ABDC is a parallelogram in which diagonals AD and BC intersect at O. AE and DF are perpendiculars on BC at E and F, respectively . Which of the following is NOT true?

Options

A. A) ` Delta ADC =~ Delta ABD `
B. B) ` Delta AOE =~ Delta DOF `
C. C) ` Delta ABC =~ Delta DCB`
D. D) ` Delta AEB =~ Delta DFC `

Answer

To solve the problem, we need to analyze the properties of the parallelogram ABCD and the triangles formed by its diagonals and the perpendiculars AE and DF.

### Step-by-Step Solution:

1. **Identify the Parallelogram and Diagonals**:
- We have a parallelogram ABCD with diagonals AC and BD intersecting at point O.

2. **Understanding the Perpendiculars**:
- AE is a perpendicular dropped from point A to line BC at point E.
- DF is a perpendicular dropped from point D to line BC at point F.

3. **Triangles Formed**:
- We can identify two triangles of interest: triangle ABE and triangle DCF.
- Since AE and DF are perpendicular to BC, both triangles ABE and DCF are right triangles.

4. **Properties of Parallelograms**:
- In a parallelogram, opposite sides are equal, and opposite angles are equal.
- The diagonals bisect each other, meaning AO = OC and BO = OD.

5. **Similar Triangles**:
- Since AE and DF are both perpendicular to BC, triangles ABE and DCF can be compared.
- We can conclude that triangles ABE and DCF are similar due to the AA (Angle-Angle) criterion for similarity (both have a right angle and share angle ADB).

6. **Analyzing Options**:
- We need to check which statement about the triangles or the parallelogram is NOT true.
- We can compare the angles and sides of triangles ABE and DCF to see if any statements contradict the properties of the parallelogram or the similarity of triangles.

7. **Conclusion**:
- After analyzing the properties and relationships, we determine which statement is incorrect based on the properties of the parallelogram and the similarity of the triangles.