Question

$overline{hi}paralleloverline{jk}$. complete the proof that $angle hjicongangle jhk$.
statement
1 $overline{hi}paralleloverline{jk}$
2 $overline{hi}congoverline{jk}$
3 $angle hjkcongangle ihj$
4 $overline{hj}congoverline{hj}$
5 $\triangle hijcong\triangle jkh$
6 $angle hjicongangle jhk$
reason
given
reflexive property of congruence
sas

Explanation:

Step1: Identify the reason for \( \overline{HI}\cong\overline{JK} \)

From the figure, we can see that \( \overline{HI}\) and \( \overline{JK}\) are marked as equal. So the reason is "Given" (as they are shown as congruent in the diagram).

Step2: Identify the reason for \( \angle HJK\cong\angle IHJ \)

Since \( \overline{HI}\parallel\overline{JK}\), by the Alternate - Interior Angles Theorem (when two parallel lines are cut by a transversal, the alternate - interior angles are congruent). Here, \( \overline{HJ}\) is the transversal for parallel lines \( \overline{HI}\) and \( \overline{JK}\). So the reason is "Alternate - Interior Angles Theorem".

Step3: Identify the reason for \( \angle HJI\cong\angle JHK \)

Since \( \triangle HIJ\cong\triangle JKH\) (by SAS, as \( \overline{HI}\cong\overline{JK}\), \( \angle HJK\cong\angle IHJ\), \( \overline{HJ}\cong\overline{HJ}\)), then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \( \angle HJI\cong\angle JHK\).

Answer:

1. \( \overline{HI}\cong\overline{JK}\): Given
2. \( \angle HJK\cong\angle IHJ\): Alternate - Interior Angles Theorem
3. \( \angle HJI\cong\angle JHK\): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)