Question

consider triangles ejh and eih. $overline{hj} perp overline{je}$, $overline{hi} perp overline{ie}$, $overline{je} cong overline{ie}$.

image of quadrilateral ejhi with diagonals eh

which criteria for triangle congruence could be used to show that $\triangle ejh cong \triangle eih$?

\bigcirc a) side-side-side
\bigcirc b) side-side-angle
\bigcirc c) hypotenuse-leg
\bigcirc d) there is not enough information to show the triangles are congruent.

Explanation:

Step1: Identify right triangles

Since \(\overline{HJ} \perp \overline{JE}\) and \(\overline{HI} \perp \overline{IE}\), \(\triangle EJH\) and \(\triangle EIH\) are right triangles with right angles at \(J\) and \(I\) respectively.

Step2: Analyze given congruent sides

We know \(\overline{JE} \cong \overline{IE}\) (given). Also, the hypotenuse \(\overline{EH}\) is common to both triangles, so \(\overline{EH} \cong \overline{EH}\) (reflexive property).

Step3: Apply Hypotenuse-Leg (HL) criterion

For right triangles, if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent (HL criterion). Here, hypotenuse \(EH\) is congruent, and leg \(JE\) (or \(IE\)) is congruent. So HL applies.

Answer:

c) Hypotenuse-Leg