In Fig. (ii), AD = BC and AD || BC. IS AB = DC? Give reasons in support of your answer.Hint. Prove that `/_\ ABC = /_\ CDA`.

Answer

To determine whether AB is equal to DC given that AD = BC and AD || BC, we can use the properties of triangles and congruence. Here’s a step-by-step solution:

### Step 1: Identify the triangles
We have two triangles: Triangle ABC and Triangle ADC.

### Step 2: State the given information
We know that:
- AD = BC (given)
- AD || BC (given)

### Step 3: Use the property of parallel lines
Since AD is parallel to BC, we can use the property of alternate interior angles. This means:
- Angle DAC = Angle ACB (alternate angles)

### Step 4: Identify the corresponding angles
In triangles ABC and ADC, we can identify the angles:
- Angle ABC and Angle ADC are corresponding angles because they are both angles formed by the transversal AC intersecting the parallel lines AD and BC.

### Step 5: Write down the angles
So we have:
- Angle DAC = Angle ACB (from step 3)
- Angle ABC = Angle ADC (corresponding angles)

### Step 6: Conclude about the triangles
Now we have:
- AD = BC (given)
- Angle DAC = Angle ACB
- Angle ABC = Angle ADC

This means that the two triangles (ABC and ADC) have:
- One pair of equal sides (AD = BC)
- Two pairs of equal angles (Angle DAC = Angle ACB and Angle ABC = Angle ADC)

### Step 7: Apply the criteria for triangle congruence
By the Angle-Side-Angle (ASA) criterion for triangle congruence, we can conclude that:
- Triangle ABC is congruent to Triangle ADC.

### Step 8: State the conclusion
Since the triangles are congruent, their corresponding sides are equal. Therefore:
- AB = DC.

### Final Answer
Yes, AB is equal to DC because Triangle ABC is congruent to Triangle ADC.

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