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Honors Geometry
Unit #2 Lesson #3
“iff” means
Def: A point is the midpoint of a segment if and only if (iff) it divides the segment into
two congruent segments.
Draw and label a diagram illustrating the following statement: M is the midpoint
of AB iff M is between A and B and AM  MB .
Midpoint Formula: The midpoint of the segment with endpoints ( x1 , y1 ) and ( x2 , y2 ) is
 x1  x2 y1  y2 
,


2 
 2
Def: A geometric figure is a segment bisector iff it intersects the segment at its
midpoint.
Draw and label a diagram illustrating a segment bisector.
Def: A geometric figure is an angle bisector iff it divides the angle into two congruent
angles.
Draw and label a diagram illustrating an angle bisector.
Def: A geometric figure is a trisector iff it divides a segment (or angle) into three
congruent segments (or angles).
Def: A counterexample is an example that shows a statement is false.
Honors Geometry
Unit #2 Assignment #3
1. True or False. If a statement is true, explain why it is true. If a statement is false, draw a
counterexample
or change a word in the statement to make it true.

a. If AX bisects BAC, then mBAX = mXAC.
W
X
b. In the diagram to the right, XY bisects VW .
c. In the diagram to the right, VW bisects XY .
d. If AB = BC, then B is the midpoint of AC.
Y
V
e. If B is the midpoint of AC, then AB = BC.
f. If M is the midpoint of PQ , the ratio of PM to PQ is 1:2.
g. If a straight angle is bisected the angles formed are right angles.
h. For every line segment there is exactly one midpoint.
i. For every angle there is exactly one angle bisector.
j. If M is not the midpoint of CD , then CM  MD.
2. Draw a segment with midpoint N(-2,3) and label it PQ .
3. Draw a segment with midpoint M(4,-1). Label it AB .
4. Let CD have endpoints C(-3,5) and D(4,-1). Find the midpoint of CD .
5. If AB has midpoint M(7,-3) and endpoint A(2,1), find the coordinates of B.
6. Draw quadrilateral (4-sided figure) with vertices P(-2,5)Q(2,9)R(2,1)S(-2,-3).
a. Find the length of each side. What do you notice?
b. Find the midpoint of PR and the midpoint of QS . What do you notice?
7. If X and Y trisect CD , what are the following ratios?
a. XY to CD
b. XD to CD
C
c. CX to CY
X

Y

D



8. In the diagram below XB bisects AXE, XC and XD trisect BXE, and mAXE =
120°. Find mBXD.
B
A
C
D
E
X
9. TRY is an obtuse angle. If mTRY  16 x  20 , find the restrictions on x.
Questions 10 – 15: Draw and label a sketch for each of the following before trying to answer the
question.
10. M is the midpoint of AB with AM = 2x + 1 and MB = 13x – 10. Find x and AB.

2
5
11. AOB below is bisected by OX . If mAOX  x  2 and mAOB  x  45 , find
3
2
mXOB .
12. LM is trisected by points P and Q. If PQ = 7x – 1 and LM = 81, find PM.


5
13. PXQ is bisected by XR and then RXQ is bisected by XS . If mRXS  x  1 and
12
mPXQ  2 x  20 , find mPXR.

14. CAT is right and is bisected by AB . If mCAB  7 x  4 , find x.

15. AB intersects CD at (-1-3,). If the coordinates of C are (4,-1) and the coordinates of D

are (-6,-5) is AB a bisector of CD ? Show work to support your answer.