Question

State the missing reasons. Given that$\overline{AB}\parallel \overline{DC}$ and$\overline{AB}\cong \overline{DC}$Prove that$\Delta ABC\cong \Delta CDA$.

Short answer

$\mathit{\Delta }\mathit{A}\mathit{B}\mathit{C}\mathbf{\cong }\mathit{\Delta }\mathit{C}\mathit{D}\mathit{A}$by SAS postulates.

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Apply the concept of ASA, SSS, and SAS postulates.

The Angle-Side-Angle Postulate (ASA) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

The Side-Side-Side Postulate (SSS) says triangles are congruent if three sides of one triangle are congruent to the corresponding sides of the other triangle.

SAS Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

Step 3. Step description.

Consider that $\overline{AB}\left|\right|\overline{DC};\overline{AB}\cong \overline{DC}$.

Reason 1. It is given that $\overline{AB}\cong \overline{DC}$.

Reason 2. Reflexive property such that $\overline{AC}\cong \overline{DC}$.

Reason 3. It is given that $\overline{AB}\left|\right|\overline{DC}$.

Reason 4. Alternate interior angles are congruent such that $\angle BAC\cong \angle DCA$.

Reason 5. SAS postulate $\Delta ABC\cong \Delta CDA$

Thus, the triangles are congruent with the SAS postulate.