20. Think About a Plan Draw an ∠ A . angle eh .  Construct an angle whose measure is 1 4 m ∠ A . 1 fourth , m angle eh .
    • How is the angle you need to construct related to the angle bisector of ∠ A ? angle eh question mark
    • How can you use previous constructions to help you?
21. Answer the questions about a segment in a plane. Explain each answer.
    1. How many midpoints does the segment have?
    2. How many bisectors does it have?
    3. How many lines in the plane are its perpendicular bisectors?
    4. How many lines in space are its perpendicular bisectors?

For Exercises 22–24, copy ∠ 1  and  ∠ 2 . angle 1 , and , angle 2 .  Construct each angle described.

22. ∠ B ; m ∠ B = m ∠ 1 + m ∠ 2 angle b semicolon m angle , b equals , m angle 1 , plus m angle 2
23. ∠ C ; m ∠ C = m ∠ 1 − m ∠ 2 angle c semicolon m angle , c equals , m angle 1 minus m angle 2

24. ∠ D ; m ∠ D = 2 m ∠ 2 angle d semicolon m angle . d equals 2 m angle 2

    

25. Writing Explain how to do each construction with a compass and straightedge.
    1. Draw a segment P Q ¯ . p q bar , .  Construct the midpoint of P Q ¯ . p q bar , .
    2. Divide P Q ¯ p q bar  into four congruent segments.
26. 1. Draw a large triangle with three acute angles. Construct the bisectors of the three angles. What appears to be true about the three angle bisectors?
    2. Repeat the constructions with a triangle that has one obtuse angle.
    3. Make a Conjecture What appears to be true about the three angle bisectors of any triangle?

Use a ruler to draw segments of 2 cm, 4 cm, and 5 cm. Then construct each triangle with the given side measures, if possible. If it is not possible, explain why not.

27. 4 cm, 4 cm, and 5 cm
28. 2 cm, 5 cm, and 5 cm
29. 2 cm, 2 cm, and 5 cm
30. 2 cm, 2 cm, and 4 cm
31. 1. Draw a segment, X Y ¯ . x y bar , .  Construct a triangle with sides congruent to X Y ¯ . x y bar , .
    2. Measure the angles of the triangle.
    3. Writing Describe how to construct a 60 ° 60 degrees  angle using what you know. Then describe how to construct a 30 ° 30 degrees  angle.

32. Which steps best describe how to construct the pattern below?

    

    1. Use a straightedge to draw the segment and then a compass to draw five half circles.
    2. Use a straightedge to draw the segment and then a compass to draw six half circles.
    3. Use a compass to draw five half circles and then a straightedge to join their ends.
    4. Use a compass to draw six half circles and then a straightedge to join their ends.

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