Question

1. basic triangle proofs (congruence only - no cpctc)

this is the only question in this section.
question
given: \\( \overline { d e } \cong \overline { d f } \\), \\( \overline { a e } \perp \overline { e d } \\) and \\( \overline { c f } \perp \overline { f d } \\).
prove: \\( \triangle a d e \cong \triangle c d f \\).
step
statement
reason
\\( d e \cong d f \\)
\\( \overline { a e } \perp \overline { e d } \\)
given
\\( \overline { c f } \perp \overline { f d } \\)
2
\\( \angle a d e \cong \angle c d f \\)
vertical angles are congruent
type of statement
answer

Explanation:

Step1: Determine the right angles

Since \(AE\perp ED\) and \(CF\perp FD\), \(\angle AED=\angle CFD = 90^{\circ}\) (Definition of perpendicular lines).

Step2: Apply the ASA (Angle - Side - Angle) congruence criterion

We know that \(DE\cong DF\) (given), \(\angle ADE\cong\angle CDF\) (vertical angles are congruent), and \(\angle AED=\angle CFD\) (right angles).
By the ASA congruence criterion, \(\triangle ADE\cong\triangle CDF\).

Answer:

\(\triangle ADE\cong\triangle CDF\) by the ASA (Angle - Side - Angle) congruence criterion.