Question

j is the midpoint of \\( \overline{hi} \\) and \\( \overline{hi} \perp \overline{gj} \\). complete the proof that \\( \angle h \cong \angle i \\).

Explanation:

Step1: Identify given info

Given \( J \) is midpoint of \( \overline{HI} \), \( \overline{HI} \perp \overline{GJ} \).

Step2: Midpoint definition

\( \overline{HJ} \cong \overline{IJ} \) (midpoint divides segment equally).

Step3: Perpendicular angles

\( \angle GJH \cong \angle GJI = 90^\circ \) (perpendicular lines form right angles).

Step4: Reflexive property

\( \overline{GJ} \cong \overline{GJ} \) (common side).

Step5: SAS congruence

\( \triangle GHJ \cong \triangle GIJ \) (SAS: \( \overline{HJ} \cong \overline{IJ} \), \( \angle GJH \cong \angle GJI \), \( \overline{GJ} \cong \overline{GJ} \)).

Step6: Corresponding parts

\( \angle H \cong \angle I \) (CPCTC: congruent triangles have congruent corresponding angles).

Answer:

\( \angle H \cong \angle I \) is proven via SAS congruence of \( \triangle GHJ \) and \( \triangle GIJ \), then CPCTC.