Question

triangle similarity: aa
proving triangle similarity
given: $overline{fh} \perp overline{gh}$, $overline{kj} \perp overline{gj}$
prove: $\triangle fhg \sim \triangle kjg$
identify the steps that complete the proof.

statements	reasons
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. $\clubsuit$
5. $\angle fgh$ and $\angle kgj$ are vertical angles	5. def. of vertical angles
6. $\diamondsuit$	6. vertical angles theorem
7. $\triangle fhg \sim \triangle kjg$	7. $\spadesuit$

(image of triangles with right angles at h and j, and point g on the line between h and ks vertical segment)

Explanation:

Step1: Analyze Step 4 (Reason)

We know that \(\angle FHG\) and \(\angle GJK\) are right angles (from Step 3), and we need to relate \(\angle FHG\) and \(\angle GJK\). Since all right angles are congruent, the reason for \(\angle FHG\cong\angle GJK\) is "all right angles are congruent".

Step2: Analyze Step 6 (Statement)

We have vertical angles \(\angle FGH\) and \(\angle KGJ\) (from Step 5), and by the vertical angles theorem, they are congruent. So the statement for Step 6 should be \(\angle FGH\cong\angle KGJ\).

Step3: Analyze Step 7 (Reason)

To prove \(\triangle FHG\sim\triangle KJG\), we use the AA (Angle - Angle) similarity criterion. We have \(\angle FHG\cong\angle GJK\) (right angles) and \(\angle FGH\cong\angle KGJ\) (vertical angles), so the reason is "AA (Angle - Angle) similarity criterion".

Answer:

• Step 4 Reason: all right angles are congruent
• Step 6 Statement: \(\angle FGH\cong\angle KGJ\)
• Step 7 Reason: AA (Angle - Angle) similarity criterion