Question

i is the midpoint of \\(\overline{eg}\\). complete the proof that \\(\overline{eh} \cong \overline{fg}\\).

(image of two triangles with vertices h, e, i and g, f, i, with i being the intersection point of eg and hf. the triangles have marked congruent angles.)

	statement	reason
2	\\(\overline{fg} \parallel \overline{eh}\\)
3	\\(\angle e \cong \angle g\\)
4	\\(\angle f \cong \angle h\\)
5	\\(\overline{ei} \cong \overline{gi}\\)
6	\\(\triangle ehi \cong \triangle gfi\\)
7	\\(\overline{eh} \cong \overline{fg}\\)

Explanation:

Step1: Reason for Statement 1

The definition of a midpoint states that a midpoint of a segment divides the segment into two congruent segments. So the reason for "I is the midpoint of $\overline{EG}$" is "Given" (since it's provided in the problem).

Step2: Reason for Statement 2

From the diagram, the arrows on $\overline{FG}$ and $\overline{EH}$ indicate they are parallel. So the reason for $\overline{FG} \parallel \overline{EH}$ is "Given (from diagram, indicated by arrows)".

Step3: Reason for Statement 3

When two parallel lines are cut by a transversal (here $\overline{EG}$), alternate interior angles are congruent. So $\angle E \cong \angle G$ because they are alternate interior angles (due to $\overline{FG} \parallel \overline{EH}$ and transversal $\overline{EG}$).

Step4: Reason for Statement 4

Similarly, $\angle F \cong \angle H$ are alternate interior angles (due to $\overline{FG} \parallel \overline{EH}$ and transversal $\overline{FH}$ or the other transversal formed by the intersection at $I$).

Step5: Reason for Statement 5

Since $I$ is the midpoint of $\overline{EG}$, by the definition of midpoint, $\overline{EI} \cong \overline{GI}$.

Step6: Reason for Statement 6

For $\triangle EHI \cong \triangle GFI$, we can use the AAS (Angle - Angle - Side) congruence criterion. We have $\angle E \cong \angle G$, $\angle H \cong \angle F$, and $\overline{EI} \cong \overline{GI}$, so AAS applies.

Step7: Reason for Statement 7

Corresponding parts of congruent triangles are congruent (CPCTC). Since $\triangle EHI \cong \triangle GFI$, their corresponding sides $\overline{EH}$ and $\overline{FG}$ are congruent.

Answer:

1. Given
2. Given (from diagram, parallel lines indicated by arrows)
3. Alternate Interior Angles Theorem (due to $\overline{FG} \parallel \overline{EH}$ and transversal $\overline{EG}$)
4. Alternate Interior Angles Theorem (due to $\overline{FG} \parallel \overline{EH}$ and transversal $\overline{FH}$)
5. Definition of Midpoint
6. AAS (Angle - Angle - Side) Congruence Criterion
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)