Question

given: \\( \overline { a d } \cong \overline { c f } \\), \\( \overline { a e } \cong \overline { g c } \\), \\( \overline { a c } \perp \overline { d e } \\) and \\( \overline { a c } \perp \overline { f g } \\).
prove: \\( \angle a \cong \angle c \\).
note: quadrilateral properties are not permitted in this proof.

Explanation:

Step1: Define right angles

Since \(\overline{AC}\perp\overline{DE}\) and \(\overline{AC}\perp\overline{FG}\), \(\angle AED = 90^{\circ}\) and \(\angle CGF=90^{\circ}\) (Definition of perpendicular lines).

Step2: Use Hypotenuse - Leg (HL) congruence

In right triangles \(\triangle ADE\) and \(\triangle CFG\), we have \(\overline{AD}\cong\overline{CF}\) (given) and \(\overline{AE}\cong\overline{GC}\) (given). By the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles, \(\triangle ADE\cong\triangle CFG\).

Step3: Conclude angle congruence

If \(\triangle ADE\cong\triangle CFG\), then their corresponding angles are congruent. So, \(\angle A\cong\angle C\) (Corresponding parts of congruent triangles are congruent - CPCTC).

Answer:

\(\angle A\cong\angle C\)