`Delta`ABC and `Delta`ADB are on the common base AB and on the same side of `AB.DA bot AB, CB bot AB` and AC = BD. Which of the following is true?

Options

A. `DeltaABCcongDeltaABD`
B. `DeltaABCcongDeltaADB`
C. `DeltaABCcongDeltaBAD`
D. `DeltaABCcongDeltaBDA`

Answer

To solve the problem, we need to analyze the given triangles, ΔABC and ΔADB, based on the conditions provided:

1. **Identify the Given Information**:
- Both triangles share a common base AB.
- Points C and D are on the same side of line AB.
- DA is perpendicular to AB (i.e., angle DAB = 90°).
- CB is also perpendicular to AB (i.e., angle CBA = 90°).
- AC = BD.

2. **Draw the Diagram**:
- Draw line segment AB.
- From point A, draw a perpendicular line to AB and label the intersection point as D.
- From point B, draw a perpendicular line to AB and label the intersection point as C.
- Mark the lengths AC and BD equal as per the given condition.

3. **Analyze the Triangles**:
- We have two right triangles: ΔADB and ΔABC.
- In ΔADB:
- Angle DAB = 90° (from DA being perpendicular to AB).
- AD is one leg and AB is the other leg.
- In ΔABC:
- Angle CBA = 90° (from CB being perpendicular to AB).
- BC is one leg and AB is the other leg.

4. **Use the RHS (Right angle-Hypotenuse-Side) Congruence Criterion**:
- For triangles ΔADB and ΔABC:
- Both triangles have a right angle (DAB and CBA).
- They share the common side AB.
- The hypotenuse lengths are equal (AC = BD).
- Therefore, by the RHS congruence criterion, we can conclude that:
- ΔADB ≅ ΔABC.

5. **Conclusion**:
- Since the two triangles are congruent, all corresponding angles and sides are equal.
- Thus, the statement that can be derived from this is that the angles and sides of both triangles are equal.

### Final Result:
The correct conclusion is that ΔABC is congruent to ΔADB (ΔABC ≅ ΔADB).