Question

4 finish the triangle proof by dragging the correct reasons to their box. there will be two extra unused reasons! given: △def and △hgf are right triangles, eg and dh bisect each other at f prove ∠edf ≅ ∠ghf

Explanation:

Step1: Fill in the first statement

The first statement should be " \(\triangle DEF\) and \(\triangle HGF\) are right - triangles". The reason is "Given" as it is provided in the problem statement.

Step2: Analyze the second reason

Since \(EG\) and \(DH\) bisect each other at \(F\) (statement 2), the reason is "Definition of Bisect".

Step3: Analyze the third reason

If \(EG\) and \(DH\) bisect each other at \(F\), then \(DF = HF\) (statement 3) and \(EF=GF\) (another statement which can be filled). The reason for \(DF = HF\) (and \(EF = GF\)) is "Definition of Bisect".

Step4: Find the angle - related reason

\(\angle EFD=\angle HFG\) (vertical angles). The reason is "Vertical Angles".

Step5: Prove triangle congruence

We have two right - triangles (\(\triangle DEF\) and \(\triangle HGF\)), \(DF = HF\), \(EF = GF\), and \(\angle EFD=\angle HFG\). By \(SAS\) (Side - Angle - Side) congruence criterion (\(EF = GF\), \(\angle EFD=\angle HFG\), \(DF = HF\)), \(\triangle DEF\cong\triangle HGF\) (statement 5). The reason is \(SAS\).

Step6: Prove the angle congruence

Since \(\triangle DEF\cong\triangle HGF\), then \(\angle EDF\cong\angle GHF\) (statement 6). The reason is \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

Statement	Reason
2. \(EG\) and \(DH\) bisect each other at \(F\)	2. Definition of Bisect
3. \(DF = HF\), \(EF = GF\)	3. Definition of Bisect
4. \(\angle EFD=\angle HFG\)	4. Vertical Angles
5. \(\triangle DEF\cong\triangle HGF\)	5. \(SAS\)
6. \(\angle EDF\cong\angle GHF\)	6. \(CPCTC\)