Question

given: hl ⊥ gi, jk ⊥ gi, gh = ji, gk = li
prove: △glh ≅ △ikj

1. given
2. perpendicular lines intersect to form right angles.
3. △glh and △ikj are right triangles.
4. gk = gl + lk and li = ik + lk

Explanation:

Step1: State given information

Given: $\overline{HL}\perp\overline{GI}$, $\overline{JK}\perp\overline{GI}$, $GH = JI$, $GK=LI$

Step2: Define right - angles

Since $\overline{HL}\perp\overline{GI}$ and $\overline{JK}\perp\overline{GI}$, by the definition of perpendicular lines, $\angle GLH=\angle IKJ = 90^{\circ}$

Step3: Identify right - triangles

$\triangle GLH$ and $\triangle IKJ$ are right - triangles because they each have a right angle (by the definition of a right - triangle)

Step4: Use segment addition postulate

We know that $GK = GL+LK$ and $LI=IK + LK$. Since $GK = LI$, then $GL+LK=IK + LK$. By the subtraction property of equality, $GL=IK$

Step5: Prove congruence

In right - triangles $\triangle GLH$ and $\triangle IKJ$, we have $GH = JI$ (given) and $GL=IK$ (proven above). By the Hypotenuse - Leg (HL) congruence theorem for right - triangles, $\triangle GLH\cong\triangle IKJ$

Answer:

1. Given: $\overline{HL}\perp\overline{GI}$, $\overline{JK}\perp\overline{GI}$, $GH = JI$, $GK = LI$
2. $\angle GLH=\angle IKJ = 90^{\circ}$ (Perpendicular lines intersect to form right angles)
3. $\triangle GLH$ and $\triangle IKJ$ are right - triangles (Definition of a right - triangle)
4. $GL = IK$ (Segment addition postulate and subtraction property of equality)
5. $\triangle GLH\cong\triangle IKJ$ (HL congruence theorem for right - triangles)