Given AB || DC, complete the flowchart proof.
Understand the Problem
The question asks to complete a flowchart proof given that line AB is parallel to line DC. This involves identifying the reasons for statements such as the congruence of angles and sides to prove that triangles ABE and CDE are congruent.
Answer
$\angle A \cong \angle C$ Reason: Alternate Interior Angles Theorem $\angle B \cong \angle D$ Reason: Alternate Interior Angles Theorem $\overline{AB} \cong \overline{CD}$ Reason: Given $\triangle ABE \cong \triangle CDE$ Reason: Angle-Angle-Side (AAS)
Answer for screen readers
$\angle A \cong \angle C$ Reason: Alternate Interior Angles Theorem $\angle B \cong \angle D$ Reason: Alternate Interior Angles Theorem $\overline{AB} \cong \overline{CD}$ Reason: Given $\triangle ABE \cong \triangle CDE$ Reason: Angle-Angle-Side (AAS)
Steps to Solve
- Determine the reason for $\angle A \cong \angle C$
Since $\overline{AB} \parallel \overline{DC}$, $\angle A$ and $\angle C$ are alternate interior angles, and alternate interior angles are congruent.
- Determine the reason for $\angle B \cong \angle D$
Since $\overline{AB} \parallel \overline{DC}$, $\angle B$ and $\angle D$ are alternate interior angles, and alternate interior angles are congruent.
- Determine the reason for $\overline{AB} \cong \overline{CD}$
The diagram shows that $\overline{AB}$ and $\overline{CD}$ have the same markings, which indicates that they are congruent. Thus, this is given.
- Determine the reason for $\triangle ABE \cong \triangle CDE$
We have $\angle A \cong \angle C$, $\angle B \cong \angle D$, and $\overline{AB} \cong \overline{CD}$. Therefore, by the Angle-Angle-Side (AAS) congruence theorem, $\triangle ABE \cong \triangle CDE$.
$\angle A \cong \angle C$ Reason: Alternate Interior Angles Theorem $\angle B \cong \angle D$ Reason: Alternate Interior Angles Theorem $\overline{AB} \cong \overline{CD}$ Reason: Given $\triangle ABE \cong \triangle CDE$ Reason: Angle-Angle-Side (AAS)
More Information
The Angle-Angle-Side (AAS) theorem states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the two triangles are congruent.
Tips
A common mistake is confusing Alternate Interior Angles with Corresponding Angles. Also, students might mix up AAS, ASA, and SAA congruence theorems.
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