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Let's solve the areas for each of the given figures one by one. Since these figures involve triangles and a parallelogram, we will use the relevant formulas to calculate the areas.

### (a) Triangle \( \triangle ABC \)
- Given: 
  - \( AB = 8 \, \text{cm}, AC = 9 \, \text{cm}, \angle A = 72^\circ \)
- Formula for the area of a triangle using two sides and the included angle:
  \[
  \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(\theta)
  \]
  where \( a \) and \( b \) are the sides and \( \theta \) is the included angle.
  \[
  \text{Area} = \frac{1}{2} \cdot 8 \cdot 9 \cdot \sin(72^\circ)
  \]

### (b) Triangle \( \triangle PQR \)
- Given: 
  - \( PQ = 7 \, \text{cm}, PR = 9 \, \text{cm}, \angle P = 112^\circ \)
- Using the same formula for the area of a triangle:
  \[
  \text{Area} = \frac{1}{2} \cdot 7 \cdot 9 \cdot \sin(112^\circ)
  \]

### (c) Triangle \( \triangle LMN \)
- Given: 
  - \( LM = 12 \, \text{cm}, LN = 10 \, \text{cm}, \angle L = 62^\circ \)
- The area of this triangle will be:
  \[
  \text{Area} = \frac{1}{2} \cdot 12 \cdot 10 \cdot \sin(62^\circ)
  \]

### (d) Triangle \( \triangle STU \)
- Given: 
  - \( ST = 6.5 \, \text{cm}, SU = 7.3 \, \text{cm}, \angle S = 105^\circ \)
- The area of this triangle is:
  \[
  \text{Area} = \frac{1}{2} \cdot 6.5 \cdot 7.3 \cdot \sin(105^\circ)
  \]

### (e) Parallelogram \( ABCD \)
- Given: 
  - \( AB = 5.8 \, \text{cm}, AD = 5.3 \, \text{cm}, \angle A = 117^\circ \)
- The formula for the area of a parallelogram is:
  \[
  \text{Area} = a \cdot b \cdot \sin(\theta)
  \]
  where \( a \) and \( b \) are the sides and \( \theta \) is the included angle. Thus:
  \[
  \text{Area} = 5.8 \cdot 5.3 \cdot \sin(117^\circ)
  \]

Let me compute the values for these areas.Here are the computed areas for each of the figures:

- **(a) Triangle ABC:** \( 34.24 \, \text{cm}^2 \)
- **(b) Triangle PQR:** \( 29.21 \, \text{cm}^2 \)
- **(c) Triangle LMN:** \( 52.98 \, \text{cm}^2 \)
- **(d) Triangle STU:** \( 22.92 \, \text{cm}^2 \)
- **(e) Parallelogram ABCD:** \( 27.39 \, \text{cm}^2 \)

Would you like a detailed breakdown of any of the calculations?

Here are 5 related questions to expand on the information:
1. How do we calculate the area of a triangle when only sides are known (without angles)?
2. What is the formula for calculating the area of a triangle using Heron's formula?
3. How can the sine rule be used to find unknown angles or sides in a triangle?
4. How do you determine the height of a triangle if the area and base are given?
5. What are the conditions for a quadrilateral to be classified as a parallelogram?

**Tip:** When dealing with triangles, knowing different ways to calculate area (based on available information) can help handle various geometric problems efficiently.

Find the area of each of the given figures involving triangles and a parallelogram.

Learn how to calculate the areas of various geometric figures, including triangles and parallelograms, using trigonometry. This includes solving for area when given side lengths and angles. Suitable for students in Grades 9-12.

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