Question

copy the segments. see example 1
15.
16.
copy the angles. see example 2
17.
18.
copy and bisect the segments.
see example 3
19.
20.
copy and bisect the angles. see example 4
21.
22.

1. where is the intersection of the perpendicular bisector of \\(\overline{gf}\\) and the angle bisector of \\(\angle e\\)? see example 5

Answer:

To solve these geometry construction problems, we follow standard geometric construction techniques (using a compass and straightedge for accuracy). Here's how to approach a few key types:

1. Copying a Segment (e.g., Problem 15: Segment \( \overline{GH} \))

Step 1: Draw a new line.

Draw a ray (or line segment) longer than \( \overline{GH} \). Label its starting point (e.g., \( G' \)).

Step 2: Measure \( \overline{GH} \) with a compass.

Open the compass to the length of \( \overline{GH} \) (place the needle at \( G \), pencil at \( H \)).

Step 3: Transfer the length.

Without changing the compass width, place the needle at \( G' \) and mark a point \( H' \) on the new ray. Now \( \overline{G'H'} \cong \overline{GH} \).

2. Copying an Angle (e.g., Problem 17: \( \angle D \))

Step 1: Draw a new ray.

Draw a ray (e.g., \( \overrightarrow{D'} \)) to serve as one side of the copied angle.

Step 2: Measure the original angle.

Place the compass needle at \( D \), draw an arc intersecting both sides of \( \angle D \) (label the intersections \( A \) and \( B \)).

Step 3: Transfer the arc to the new ray.

Without changing the compass width, place the needle at \( D' \) and draw an arc intersecting \( \overrightarrow{D'} \) at \( A' \).

Step 4: Match the arc length.

Adjust the compass to the distance \( AB \) (from the original angle’s arc). Place the needle at \( A' \) and draw an arc intersecting the previous arc at \( B' \).

Step 5: Draw the second side.

Draw a ray from \( D' \) through \( B' \). Now \( \angle D' \cong \angle D \).

3. Bisecting a Segment (e.g., Problem 19: \( \overline{AB} \))

Step 1: Open the compass.

Set the compass width to more than half of \( \overline{AB} \).

Step 2: Draw arcs from \( A \) and \( B \).

Place the needle at \( A \), draw arcs above and below \( \overline{AB} \). Repeat with the needle at \( B \), intersecting the first arcs at two points (e.g., \( X \) and \( Y \)).

Step 3: Draw the perpendicular bisector.

Draw a line through \( X \) and \( Y \). This line bisects \( \overline{AB} \) at its midpoint \( M \), so \( AM = MB \).

4. Bisecting an Angle (e.g., Problem 21: \( \angle K \))

Step 1: Draw an arc from the vertex.

Place the compass needle at \( K \), draw an arc intersecting both sides of \( \angle K \) (label the intersections \( A \) and \( B \)).

Step 2: Draw arcs from \( A \) and \( B \).

Set the compass to a width greater than \( \frac{1}{2}AB \). Draw arcs from \( A \) and \( B \) that intersect at a point \( P \).

Step 3: Draw the angle bisector.

Draw a ray from \( K \) through \( P \). This ray bisects \( \angle K \), so \( \angle AKP = \angle BKP \).

5. Intersection of Bisectors (Problem 23)

To find the intersection of the perpendicular bisector of \( \overline{GF} \) and the angle bisector of \( \angle E \):

• First, construct the perpendicular bisector of \( \overline{GF} \) (as in Step 3 above).
• Then, construct the angle bisector of \( \angle E \) (as in Step 3 above).
• The point where these two bisectors meet is their intersection.

Final Answer (for Construction Problems)

These problems require hands - on geometric construction using a compass and straightedge. Follow the step - by - step methods above for each type of construction (copying segments/angles, bisecting segments/angles, or finding intersections of bisectors). For example, when bisecting \( \overline{AB} \), the midpoint \( M \) satisfies \( AM = MB \); when bisecting \( \angle K \), the bisector creates two equal angles. The intersection in Problem 23 is the point common to both the perpendicular bisector of \( \overline{GF} \) and the angle bisector of \( \angle E \).