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

Explanation:

Step1: Define right - angles

Since $\overline{HL}\perp\overline{GI}$ and $\overline{JK}\perp\overline{GI}$, we get $\angle HLG=\angle JKI = 90^{\circ}$ based on the definition of perpendicular lines.

Step2: Segment addition

Using the segment - addition postulate, we express $GL$ and $KI$ in terms of the given segments.

Step3: Substitute segments

Substitute the given equal segments ($GK = LI$) into the segment - addition expressions to show $GL = KI$.

Step4: List congruent parts

List the congruent parts of the two triangles: a pair of right - angles, and two pairs of equal sides.

Step5: Apply congruence criterion

Apply either the HL (if right - triangles) or SSS congruence criterion to prove the triangles congruent.

Answer:

1. Since $\overline{HL}\perp\overline{GI}$ and $\overline{JK}\perp\overline{GI}$, $\angle HLG = \angle JKI=90^{\circ}$ (Definition of perpendicular lines).
2. Given $GH = JI$ and $GK = LI$.
3. $GK + KL=GL$ and $LI + KL = KI$ (Segment - addition postulate).
4. By substitution, $GL = KI$ (Substitution property, using step 2 and 3).
5. In $\triangle GLH$ and $\triangle IKJ$, we have:

• $\angle HLG=\angle JKI$ (from step 1).
• $GL = KI$ (from step 4).
• $GH = JI$ (given).

1. So, $\triangle GLH\cong\triangle IKJ$ by the Hypotenuse - Leg (HL) congruence criterion (if we consider right - triangles, and if the problem is about right - triangles; or by SSS congruence criterion if we just consider the side - side - side relationship).