Given a trapezium ABCD in which AB||CD and AD = BC. If `angle C = 76^(@)`, then `angle D` equals

Options

A. `14^(@)`
B. `104^(@)`
C. `76^(@)`
D. none of these

Answer

To find the measure of angle D in trapezium ABCD where AB || CD and AD = BC, follow these steps:

### Step 1: Understand the properties of trapezium
In trapezium ABCD, since AB is parallel to CD (AB || CD), we can use the properties of angles formed by parallel lines.

**Hint:** Remember that when two parallel lines are cut by a transversal, the alternate interior angles are equal.

### Step 2: Identify given information
We are given that angle C = 76° and AD = BC, which indicates that trapezium ABCD is an isosceles trapezium.

**Hint:** An isosceles trapezium has non-parallel sides that are equal in length.

### Step 3: Draw perpendiculars from points A and B to line CD
Drop perpendiculars from points A and B to line CD. Let these perpendiculars meet CD at points M and N respectively. This creates two right triangles: triangle ADM and triangle BNC.

**Hint:** Visualizing the trapezium with perpendiculars helps in analyzing the triangles formed.

### Step 4: Analyze triangles ADM and BNC
In triangles ADM and BNC:
- Angle AMD = 90° (perpendicular from A)
- Angle BNC = 90° (perpendicular from B)
- AD = BC (given)

**Hint:** Use the properties of congruent triangles to find relationships between angles.

### Step 5: Apply the congruence criteria
Since both triangles ADM and BNC have:
- Right angles (90°)
- One pair of equal sides (AD = BC)
- The distance between the parallel lines (AM = BN)

We can conclude that triangles ADM and BNC are congruent by the Hypotenuse-Leg (HL) theorem.

**Hint:** Congruent triangles have equal corresponding angles.

### Step 6: Conclude that angle D equals angle C
Since triangles ADM and BNC are congruent, we have:
- Angle D = Angle C

Given that angle C = 76°, it follows that:
- Angle D = 76°

**Hint:** Use the property of corresponding parts of congruent triangles (CPCT).

### Final Answer:
Thus, angle D equals 76°.

---