Question

Reorder the steps of the proof to make sure that steps that are logically dependent on prior steps are in the proper order. Given: angle, A, C, D, \cong, angle, B, C, D, comma∠ACD≅∠BCD, start overline, E, B, end overline, \parallel, start overline, A, C, end overline EB ∥ AC and start overline, A, B, end overline, is perpendicular to, start overline, C, E, end overline, . AB ⊥ CE . Prove: start overline, C, D, end overline, \cong, start overline, D, E, end overline CD ≅ DE

Answer 1

By the definition of congruent segments, we can state that CD is congruent to DE.

To prove that CD is congruent to DE, we need to follow a step-by-step approach and provide a clear explanation.

Proof:

Given: angle ACD ≅ angle BCD
Given: EB ∥ AC and AB ⊥ CE

1. We are given that angle ACD is congruent to angle BCD. This means that the two angles have the same measure.

2. We are also given that EB is parallel to AC and AB is perpendicular to CE. This implies that angle CEB is a right angle.

3. By the definition of perpendicular lines, we know that AB is perpendicular to CE, which means that the two lines intersect at a 90-degree angle.

4. Since angle CEB is a right angle, it is congruent to itself.

5. Using the Alternate Interior Angles Theorem, we can conclude that angles ACD and CEB are congruent. This is because they are alternate interior angles formed by the transversal line EB.

6. According to the Corresponding Angles Postulate, angles ACD and CEB are also congruent.

7. By the Angle Congruence Theorem, we can conclude that angles CEB and CDE are congruent.

8. Using the Transitive Property of Congruence, we can deduce that angles ACD and CDE are congruent.

9. By the definition of congruent angles, we can state that angle CDE is congruent to angle ACD.

10. We are given that CD is congruent to CD. This is the Reflexive Property of Congruence, which states that any segment is congruent to itself.

11. Using the Transitive Property of Congruence again, we can conclude that CD is congruent to DE.

12. Finally, by the definition of congruent segments, we can state that CD is congruent to DE.

In summary, we have proved that CD is congruent to DE by following a logical sequence of steps and using relevant postulates and theorems.

Answer 2

To prove that CD ≅ DE, we can use angle-angle-side congruence to show that the triangles ▱ABCD and ▱CDE are congruent.

Step-by-step explanation:

To prove: CD ≅ DE

1. Given: ∠ACD ≅ ∠BCD, EB ∥ AC, AB ⊥ CE
2. Construct: ▱ABCD and ▱CDE
3. Since EB ∥ AC and AB ⊥ CE, we have ∠AEB ≅ ∠C (Alternate Interior Angles) and ∠EBC ≅ ∠CDE (Alternate Interior Angles)
4. ∠ACD ≅ ∠BCD (Given)
5. ∠AEB ≅ ∠C (Alternate Interior Angles)
6. Therefore, ∠AEB ≅ ∠EBC ≅ ∠CDE (All corresponding angles of congruent triangles are congruent)
7. ▱ABCD ≅ ▱CDE (Angle-Angle-Side)
8. So, CD ≅ DE (Corresponding sides of congruent triangles are congruent)