Math 611 Geometry Chapter 4 Review Problems

Math 611 Geometry Chapter 4 Review Problems

1. Fill in the blanks of the proof with the correct statements and reasons:

Given: P is the midpoint of .

Prove:

Statements / Reasons
1. P is the midpt. of . / 1. Given
2. ZP = XP / 2. If a point is the midpoint of a segment, then it divides the segment into two congruent segments
3. / 3. Given
4. WX = WX / 4. If angles, then sides.
5. / 5. If two points are each equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.
6. / 6. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.

2. a) Identify a pair of corresponding angles

formed by and with transversal .

Angles ABE and ACD

b) Identify a pair of alternate interior angles

formed by and with transversal

Angles EBD and CDB

3. Given:

Prove: is an altitude.

Statements / Reasons
1. Angle ADB = Angle CDB
2. Angle ADB and Angle CDB are right angles
3. BD is an altitude / 1. Given
2. If two angles are both supplementary and congruent, then they are right angles.
3. An altitude forms right angles with the side to which is is drawn.

4. Given:

bisects

Prove:

Statements / Reasons
1. Angle 1 = Angle 4
2. FC bisects Angle BFD
3. Angle 2 = Angle 3
4. Angle CFA = Angle CFE
5. Angle CFA and Angle CFE are right angles
6. CF is perpendicular to AE / 1. Given
2. Given
3. If a ray bisects an angle then it divides the angle into two congruent angles
4. Addition Property
5. If two angles are both supplementary and congruent, then they are right angles.
6. If two lines intersect to form right angles, then they are perpendicular.

5. Set up and complete a proof for the following:

If two isosceles triangles share the same base, then the line joining the vertex angles of the triangles is the perpendicular bisector of the base.

Statements / Reasons
1. Triangles ABC and DBC are isosceles with base BC
2. AB = AC
and DB = DC
3. AD is the perpendicular bisector of BC / 1. Given
2. The legs of an isosceles triangle are congruent.
3. If two points are each equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.

Given: Triangles ABC and DBC are

isosceles with base BC

Prove: AD is the perpendicular bisector of BC

6. Given: .

Prove:

Statements / Reasons
1.  Triangle ABC is isosceles with base AC
2.  BA = BC
3.  Angle 1 = Angle 2
4.  DA = DC
5.  BD is the perpendicular bisector of AC / 1.  Given
2.  The legs of an isosceles triangle are congruent
3.  Given
4.  If angles, then sides
5.  If two points are each equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.

7. Given:

.

Prove:

Statements / Reasons
1.  Circle O
2.  Draw AO and BO
3.  AO = BO
4.  M is the midpoint of AB
5.  AM = BM
6.  OM perpendicular bisector of AB / 1.  Given
2.  Two points determine a segment
3.  All radii of a circle are congruent
4.  Given
5.  If a point is the midpoint of a segment, then it divides the segment into two congruent segments.
6.  If two points are each equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.

8. Set up a proof for the statement, “If two chords of a circle are congruent, then the segments

joining the midpoints of the chords to the center of the circle are congruent.” (A chord is a segment whose endpoints are on the circle.)

Given: Circle O

AB = CD

M and N are midpoints

Prove: MO = NO

9. a) If the median from A intersects at M,

what are the coordinates of M?

(9, 4)

b)  Find the slope of .

1/2

c)  Is parallel to ? Why or why not?

No, they have different slopes

d) Find the slope of the altitude from A to .

-2

10. Given:

Prove:

11. Given:

Prove:

Statements / Reasons
1.  Angle 1 = Angle 2 = Angle 3 = Angle 4
2.  BE = BF
3.  BD = BD
4.  Triangle BED = Triangle BFD
5.  Angle BED = Angle BFD
6.  Angle AEB = Angle CFB
7.  Triangle ABE = Triangle CBF / 1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC
6. If angles are supplementary to congruent angles, then they are congruent
7. ASA

12. Given:

Prove: .

Statements / Reasons
1. Circle O
2. Draw OY and OX
3. OY = OX
4. DX = DY
5. DO (which is also DZ) is the perpendicular bisector of XY / 1. Given
2. Two points determine a line
3. All radii of a circle are congruent
4. Given
5. If two points are each equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.

13. Given:

Prove:

Statements / Reasons
1.  Angle XWY = Angle ZYX
2.  WX = ZY
3.  XY = YX
4.  Triangle WXY = Triangle ZYX
5.  Angle RXY = Angle RYX
6.  WY = ZX
7.  XR = YR
8.  WR = ZR / 1. Given
2. Given
3. Reflexive Property
4. SAS
5. CPCTC
6. CPCTC
7. If angles, then sides
8. Subtraction Property

14. Given:

Prove:

Statements / Reasons
1.  AB = AF
2.  BD = FD
3.  AD = AD
4.  Triangle AFD = Triangle ABD
5.  Angle FDA = Angle BDA
6.  Angle 1 = Angle 2
7.  Angle EDA = Angle CDA
8.  Angle EDA and Angle CDA are right angles
9.  AD perpendicular to CE / 1.  Given
2.  Given
3.  Reflexive Property
4.  SSS
5.  CPCTC
6.  Given
7.  Addition Property
8.  If two angles are both supplementary and congruent, then they are right angles.
9.  If two segments intersect to form right angles, then they are perpendicular.

15. Given:

Prove:

Statements / Reasons
1.  AD is the perpendicular bisector of BC
2.  FC = FB and AC = AB
3.  AF = AF
4.  Triangle AFC = Triangle AFB
5.  Angle 1 = Angle 2 / 1. Given
2. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
3. Reflexive Property
4. SSS
5. CPCTC

16. is a right angle. Explain why (9, 6)

could not be the coordinates of H.

The slope of EF is -2.

If H was at (9, 6), then the slope of FH would be 3/5,

Which is not the opposite reciprocal of -2.

17. Prove: If the bisector of an angle whose vertex lies on a circle passes through the center of

the circle, then it is the perpendicular bisector of the segment that joins the points where the sides of the angle intersect the circle.