Question

given: $overline{fh} perp overline{gh}$, $overline{kj} perp overline{gj}$
prove: $\triangle fhg sim \triangle kjg$
diagram: f, h (right angle), g, j (right angle), k
identify the steps that complete the proof.
dropdowns: sss similarity theorem, asa similarity theorem, aa similarity theorem, sas similarity theorem
statements | reasons
--- | ---

1. $overline{fh} perp overline{gh}$ | 1. given
2. $overline{kj} perp overline{gj}$ | 2. given
3. $angle fhg$ and $angle gjk$ are right angles | 3. def. of perpendicular lines
4. $angle fhg cong angle gjk$ | 4. ♦
5. $angle fgh$ and $angle kgj$ are vertical angles | 5. def. of vertical angles
6. ♦ | 6. vertical angles theorem
7. $\triangle fhg sim \triangle kjg$ | 7. ♦

Explanation:

Step 1: Reason for ∠FHG ≅ ∠GJK

All right angles are congruent. So the reason for \( \angle FHG \cong \angle GJK \) is "all right angles are congruent".

Step 2: Statement for Vertical Angles Theorem

The vertical angles theorem states that vertical angles are congruent. So statement 6 should be \( \angle FGH \cong \angle KGJ \).

Step 3: Reason for Triangle Similarity

We have two pairs of congruent angles (\( \angle FHG \cong \angle GJK \) and \( \angle FGH \cong \angle KGJ \)). By the AA (Angle - Angle) similarity theorem, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. So the reason for \( \triangle FHG \sim \triangle KJG \) is the AA similarity theorem.

Answer:

• For step 4 reason: all right angles are congruent
• For step 6 statement: \( \angle FGH \cong \angle KGJ \)
• For step 7 reason: AA similarity theorem