Question

8
given: \\( \overline{hd} \parallel \overline{tr} \\), \\( \overline{hi} \parallel \overline{jk} \\)
prove: \\( \triangle hkl \cong \triangle khj
\

$$\begin{array}{|l|l|} \\hline \\text{statements} & \\text{reasons} \\\\ \\hline 1. \\overline{hd} \\parallel \\overline{tr}, \\overline{hi} \\parallel \\overline{jk} & 1. \\text{given} \\\\ \\hline 2. & 2. \\text{alternate interior angles} \\\\ \\hline 3. \\angle jkh \\cong \\angle lhk & 3. \\\\ \\hline 4. & 4. \\text{reflexive property} \\\\ \\hline 5. \\triangle hkl \\cong \\triangle khj & 5. \\\\ \\hline\\end{array}$$

Explanation:

Step1: Analyze Given Information

We know \( \overline{HJ} \parallel \overline{LK} \) and \( \overline{HL} \parallel \overline{JK} \) (given). Also, we need to prove \( \triangle HKL \cong \triangle KHJ \) (assuming typo, should be \( \triangle HKL \cong \triangle KHJ \) or similar, but let's proceed with the table).

Step2: Fill Statement 2

From the reason "Alternate Interior Angles", the statement should be about alternate interior angles. Since \( \overline{HJ} \parallel \overline{LK} \) and transversal \( HK \), we get \( \angle KHL \cong \angle H K J \) (or similar alternate interior angles). Wait, maybe \( \angle HKL \) and \( \angle KHJ \) related? Wait, the reason is "Alternate Interior Angles", so the statement is \( \angle HKL \cong \angle KHJ \) (if \( \overline{HJ} \parallel \overline{LK} \) and transversal \( HK \))? Wait, no, let's check the lines. Given \( \overline{HJ} \parallel \overline{LK} \) (wait, original given is \( \overline{HD} \parallel \overline{TR} \)? Wait, maybe typo, but the table has "Alternate Interior Angles" as reason 2, so statement 2 should be an angle congruence from parallel lines. So Statement 2: \( \angle HKL \cong \angle KHJ \) (Alternate Interior Angles, since \( \overline{HJ} \parallel \overline{LK} \), transversal \( HK \)).

Step3: Fill Reason 3

Statement 3 is \( \angle JKH \cong \angle LHK \). This is also from alternate interior angles, because \( \overline{HL} \parallel \overline{JK} \) (given \( \overline{HL} \parallel \overline{JK} \)) and transversal \( HK \), so Reason 3: Alternate Interior Angles.

Step4: Fill Statement 4

Reason 4 is "Reflexive Property", so the statement is \( HK \cong HK \) (common side, reflexive).

Step5: Fill Reason 5

Statement 5 is \( \triangle HKL \cong \triangle KHJ \). The congruence criteria: we have two angles and included side? Wait, from Step2: \( \angle HKL \cong \angle KHJ \), Step4: \( HK \cong HK \), Step3: \( \angle JKH \cong \angle LHK \). So by ASA (Angle-Side-Angle) congruence, Reason 5: ASA (Angle-Side-Angle) Congruence Postulate.

Wait, let's re-express the table properly:

1. Statement 1: \( \overline{HJ} \parallel \overline{LK} \), \( \overline{HL} \parallel \overline{JK} \) (Given)
2. Statement 2: \( \angle HKL \cong \angle KHJ \) (Alternate Interior Angles, from \( \overline{HJ} \parallel \overline{LK} \), transversal \( HK \))
3. Statement 3: \( \angle JKH \cong \angle LHK \) (Alternate Interior Angles, from \( \overline{HL} \parallel \overline{JK} \), transversal \( HK \))
4. Statement 4: \( HK \cong HK \) (Reflexive Property)
5. Statement 5: \( \triangle HKL \cong \triangle KHJ \) (ASA Congruence, using \( \angle HKL \cong \angle KHJ \), \( HK \cong HK \), \( \angle JKH \cong \angle LHK \))

So filling the table:

• Statement 2: \( \angle HKL \cong \angle KHJ \)
• Reason 3: Alternate Interior Angles
• Statement 4: \( HK \cong HK \)
• Reason 5: ASA (Angle - Side - Angle) Congruence Postulate

But maybe the original problem has different labels, but based on the table:

1. Reason 1: Given (correct as filled)
2. Statement 2: \( \angle HKL \cong \angle KHJ \) (Alternate Interior Angles)
3. Reason 3: Alternate Interior Angles (for \( \angle JKH \cong \angle LHK \))
4. Statement 4: \( HK \cong HK \) (Reflexive Property)
5. Reason 5: ASA (or AAS, but with two angles and included side, ASA)

Answer:

Filling the table:

Statements	Reasons
2. \( \angle HKL \cong \angle KHJ \)	2. Alternate Interior Angles
3. \( \angle JKH \cong \angle LHK \)	3. Alternate Interior Angles
4. \( HK \cong HK \)	4. Reflexive Property
5. \( \triangle HKL \cong \triangle KHJ \)	5. ASA (Angle - Side - Angle) Congruence Postulate

(Note: Depending on the actual triangle labels, adjust the angle and side names, but this is the general approach for the proof table.)