Given: \(\overline{AB} \perp \overline{BC}\), \(\overline{AD} \perp \overline{DC}\), and \(\overline{BC} \parallel \overline{AD}\). Prove: \(\triangle ABC \cong \triangle CDA\).
Understand the Problem
The question asks us to complete a geometric proof showing that triangle ABC is congruent to triangle CDA. We are given that AB is perpendicular to BC, AD is perpendicular to DC, and BC is parallel to AD. The first step, along with the given information, is already provided. We need to determine the subsequent steps and reasons to arrive at the conclusion that the two triangles are congruent.
Answer
See table in "answer".
Answer for screen readers
| Step | Statement | Reason |
|---|---|---|
| 1 | $\overline{AB} \perp \overline{BC}$, $\overline{AD} \perp \overline{DC}$, $\overline{BC} \parallel \overline{AD}$ | Given |
| 2 | $\angle ABC$ and $\angle CDA$ are right angles | Definition of perpendicular lines |
| 3 | $\angle ABC \cong \angle CDA$ | All right angles are congruent |
| 4 | $\angle BCA \cong \angle DAC$ | Alternate Interior Angles Theorem |
| 5 | $\overline{AC} \cong \overline{AC}$ | Reflexive Property |
| 6 | $\triangle ABC \cong \triangle CDA$ | AAS |
Steps to Solve
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Angles are Right Angles Since $\overline{AB} \perp \overline{BC}$ and $\overline{AD} \perp \overline{DC}$, we know that $\angle ABC$ and $\angle CDA$ are right angles.
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Right Angles are Congruent Since $\angle ABC$ and $\angle CDA$ are right angles, they are congruent. Thus, $\angle ABC \cong \angle CDA$.
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Alternate Interior Angles Since $\overline{BC} \parallel \overline{AD}$, then $\angle BCA$ and $\angle DAC$ are alternate interior angles and are therefore congruent. So, $\angle BCA \cong \angle DAC$.
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Reflexive Property $\overline{AC} \cong \overline{AC}$ by the Reflexive Property.
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Angle-Angle-Side (AAS) Congruence Now we have $\angle ABC \cong \angle CDA$, $\angle BCA \cong \angle DAC$, and $\overline{AC} \cong \overline{AC}$. Therefore, $\triangle ABC \cong \triangle CDA$ by AAS.
| Step | Statement | Reason |
|---|---|---|
| 1 | $\overline{AB} \perp \overline{BC}$, $\overline{AD} \perp \overline{DC}$, $\overline{BC} \parallel \overline{AD}$ | Given |
| 2 | $\angle ABC$ and $\angle CDA$ are right angles | Definition of perpendicular lines |
| 3 | $\angle ABC \cong \angle CDA$ | All right angles are congruent |
| 4 | $\angle BCA \cong \angle DAC$ | Alternate Interior Angles Theorem |
| 5 | $\overline{AC} \cong \overline{AC}$ | Reflexive Property |
| 6 | $\triangle ABC \cong \triangle CDA$ | AAS |
More Information
The Angle-Angle-Side (AAS) theorem is crucial here. It 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 could be assuming that because the problem involves parallel lines, the sides are also congruent without proper justification. Another mistake could be using ASA instead of AAS, or confusing the order of the angles and sides. Also, students might forget the reflexive property, which is essential for identifying the congruent side in this proof.
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