | 3. Definition of midpoint | | 4. EK¯¯¯¯¯¯¯¯≅EK¯¯¯¯¯¯¯¯¯ | 4. Reflexive property of congruence | | 5. △EKG≅△EKF | 5. SSS Congruence Theorem | | 6. ∠G≅∠F | 6. CPCTC Theorem |

This ensures the proof follows correctly to show that the angles opposite the congruent sides of the isosceles triangle are, indeed, congruent as stated by the Base Angles Theorem.