Question

given: $overline{hl}perpoverline{gi}$, $overline{jk}perpoverline{gi}$, $gh = ji$, $gk = li$. prove: $\triangle glhcong\triangle ikj$. 1. given 2. perpendicular lines intersect to form right angles. 3. $\triangle glh$ and $\triangle ikj$ are right triangles. 3. definition of a right triangle 4. $gk = gl + lk$ and $li = ik + lk$

Explanation:

Step1: State given information

Given the perpendicularity and side - length equalities.

Step2: Determine right - angles

Use the property of perpendicular lines.

Step3: Identify right - triangles

Apply the definition of a right triangle.

Step4: Prove equal side - lengths

Use the segment addition postulate and subtraction property of equality.

Step5: Prove triangle congruence

Apply the HL (Hypotenuse - Leg) congruence theorem for right - triangles.

Answer:

1. $\overline{HL}\perp\overline{GI},\overline{JK}\perp\overline{GI}, GH = JI, GK = LI$
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. Since $GK = GL+LK$ and $LI = IK + LK$ and $GK = LI$, then $GL+LK=IK + LK$, so $GL = IK$ (Subtraction property of equality)
5. In right - triangles $\triangle GLH$ and $\triangle IKJ$, $GH = JI$ (given) and $GL = IK$ (proven above), so $\triangle GLH\cong\triangle IKJ$ (HL congruence theorem for right - triangles)