Explain how you can use a straightedge and a compass to construct an angle that is both congruent and adjacent to a given angle.
To construct an angle that is both congruent and adjacent to a given angle, follow these steps:
1. Start by drawing the given angle using a pencil and ruler.
2. Place the pointed end of the compass at the vertex of the given angle and draw an arc that intersects both sides of the angle.
3. Without changing the compass width, reposition the pointed end at one of the intersection points on one side of the given angle and draw another arc that intersects the angle.
4. Without changing the compass width, reposition the pointed end at the second intersection point on the opposite side of the given angle and draw a final arc that intersects the angle.
5. Now, use your straightedge to draw a line connecting the vertex of the given angle with the last two intersection points of the arcs you drew. This line will be your constructed angle.
6. Finally, erase the construction arcs and unnecessary lines to reveal the constructed angle, which is both congruent and adjacent to the given angle.
By using a straightedge and compass, you can ensure that the constructed angle will be congruent to the given angle because the compass width remains the same throughout the construction process. Additionally, the construction guarantees adjacency as the constructed angle shares a common vertex with the given angle.
When constructing a perpendicular bisector, why must the compass opening be greater than one-half the length of the segment?
When constructing a perpendicular bisector, the compass opening must be greater than one-half the length of the segment because of the nature of the construction process.
To construct a perpendicular bisector, follow these steps:
1. Draw the given line segment using a pencil and ruler.
2. Place the pointed end of the compass at one endpoint of the segment and adjust the opening to be greater than one-half the length of the segment.
3. Draw two arcs, one above and one below the line segment, intersecting both sides of the segment.
4. Without changing the compass width, place the pointed end at the other endpoint of the segment and draw two arcs intersecting both sides of the segment.
5. Use the straightedge to draw a line connecting the two intersection points of the arcs. This line will be your perpendicular bisector.
The reason the compass opening must be greater than one-half the length of the segment is because we need to ensure that the arcs intersect above and below the segment. If the compass opening is exactly one-half the length of the segment, the arcs will only intersect at the endpoints of the segment, resulting in a line that does not perfectly bisect the segment.
By making the compass opening slightly larger than one-half the length of the segment, we create enough space for the arcs to intersect above and below the segment, ensuring that the perpendicular bisector line accurately bisects the segment at its midpoint.
When constructing an angle bisector, why must the arcs intersect?
To construct an angle that is both congruent and adjacent to a given angle, follow these steps using a straightedge and compass:
Step 1: Draw a line segment AB, representing the given angle.
Step 2: Place the compass at point A and draw an arc that intersects line AB at point C.
Step 3: Without changing the compass width, place the compass at point C and draw another arc that intersects the first arc at point D.
Step 4: Use the straightedge to draw a line segment from point A to point D.
Step 5: Place the compass at point D and draw an arc that intersects line AD at point E.
Step 6: Place the compass at point A and draw an arc with the same width as earlier. This arc should intersect the extension of line AD at point F.
Step 7: Use the straightedge to draw a line segment from point E to point F.
The resulting angle DEF is both congruent and adjacent to the given angle.
To construct an angle that is both congruent and adjacent to a given angle using a straightedge and a compass, follow these steps:
1. Start with the given angle, which we'll call angle ABC. Draw a ray with endpoint A, which will serve as the initial side of the new angle we want to construct.
2. Place the compass at point A and draw an arc that intersects both sides of angle ABC. Label the points where the arc intersects the sides of the given angle as D and E.
3. Without changing the radius of the compass, place the compass at point D and draw an arc that crosses the ray you drew in step 1, marking a point F on that ray.
4. Keeping the same radius, place the compass at point E and draw another arc that intersects the arc drawn in step 3. Label the intersection of these two arcs as G.
5. Use your straightedge to draw a straight line connecting point A and point G.
The angle formed by ray AG and the given ray AB is both congruent and adjacent to the given angle ABC.