Answer :
Certainly! Let's go through the detailed steps to construct each of these angles using a compass and straightedge.
a) Constructing a 60° angle:
1. Draw a straight line segment. This will be one arm of the angle.
2. Place the compass point on the end of the line segment.
3. Adjust the compass to a convenient radius and draw an arc that intersects the line segment.
4. Without changing the radius, place the compass point at the intersection of the arc and line segment. Draw another arc that intersects the first arc.
5. Draw a straight line from the end of the original line segment through the new arc intersection. This line forms a 60° angle with the original line segment.
b) Constructing a 120° angle:
1. Follow the steps to construct a 60° angle.
2. Extend the arm of the 60° angle to form a straight line segment.
3. Construct another 60° angle on the opposite side of the straight line.
4. The angle opposite the first 60° angle will be 120°.
c) Constructing a 90° angle:
1. Draw a straight line segment.
2. Place the compass at the midpoint of the line segment.
3. Draw arcs above and below the line segment using the same radius.
4. Without changing the radius, place the compass at each end of the segment and draw arcs that intersect the previous arcs.
5. Draw a straight line through the intersection points of the arcs. This line is perpendicular to the initial segment, forming a 90° angle.
d) Constructing a 30° angle:
1. Follow the steps to construct a 60° angle.
2. Bisect the 60° angle. To do this,:
a. Draw an arc from the vertex of the angle that intersects both arms.
b. From both intersection points, draw arcs that intersect each other.
c. Draw a line through the vertex and the intersection of arcs. This line bisects the 60° angle, forming two 30° angles.
e) Constructing a 75° angle:
1. Follow the steps to construct a 60° angle.
2. Follow the steps to construct a 90° angle.
3. Bisect the angle between 60° and 90°:
a. Draw arcs from the vertex intersecting both angle arms.
b. From these intersections, draw arcs intersecting each other.
c. Draw a line through these new intersection points and the vertex, giving a 75° angle.
f) Constructing a 105° angle:
1. Construct a 60° angle.
2. Construct a 90° angle.
3. Bisect the angle between 90° and 120° to get a 45° angle.
4. Add 60° and 45° angles to get a 105° angle.
g) Constructing a 150° angle:
1. Construct a 30° angle.
2. Extend the opposite arm of the 30° angle to form a straight line.
3. The angle between the extended arm and the initial arm is 150°.
h) Constructing a 45° angle:
1. Follow the steps to construct a 90° angle.
2. Bisect the 90° angle to get two 45° angles.
i) Constructing a 15° angle:
1. Construct a 30° angle.
2. Bisect the 30° angle to form two 15° angles.
j) Constructing a 22.5° angle:
1. Follow the steps to construct a 45° angle.
2. Bisect the 45° angle to form two 22.5° angles.
These steps provide a clear and systematic way to construct each of the specified angles using a compass and straightedge.
a) Constructing a 60° angle:
1. Draw a straight line segment. This will be one arm of the angle.
2. Place the compass point on the end of the line segment.
3. Adjust the compass to a convenient radius and draw an arc that intersects the line segment.
4. Without changing the radius, place the compass point at the intersection of the arc and line segment. Draw another arc that intersects the first arc.
5. Draw a straight line from the end of the original line segment through the new arc intersection. This line forms a 60° angle with the original line segment.
b) Constructing a 120° angle:
1. Follow the steps to construct a 60° angle.
2. Extend the arm of the 60° angle to form a straight line segment.
3. Construct another 60° angle on the opposite side of the straight line.
4. The angle opposite the first 60° angle will be 120°.
c) Constructing a 90° angle:
1. Draw a straight line segment.
2. Place the compass at the midpoint of the line segment.
3. Draw arcs above and below the line segment using the same radius.
4. Without changing the radius, place the compass at each end of the segment and draw arcs that intersect the previous arcs.
5. Draw a straight line through the intersection points of the arcs. This line is perpendicular to the initial segment, forming a 90° angle.
d) Constructing a 30° angle:
1. Follow the steps to construct a 60° angle.
2. Bisect the 60° angle. To do this,:
a. Draw an arc from the vertex of the angle that intersects both arms.
b. From both intersection points, draw arcs that intersect each other.
c. Draw a line through the vertex and the intersection of arcs. This line bisects the 60° angle, forming two 30° angles.
e) Constructing a 75° angle:
1. Follow the steps to construct a 60° angle.
2. Follow the steps to construct a 90° angle.
3. Bisect the angle between 60° and 90°:
a. Draw arcs from the vertex intersecting both angle arms.
b. From these intersections, draw arcs intersecting each other.
c. Draw a line through these new intersection points and the vertex, giving a 75° angle.
f) Constructing a 105° angle:
1. Construct a 60° angle.
2. Construct a 90° angle.
3. Bisect the angle between 90° and 120° to get a 45° angle.
4. Add 60° and 45° angles to get a 105° angle.
g) Constructing a 150° angle:
1. Construct a 30° angle.
2. Extend the opposite arm of the 30° angle to form a straight line.
3. The angle between the extended arm and the initial arm is 150°.
h) Constructing a 45° angle:
1. Follow the steps to construct a 90° angle.
2. Bisect the 90° angle to get two 45° angles.
i) Constructing a 15° angle:
1. Construct a 30° angle.
2. Bisect the 30° angle to form two 15° angles.
j) Constructing a 22.5° angle:
1. Follow the steps to construct a 45° angle.
2. Bisect the 45° angle to form two 22.5° angles.
These steps provide a clear and systematic way to construct each of the specified angles using a compass and straightedge.
