Question

write an informal proof for each of the following problems.
1.) prove that triangles abc and edc are congruent.
given:
angle abc is a right angle.
angle edc is a right angle.
c is the midpoint of \\( \overline{bd} \\).
\\( \overline{ab} \cong \overline{ed} \\)

Explanation:

Step1: Analyze the given information

We know that \(\angle ABC = \angle EDC=90^{\circ}\), \(C\) is the mid - point of \(\overline{BD}\), so \(BC = DC\), and \(\overline{AB}\cong\overline{ED}\).

Step2: Apply the congruence criterion

For right - angled triangles, we can use the Hypotenuse - Leg (HL) congruence criterion. In right triangles \(ABC\) and \(EDC\), the hypotenuse - leg congruence states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. Here, \(AB = ED\) (given as \(\overline{AB}\cong\overline{ED}\)) and \(BC = DC\) (since \(C\) is the mid - point of \(BD\)).

Answer:

Triangles \(ABC\) and \(EDC\) are congruent by the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles.