In parallelogram PQRS, `angleQ=(4x-5)^(@) and angleS=(3x+10)^(@)`. Calculate: `angleQ and angleR`.

Answer

To solve the problem, we need to find the values of angle Q and angle R in the parallelogram PQRS, given that angle Q = (4x - 5)° and angle S = (3x + 10)°.

### Step-by-Step Solution:

1. **Set up the equation using the property of parallelograms:**
In a parallelogram, opposite angles are equal. Therefore, we can set angle Q equal to angle S:
\[ 4x - 5 = 3x + 10 \]

**Hint:** Remember that in a parallelogram, opposite angles are equal.

2. **Rearrange the equation to isolate x:**
Subtract 3x from both sides:
\[ 4x - 3x - 5 = 10 \]
This simplifies to:
\[ x - 5 = 10 \]

**Hint:** Combine like terms to simplify the equation.

3. **Solve for x:**
Add 5 to both sides:
\[ x = 10 + 5 \]
Thus, we find:
\[ x = 15 \]

**Hint:** Make sure to perform the same operation on both sides of the equation to maintain equality.

4. **Substitute x back into the expression for angle Q:**
Now that we have the value of x, we can find angle Q:
\[ \text{Angle Q} = 4x - 5 = 4(15) - 5 \]
Calculate:
\[ = 60 - 5 = 55° \]

**Hint:** Substitute the value of x carefully into the angle expression.

5. **Find angle R using the property of adjacent angles:**
In a parallelogram, adjacent angles sum up to 180°. Therefore:
\[ \text{Angle Q} + \text{Angle R} = 180° \]
Substitute the value of angle Q:
\[ 55° + \text{Angle R} = 180° \]
To find angle R, subtract 55° from 180°:
\[ \text{Angle R} = 180° - 55° = 125° \]

**Hint:** Remember that adjacent angles in a parallelogram are supplementary, meaning they add up to 180°.

### Final Answers:
- Angle Q = 55°
- Angle R = 125°