A square ABCD has an equilateral triangle drawn on the side AB ( interior of the square ) . The triangle has vertex at G. What is the measure of the angle CGB ?

Options

A. `60^(@)`
B. `80^(@) `
C. `75^(@) `
D. `90^(@)`

Answer

To find the measure of angle CGB in the given square ABCD with an equilateral triangle drawn on side AB, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a square ABCD with vertices A, B, C, and D.
- An equilateral triangle AGB is drawn inside the square on side AB, with vertex G.

2. **Identify the Angles in the Triangle**:
- Since triangle AGB is equilateral, all its angles are equal to 60 degrees. Therefore:
- Angle AGB = 60 degrees
- Angle ABG = 60 degrees
- Angle GAB = 60 degrees

3. **Identify the Angles in the Square**:
- In square ABCD, all angles are 90 degrees. Therefore:
- Angle ABC = 90 degrees

4. **Calculate Angle CBG**:
- We know that angle ABC is composed of angle ABG and angle CBG. Thus:
- Angle ABC = Angle ABG + Angle CBG
- 90 degrees = 60 degrees + Angle CBG
- Rearranging gives:
- Angle CBG = 90 degrees - 60 degrees = 30 degrees

5. **Connect Point G to Point C**:
- Now, we connect point G to point C. We need to find angle CGB.

6. **Use Isosceles Triangle Properties**:
- In triangle CBG, we have:
- BG = BC (since AB = BC in the square and BG = AB in the equilateral triangle).
- Therefore, triangle CBG is isosceles with:
- BG = BC
- This means angle CGB = angle GCB.

7. **Sum of Angles in Triangle CBG**:
- The sum of angles in triangle CBG is 180 degrees:
- Angle CGB + Angle GCB + Angle CBG = 180 degrees
- Since angle CGB = angle GCB, we can denote them both as θ:
- θ + θ + 30 degrees = 180 degrees
- This simplifies to:
- 2θ + 30 degrees = 180 degrees
- 2θ = 180 degrees - 30 degrees
- 2θ = 150 degrees
- θ = 150 degrees / 2 = 75 degrees

8. **Conclusion**:
- Therefore, angle CGB = 75 degrees.

### Final Answer:
The measure of angle CGB is **75 degrees**.