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.

Explanation:

Step1: Recall right - angle definition

Since $\overline{HL}\perp\overline{GI}$ and $\overline{JK}\perp\overline{GI}$, $\angle HLG = \angle JKI=90^{\circ}$ (Perpendicular lines form right angles).

Step2: Use segment addition

We know that $GK = LI$. Let $GL + LK=GK$ and $LK + KI=LI$. So, $GL + LK=LK + KI$. By the Subtraction Property of Equality, we can subtract $LK$ from both sides to get $GL = KI$.

Step3: Apply Hypotenuse - Leg (HL) congruence

We are given $GH = JI$. In right - triangles $\triangle GLH$ and $\triangle IKJ$, we have:

• Hypotenuse $GH = JI$ (given)
• Leg $GL = KI$ (proven above)
• $\angle HLG=\angle JKI = 90^{\circ}$ (proven above)

So, $\triangle GLH\cong\triangle IKJ$ by the Hypotenuse - Leg (HL) congruence criterion.

Answer:

$\triangle GLH\cong\triangle IKJ$ by the Hypotenuse - Leg (HL) congruence criterion.