Question

lesson 7 practice problems

1. what triangle congruence theorem could you use to prove triangle ade is congruent to triangle cbe?
2. han wrote a proof that triangle bcd is congruent to triangle dab. hans proof is incomplete. how can han fix his proof?

• line ab is parallel to line dc and cut by transversal db. so angles cdb and abd are alternate interior angles and must be congruent.
• side db is congruent to side bd because theyre the same segment.
• angle a is congruent to angle c because theyre both right angles.
• by the angle - side - angle triangle congruence theorem, triangle bcd is congruent to triangle dab.

Explanation:

Step1: Analyze the given information for problem 1

We know that \( \angle AED=\angle BEC\) (vertical angles), \( \angle A=\angle C\) (given or can be inferred from the figure's properties), and \(AD = CB\) (given or can be inferred from the figure's markings).

Step2: Recall triangle congruence theorems

The ASA (Angle - Side - Angle) theorem 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.
For problem 2:

Step1: Check the Angle - Side - Angle (ASA) requirements

In the ASA theorem, the side must be between the two angles. In Han's proof, the side \(DB\) is between \( \angle CDB\) and \( \angle DBC\) in \(\triangle BCD\) and between \( \angle ABD\) and \( \angle BAD\) in \(\triangle DAB\). But Han used \( \angle A\) and \( \angle C\) which are not adjacent to the side \(DB\) in the correct order for ASA. He should use \( \angle ADB\) and \( \angle CBD\) (another pair of alternate interior angles since \(DC\parallel AB\) and cut by transversal \(DB\)) along with \(DB = BD\) (common side) and \( \angle A=\angle C\) (right angles) and apply the AAS (Angle - Angle - Side) theorem.

Answer:

1. ASA (Angle - Side - Angle) theorem can be used to prove \(\triangle ADE\cong\triangle CBE\).
2. Han should use the AAS (Angle - Angle - Side) theorem instead of ASA. He needs to identify another pair of angles (alternate interior angles \( \angle ADB\) and \( \angle CBD\)) along with the given \( \angle A=\angle C\) (right angles) and \(DB = BD\) (common side) to prove \(\triangle BCD\cong\triangle DAB\).