An angle of `105^(@)` is drawn using a pair of compass and ruler by bisecting angles________.

Options

A. `90^(@)` and `180^(@)`
B. `30^(@)` and `60^(@)`
C. `90^(@)` and `120^(@)`
D. `120^(@)` and `180^(@)`

Answer

To draw an angle of 105 degrees using a pair of compass and ruler by bisecting angles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Angles to Bisect**: We need to find two angles whose bisected sum equals 105 degrees. After evaluating the options, we find that the angles 90 degrees and 120 degrees will work.

2. **Draw a 90-degree Angle**:
- Start by drawing a straight line using a ruler. This will be one arm of the angle.
- Use a compass to draw a circle with a center at one end of the line.
- Mark the point where the circle intersects the line. This is point A.
- Now, without changing the compass width, place the compass on point A and draw an arc above the line.
- Now, place the compass on the intersection point of the line and the arc, and draw another arc that intersects the first arc.
- Label the intersection point as B.
- Draw a line from the center to point B. This line forms a 90-degree angle with the original line.

3. **Draw a 120-degree Angle**:
- From the same starting point, use the compass to draw another circle.
- Mark the point where the circle intersects the line as point C.
- Now, draw an arc above the line from point C, similar to the previous step.
- Place the compass on point C and draw another arc that intersects the first arc.
- Label this intersection point as D.
- Draw a line from the center to point D. This line forms a 120-degree angle with the original line.

4. **Bisect the Angles**:
- Now, we need to bisect the angles formed by the lines you just drew (90 degrees and 120 degrees).
- To bisect the 90-degree angle, place the compass at the vertex of the angle and draw arcs that intersect both arms of the angle.
- Label the intersection points as E and F.
- Now, draw a line from the vertex through the point where the arcs intersect. This line will bisect the 90-degree angle into two 45-degree angles.
- Repeat the same process for the 120-degree angle, marking the intersection points as G and H, and drawing a line from the vertex through the intersection point.

5. **Measure the Resultant Angle**:
- The angle formed between the original line and the bisector of the 90-degree angle will be 45 degrees.
- The angle formed between the original line and the bisector of the 120-degree angle will be 60 degrees.
- Therefore, the angle between the two bisectors (45 degrees + 60 degrees) will give us the required angle of 105 degrees.

### Final Result:
Thus, by bisecting the angles of 90 degrees and 120 degrees, we can successfully draw an angle of 105 degrees.