Question

given: $overline{hl}perpoverline{gi}$, $overline{jk}perpoverline{gi}$, $gh = ji$, $gk = li$. prove: $\triangle glhcong\triangle ikj$. 6. substitution property. 7. subtraction property. 8. given. 9. hypotenuse - leg

Explanation:

Step1: Find equal segments

Since \(GK = LI\), by segment - addition postulate \(GL+LK=LK + KI\), then \(GL = KI\) by substitution.

Step2: Find right - angles

Using the definition of perpendicular lines, \(\angle GLH=\angle IKJ = 90^{\circ}\) as \(\overline{HL}\perp\overline{GI}\) and \(\overline{JK}\perp\overline{GI}\).

Step3: Recall given side

Given \(GH = JI\).

Step4: Prove congruence

In right - triangles \(\triangle GLH\) and \(\triangle IKJ\), with \(GH = JI\) (hypotenuses) and \(GL = KI\) (legs), we use HL theorem to prove \(\triangle GLH\cong\triangle IKJ\).

Answer:

1. Since \(GK = LI\), we can say \(GL+LK=LK + KI\) (Segment - Addition Postulate). Then by substitution property, \(GL = KI\).
2. We know that \(\overline{HL}\perp\overline{GI}\) and \(\overline{JK}\perp\overline{GI}\), so \(\angle GLH=\angle IKJ = 90^{\circ}\) (Definition of perpendicular lines).
3. Given \(GH = JI\).
4. In right - triangles \(\triangle GLH\) and \(\triangle IKJ\), we have \(GH = JI\) (hypotenuses) and \(GL = KI\) (legs), so \(\triangle GLH\cong\triangle IKJ\) by the Hypotenuse - Leg (HL) congruence theorem.