Given that ABCD is a parallelogram whose diagonals intersect at point. `O. angle ABC = 110^@ , angle ACB = 35^(@) and angle ADB = 55^(@)`. The term that best describes ABCD is:

Options

A. Rectangle
B. Rhombus
C. Square
D. Kite

Answer

To determine the term that best describes the parallelogram ABCD, we will analyze the given angles and properties step by step.

### Step 1: Understand the properties of a parallelogram
A parallelogram has opposite sides that are equal and parallel. The diagonals of a parallelogram bisect each other.

### Step 2: Identify the given angles
We have the following angles:
- Angle ABC = 110°
- Angle ACB = 35°
- Angle ADB = 55°

### Step 3: Use the properties of angles in a parallelogram
Since ABCD is a parallelogram, we know that:
- Angle ABC + Angle ADC = 180° (consecutive angles)
- Angle ACB + Angle BDC = 180° (consecutive angles)

### Step 4: Find angle BDC
We can find angle BDC using the fact that angle ADB is equal to angle BDC (alternate interior angles):
- Angle BDC = Angle ADB = 55°

### Step 5: Find angle ADC
Now, we can find angle ADC:
- Angle ADC = 180° - Angle ABC = 180° - 110° = 70°

### Step 6: Check the properties of the diagonals
Since the diagonals of a parallelogram bisect each other, we can analyze triangle BOC:
- In triangle BOC, we have:
- Angle ACB = 35°
- Angle BDC = 55°

Using the angle sum property:
- Angle BOC = 180° - (Angle ACB + Angle BDC) = 180° - (35° + 55°) = 180° - 90° = 90°

### Step 7: Analyze triangle ABC
Now, let's analyze triangle ABC:
- We already know:
- Angle ABC = 110°
- Angle ACB = 35°

To find angle BAC:
- Angle BAC = 180° - (Angle ABC + Angle ACB) = 180° - (110° + 35°) = 180° - 145° = 35°

### Step 8: Check for equal sides
Since angle ACB = angle BAC = 35°, we can conclude that sides AB and BC are equal (by the Isosceles Triangle Theorem).

### Step 9: Conclusion
In parallelogram ABCD:
- The adjacent sides are equal (AB = BC).
- The diagonals are perpendicular (BD ⊥ AC).

These properties indicate that ABCD is not just a parallelogram, but specifically a **rhombus**.

### Final Answer
The term that best describes ABCD is **rhombus**.

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