Question

triangle congruence proofs practice
proof 1
given: \\( \overline { a d } \cong \overline { d c } \\) and \\( \angle a \cong \angle c \\)
prove: \\( \triangle a d e \cong \triangle c d f \\)
proof 2
given: \\( \overline { d c } \\) bisects \\( \angle a c b \\) and \\( \overline { a c } \cong \overline { b c } \\)
prove: \\( \triangle a c d \cong \triangle b c d \\)

Explanation:

Proof 1

Step1: Vertical angles are congruent

$\angle ADE\cong\angle CDF$ (Vertical angles are congruent)

Step2: Use ASA congruence

In $\triangle ADE$ and $\triangle CDF$, we have $\overline{AD}\cong\overline{DC}$ (given), $\angle A\cong\angle C$ (given), and $\angle ADE\cong\angle CDF$ (from Step 1). By the Angle - Side - Angle (ASA) congruence criterion, $\triangle ADE\cong\triangle CDF$

Proof 2

Step1: Definition of angle bisector

Since $DC$ bisects $\angle ACB$, then $\angle ACD\cong\angle BCD$ (Definition of angle bisector)

Step2: Common side

$\overline{CD}\cong\overline{CD}$ (Reflexive property of congruence)

Step3: Use SAS congruence

In $\triangle ACD$ and $\triangle BCD$, we have $\overline{AC}\cong\overline{BC}$ (given), $\angle ACD\cong\angle BCD$ (from Step 1), and $\overline{CD}\cong\overline{CD}$ (from Step 2). By the Side - Angle - Side (SAS) congruence criterion, $\triangle ACD\cong\triangle BCD$

Answer:

For Proof 1: $\triangle ADE\cong\triangle CDF$ by ASA. For Proof 2: $\triangle ACD\cong\triangle BCD$ by SAS.