Question

question
given: \\( \overline { a b } \cong \overline { a d } \\) and \\( a c \perp b d \\).
prove: \\( \triangle b a e \cong \triangle d a e \\)

Explanation:

Step1: Define right angles

Since \(AC\perp BD\), \(\angle AEB=\angle AED = 90^{\circ}\) (Definition of perpendicular lines: If two lines are perpendicular, the angles formed are right angles).

Step2: Use Hypotenuse - Leg (HL) criterion

In right - triangles \(\triangle BAE\) and \(\triangle DAE\):

• The hypotenuse \(AB = AD\) (Given).
• The leg \(AE=AE\) (Common side).

By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\(HL\): If the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent), \(\triangle BAE\cong\triangle DAE\).

Answer:

\(\triangle BAE\cong\triangle DAE\) by the Hypotenuse - Leg (\(HL\)) congruence criterion.