In a quadrilateral STAR, if `angleS= 120^@, and angleT: angleA : angle R= 5:3:7,` then the measure of `angleR` (in degrees) is____

Answer

To find the measure of angle R in the quadrilateral STAR, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Angles:**
- We know that angle S = 120°.
- The ratio of angles T, A, and R is given as 5:3:7.

2. **Express Angles T, A, and R in Terms of a Variable:**
- Let the common multiple for the ratios be \( x \).
- Therefore, we can express the angles as:
- Angle T = \( 5x \)
- Angle A = \( 3x \)
- Angle R = \( 7x \)

3. **Use the Property of Quadrilaterals:**
- The sum of the interior angles of a quadrilateral is 360°.
- Thus, we can set up the equation:
\[ \text{Angle S} + \text{Angle T} + \text{Angle A} + \text{Angle R} = 360° \]
- Substituting the known values:
\[ 120° + 5x + 3x + 7x = 360° \]

4. **Combine Like Terms:**
- Combine the terms involving \( x \):
\[ 120° + 15x = 360° \]

5. **Isolate the Variable \( x \):**
- Subtract 120° from both sides:
\[ 15x = 360° - 120° \]
\[ 15x = 240° \]

6. **Solve for \( x \):**
- Divide both sides by 15:
\[ x = \frac{240°}{15} = 16° \]

7. **Find the Measure of Angle R:**
- Now substitute \( x \) back into the expression for angle R:
\[ \text{Angle R} = 7x = 7 \times 16° = 112° \]

### Final Answer:
The measure of angle R is **112°**.

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