In a pentagon ABCDE, AB is parallel to DC and `angleA: angleE: angleD= 3: 4: 5`. Find angle E.

Answer

To solve the problem, we need to find the measure of angle E in the pentagon ABCDE, given that AB is parallel to DC and the angles are in the ratio 3:4:5.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a pentagon ABCDE where AB is parallel to DC.
- Since AB is parallel to DC, angles B and C are both right angles (90 degrees).

2. **Set Up the Ratio of Angles**:
- We are given that the angles A, E, and D are in the ratio 3:4:5.
- Let angle A = 3k, angle E = 4k, and angle D = 5k.

3. **Sum of Interior Angles of a Pentagon**:
- The formula for the sum of interior angles of a polygon is (n - 2) × 180 degrees, where n is the number of sides.
- For a pentagon (n = 5), the sum of the interior angles = (5 - 2) × 180 = 3 × 180 = 540 degrees.

4. **Write the Equation for the Angles**:
- The sum of the angles in the pentagon can be expressed as:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} + \text{Angle E} = 540 \]
- Substituting the known angles:
\[ 3k + 90 + 90 + 5k + 4k = 540 \]

5. **Combine Like Terms**:
- Combine the terms involving k:
\[ 3k + 5k + 4k + 180 = 540 \]
\[ 12k + 180 = 540 \]

6. **Isolate k**:
- Subtract 180 from both sides:
\[ 12k = 540 - 180 \]
\[ 12k = 360 \]
- Divide both sides by 12:
\[ k = 30 \]

7. **Find Angle E**:
- Now substitute k back to find angle E:
\[ \text{Angle E} = 4k = 4 \times 30 = 120 \]

### Final Answer:
- Angle E = 120 degrees.