CCGPS Analytic Geometry A
6-1 Perpendicular and Angle Bisector Notes
Date _______________
Define equidistant.
PERPENDICULAR BISECTOR
1. Construct a perpendicular bisector of a given line segment using patty paper
̅̅̅̅.
A: Draw a line segment on your patty paper. Label the line segment 𝐴𝐵
B: Fold the patty paper so that points A and B, the two end points of the segment you drew
on the patty paper, coincide with each other. Crease the paper along the fold.
C: Open the patty paper and draw a line on the crease. Label this line l. Label the
intersection of line l with line segment ̅̅̅̅
𝐴𝐵 as point M.
2. Draw point X on line l.
̅̅̅̅ and 𝑋𝐵
̅̅̅̅ as accurately as possible.
3. Draw 𝑋𝐴
̅̅̅̅
4. Measure 𝑋𝐴 and ̅̅̅̅
𝑋𝐵.
5. What is the relationship between XA and XB?
X, a point on the perpendicular bisector of ̅̅̅̅
𝐴𝐵 , is ______________________ from the endpoints A and B.
ANGLE BISECTOR
1. Construct the bisector of an angle using patty paper
A: Draw an angle on a sheet of patty paper. Label this angle ∠APB.
B: Fold your patty paper so that the two sides of the angle, ⃗⃗⃗⃗⃗
𝑃𝐴 and ⃗⃗⃗⃗⃗
𝑃𝐵, coincide. Crease the
paper along the fold.
C: Unfold your patty paper. Select a point on the interior of ∠ APB that lies on the crease.
⃗⃗⃗⃗⃗ .
Label this point C. Draw 𝑃𝐶
2. Draw the segments representing the distance between C and the rays ⃗⃗⃗⃗⃗
𝑃𝐴 and ⃗⃗⃗⃗⃗
𝑃𝐵.
[Remember that the distance from a point to a line is the length of the perpendicular segment from the point to the line.]
3. Measure the segments representing the distance between C and the rays ⃗⃗⃗⃗⃗
𝑃𝐴 and ⃗⃗⃗⃗⃗
𝑃𝐵.
4. What is the relationship between the distances from point C to each of the sides of the angle?
C, a point on the angle bisector of ∠𝐴𝑃𝐵, is ____________________ from the sides of ∠𝐴𝑃𝐵, ⃗⃗⃗⃗⃗
𝑃𝐴 and ⃗⃗⃗⃗⃗
𝑃𝐵.
Theorem
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a
segment, then it is equidistant from the
endpoints of the segment.
Hypothesis
Conclusion
_____ = _____
Converse of the Perpendicular Bisector
Theorem
If a point is equidistant from the endpoints of a
segment, then it is on the perpendicular
bisector of the segment.
_____ ⊥ _____
_____ ≅ _____
Proof of the Perpendicular Bisector Theorem:
̅̅̅̅
GIVEN: l is the perpendicular bisector of 𝐴𝐵
PROVE: 𝑋𝐴 = 𝑋𝐵
STATEMENTS
REASONS
̅̅̅̅ .
1. l is the perpendicular bisector of 𝐴𝐵
1. Given.
2. 𝑙 ⊥ ̅̅̅̅
𝐴𝐵 ; Y is the midpoint of ̅̅̅̅
𝐴𝐵 .
2. Definition of perpendicular bisector.
3. ∠𝐴𝑌𝑋 and ∠𝐵𝑌𝑋 are right angles.
3.
4. ∠𝐴𝑌𝑋 ≅ ∠𝐵𝑌𝑋
4. Right Angle Congruence Theorem
̅̅̅̅
5. ̅̅̅̅
𝐴𝑌 ≅ 𝐵𝑌
5.
6. ̅̅̅̅
𝑋𝑌 ≅ ̅̅̅̅
𝑋𝑌
6.
7. ∆𝐴𝑌𝑋 ≅ ∆𝐵𝑌𝑋
7.
8. ̅̅̅̅
𝑋𝐴 ≅ ̅̅̅̅
𝑋𝐵
8.
9. 𝑋𝐴 = 𝑋𝐵
9.
A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as
the locus of points in a plane that are equidistant from the endpoints of a segment.
Applying the Perpendicular Bisector Theorem and its Converse:
Find each measure.
1. YW
2. BC
3. PR
Theorem
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is
equidistant from the sides of the angle.
Hypothesis
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is
equidistant from the sides of the angle, then it
is on the bisector of the angle.
Conclusion
_____ = _____
∠ _______ ≅ ∠_______
Applying the Angle Bisector Theorem and its Converse:
Find each measure.
1. LM
2. 𝑚∠𝐴𝐵𝐷,
given that 𝑚∠𝐴𝐵𝐶 = 112°
3. 𝑚∠𝑇𝑆𝑈