Question

given: gj = 2, gk = 3, jh = 6, ki = 9
prove: △gjk ~ △ghi

1. given
2. choose your answer...
3. ∠igh ≅ ∠kgj
4. choose your answer...
5. △gjk ~ △ghi
6. given
7. choose your answer...
8. division property of equality
9. reflexive property
10. sas-

Explanation:

Step1: Find the ratio of sides

We are given \(GJ = 2\), \(GK=3\), \(JH = 6\), \(KI = 9\).
The ratio of \(\frac{GJ}{JH}=\frac{2}{6}=\frac{1}{3}\) and \(\frac{GK}{KI}=\frac{3}{9}=\frac{1}{3}\).

Step2: Check the included angle

We need to check the included angle. \(\angle JGK\) and \(\angle GHI\) are the included angles.
Since \(\angle JGK\cong\angle GHI\) (by the reflexive property of angles in similar - triangle proof setup, when we consider the angle - angle - side or side - angle - side relations for similarity).

Step3: Apply the SAS (Side - Angle - Side) similarity criterion

For two triangles \(\triangle GJK\) and \(\triangle GHI\), if \(\frac{GJ}{JH}=\frac{GK}{KI}\) (from Step 1) and \(\angle JGK\cong\angle GHI\) (Step 2), then by the SAS (Side - Angle - Side) similarity criterion, \(\triangle GJK\sim\triangle GHI\).

Answer:

\(\triangle GJK\sim\triangle GHI\) by the SAS (Side - Angle - Side) similarity criterion.