Question

in the figure above, two congruences are known.
\\( \overline{hk} \cong \overline{hk} \\) and \\( \overline{hj} \cong \overline{hl}
which additional fact is needed in order to use the sas congruence criterion to prove the two triangles are congruent?
a. \\( \angle jkh \cong \angle klh
\\)b. any pair of congruent angles can be used.
c. \\( \angle hjk \cong \angle hlk
\\)d. \\( \angle jhk \cong \angle lhk
\\)

Explanation:

Step1: Recall SAS Criterion

SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

We know that \(\overline{HK}\cong\overline{HK}\) (common side) and \(\overline{HJ}\cong\overline{HL}\). Let's consider triangles \(\triangle HJK\) and \(\triangle HLK\). The sides \(\overline{HJ}\) and \(\overline{HK}\) form \(\angle JHK\) in \(\triangle HJK\), and the sides \(\overline{HL}\) and \(\overline{HK}\) form \(\angle LHK\) in \(\triangle HLK\). For SAS, we need the included angle between the two congruent sides to be congruent. So we need \(\angle JHK\cong\angle LHK\).

Step2: Analyze the Options

• Option A: \(\angle JKH\cong\angle KLH\) - These are not the included angles between the known congruent sides, so this does not help with SAS.
• Option B: "Any pair of congruent angles" is incorrect because for SAS, the angle must be the included angle between the two sides.
• Option C: \(\angle HJK\cong\angle HLK\) - These are not the included angles between the known congruent sides, so this does not help with SAS.
• Option D: \(\angle JHK\cong\angle LHK\) - This is the included angle between \(\overline{HJ}\) and \(\overline{HK}\) (in \(\triangle HJK\)) and between \(\overline{HL}\) and \(\overline{HK}\) (in \(\triangle HLK\)), which is exactly what we need for SAS.

Answer:

D. \(\angle JHK \cong \angle LHK\)