https://file.aiquickdraw.com/mathbot/file/1725438951637yjlfwdrg.jpg

To solve the problem, we need to calculate the perimeter of the quadrilateral \( ABCD \) shown in the image.

### Steps to Solve:

1. **Identify Given Information**:
   - \( \angle DAB = 87^\circ \)
   - \( \angle ABC = 90^\circ \)
   - \( \angle BCD = 45^\circ \)
   - \( \angle ADC = 15^\circ \)
   - \( AB = 14 \, \text{cm} \)
   - \( CD = 16 \, \text{cm} \)

2. **Use Trigonometry and the Law of Cosines/Sines**:
   - The side lengths \( BC \) and \( AD \) need to be determined using the given angles and side lengths.

3. **Calculate \( BC \)**:
   - Since \( \triangle ABC \) is a right triangle, \( BC \) can be found using the sine or cosine of \( \angle ABC \).
   - \( BC = AB \cdot \tan(\angle BAC) \)

4. **Calculate \( AD \)**:
   - Apply the Law of Sines in \( \triangle ABD \) to find \( AD \).

5. **Perimeter Calculation**:
   - Add up all the side lengths: \( AB + BC + CD + DA \).

Let's go through the calculations.

### Calculation

1. **Calculate \( BC \)**:
   - \( \angle BAC = 87^\circ \)
   - \( BC = AB \cdot \tan(87^\circ) \)

   \( BC = 14 \, \text{cm} \times \tan(87^\circ) \approx 14 \times 19.0811 = 267.1354 \, \text{cm} \).

2. **Calculate \( AD \)**:
   - We need \( \angle ABD = 180^\circ - 87^\circ = 93^\circ \).
   - Use the Law of Sines in \( \triangle ABD \):
     \[
     \frac{AD}{\sin(87^\circ)} = \frac{14}{\sin(93^\circ)}
     \]
     \[
     AD = \frac{14 \times \sin(87^\circ)}{\sin(93^\circ)} \approx 14 \times 0.9998477 / 0.9998477 = 14 \, \text{cm}
     \]

3. **Perimeter**:
   - \( AB + BC + CD + DA = 14 + 267.1354 + 16 + 14 = 311.1354 \, \text{cm} \)

The perimeter of the quadrilateral \( ABCD \) is approximately **311.14 cm**.

Would you like more details or have any questions?

### Related Questions:
1. How would the perimeter change if the angle \( \angle DAB \) were different?
2. How can the Law of Cosines be used instead of the Law of Sines in this scenario?
3. What is the significance of each angle in determining the side lengths of the quadrilateral?
4. How do we handle cases where the quadrilateral is not cyclic?
5. Could the Pythagorean theorem be used to find any other sides in the problem?

### Tip:
Always double-check the angle measurements and ensure you use the correct trigonometric function based on the specific triangle you're working with.

Calculate the perimeter of the quadrilateral ABCD based on the given angles and side lengths.

Learn how to calculate the perimeter of a quadrilateral using trigonometric functions and the Law of Sines. This problem involves determining the side lengths based on angles and applying key geometric concepts. Suitable for students in Grades 10-12.

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