Geometry 5.1Polygon Interior/Exterior AnglesRecall the sum of a polygon’s interior angles:

Sum of interior angles in any triangle: _____

Sum of interior angles in any quadrilateral: _____

Sum of the angles in any n-sided figure: _____

Practice:Find the missing angles in each figuure below:

1. 2. 3.

Exterior angles:The sum of the exterior angles in apolygon always equals 360o. This iseasier to explain with a diagramthan a proof:

Concave polygons: An easy way to define an exterior angle is: The supplement of the corresponding interior angle.What must be true about the exterior anglesin a convex figure?

What is the sum of the two missing exterior angles in the figure above?

x

140o 80o

x60o

70o 80o

50o

x

120o

60o40o

100o

40o

120o40o

100o40o

60o

170o

110o

160o

Geometry 5.2Exterior AnglesAnswer:1. The exterior angles are supplementary to the interior angles in aregular polygon with n sides.360/n gives you the measure of each exterior angle in a regular n-gon.Therefore: n[180-(360/n)] can be used to find the sum of all angles in aregular n-gon. Is this the same as 180(n-2)?

2. What is the exterior angle measure of a 15-gon?

3. In regular decagon ABCDEFGHIJ, what is the measure of angle ACB?

4. The interior angles of a polygon measure 179o. How many sides doesthe figure have?

5. A regular polygon has exterior angles measuring 12o.How many sides does the polygon have?

Answer:1. Is it possible for a triangle to have an acute exterior angle?

2. A polygon has 11 sides and 10 congruent 150o interior angles.What are the measures of its exterior angles?

3. The star below has ten sides and five acute interior angles. Can youdraw a ten-sided polygon with more than five acute interior angles?

8 is not possible because 8(90)=720o

leaving 720o for the remaining 2 angles.2

1

3

4

5

6

7

Geometry 5.3Kites and TrapezoidsParts of a Kite:

1 and 3 are Vertex Angles

2 and 4 are Nonvertex Angles

More proofs!No need to write them, just explain how each could be done.

1. How could you prove that the nonvertex angles are congruent?

2. How could you prove that the diagonal connecting the vertex angles bisects them?

3. How could you prove that the two diagonals are perpendicular to each other.

4. How could you prove that the diagonal connecting the vertex angles bisects theother diagonal?

Parts of a Trapezoid:

AB and CD are bases.A and B are a pair of base angles.What is the other pair?If AD = BC, the trapezoid is an Isosceles TrapezoidBE is an altitude.

1. How can you prove that consecutive angles of a trapezoid are supplementary?

2. How can you prove that the base angles of an isosceles trapezoid are congruent?hint:

3. Using the information from #2, how can you prove that the diagonals in an isosce-les trapezoid are congruent?

4. Find the angle measures:m ABD =m ABC =m BCD =m CDB =

1

2

3

4

A B

CD E~

A B

CD

30o

A

B C

D

E

Geometry 5.4MidsegmentsMidsegments of a triangle connect the midpoints of the sides.Every triangle has three midsegments.

Discover properties of triangle midsegments by completing the followingconstruction:

1. Draw a hand-sized scalene triangle on your paper.2. Construct and connect the three triangle midsegments.

You should have created four triangles.Measure the sides of all four triangles.What do you notice?

Complete the following statements about midsegment AB:

Segments AB and YZ are ________. Explain the proof (use congruent s)

Segment AB is _____ the length of segment YZ. Explain the proof.

Determine the measure of each missing side or angle below:

A B

X

Y ZC

10cm

a

3cmb

8cm

c

38 o

d

50 o80 o

e

50o

fg

25o

Geometry 5.4MidsegmentsMidsegments of a trapezoid connect midpoints of the non parallel sides.

The midsegment of a trapezoid will always be parallel to the bases.The simplest proof that the midsegment is parallel to the bases involvescoordinate geometry and slope.

The length of the midsegment is related to its bases as well.

Can you write a formula for the length of the midsegment of a trapezoidusing B1 and B2 as the bases?

Solve:Find length x given the following midsegments:

1. 2. 3.

B1 B1 B2

B2

119

10

x

3

6.5

x

x

y-3y

Geometry 5.4ReviewDetermine the measure of each angle labeled below:

1. 2.

3. What is the maximum number of obtuseexterior angles in a hexagon?

4. Determine the measure of the interior angles in a regular 36-gon.

Determine the measure of each angle labeled below:

1. 2.

3. 4. Two angles in a kitemeasure 50 and 110degrees. List all possiblecombinations of the othertwo angles.

a

b c

d

ea

cb

ed

a

40o

70o

b

de

c

f

120o

35oa b

c

d

110o

30o

c

b

a

Geometry 5.4Review

1511.5

m

n

Determine the measure of each missing side below in eachtrapezoid. Midsegments are drawn in each.

1. 2. 3.

Determine the angle measure for each:Write ‘cannot be determined’ where applicable.

1. 2.

k

5 3

f

34

6

4

g

h

12

1889o

fd

a

c

60o

b

e

54o134o

91oa

be

f hj135o

c

d

g

k

60o

m

Geometry 5.5Parallelogram PropertiesUsing a straight edge, draw parallelogram KLMN that is not a rectangleor a rhombus.

Complete the following statements about the parallelogram:

Opposite angles are ___________.Consecutive angles are ___________.Opposite sides are __________.Diagonals of the parallelogram ______ _______ _______.

Explain the proof of each statement above using the figure below:

Solve:

K

N

L

M

J

1. Given:RS = 10cmMA = 16cmGM = 21cmFind the perimeterof GRM.G

R

M A

S

2. Given:Perimeter of GRS=21Perimeter of RAS=26Perimeter of GRA=35Find the length of RM.

GR

M A

S

3. Vectors:A plane flies north at250mph, with aheadwind as shownblowing 40mph.draw a line to repre-sent the path andspeed of the plane?

Wind

Air

Spee

d

Geometry 5.5Perimeter ProblemsSolve:

RA

DU

L

1. Given:Perimeter of RAD=24cmPerimeter of RAU=21cmAU=7cmFind the length of RD.

C

A

D

R

2. Given:Perimeter of DOC=18cmPerimeter of CAR=26cmCO=5cm CA=9cmFind the length of DO.

O

Solve:

T

A

NR

I

1. Given:Perimeter of TIN=24cmFind the perimeter of RAG

N

2. Given:GB is a midesgment oftrapezoid NRKF.Perimeter of BRKG = 31mPerimeter of BGFN = 37mRK = 10mFind BG

GB

RK

G

F

3. Given:RIA is a 5-12-13 right trianglePerimeter of BRN = 22kmFind the perimeter of kite BRAN

Challenge:CE is the midsegment of BDFBF is the midsegment of ACEGThe perimeter of CDE is 17The perimeter of ACEG is 46Find CE

B

R A

N

I

A

B

C

D

E

F

G

GeometryPolygon Properties ReviewName________________________ Period _____

Determine the measures of each missing angle or side labeled below:Not To ScaleThe hexagon and pentagon are both regular.

a= _____

b= _____

c= _____

d= _____

e= _____

f= _____

g= _____

h= _____

j= _____

k= _____

m= _____

n= _____

5cma

b

c

d

e

f

hg

j

4cm

6cmk

m

Given:Perimeter of WXY = 42cmPerimeter of YXZ = 33cm

W

V

X

Y

Z

n

5.5

Geometry 5.5Polygon Properties PracticeDetermine the measures of each missing angle below:The decagon in the figure is regular.

1. a= ______

2. b= ______

3. c= ______

4. d= ______

5. e= ______

6. Two of the angles in a convex kite measure 105o and 145o. What is the SMALLESTpossible angle measure in the kite?

6. ______

Determine the measures or length for each:

7. A= ______

8. BK= ______

9. ALN= ______

10. SK= ______

11. NK= ______

12. BSF= ______

13. GF= ______

14. Use the given information to find the perimeter of parallelogram PAKE:14. ______

Name________________________ Period _____

a c

db

e

B

A

EL N

KS

Given:ES = 10cm SEN = 40o

LN GF

FG

5cm 7cm

Given:Perimeter of ARK = 16cmPerimeter of KER = 13cmAK = 7cm

P

AR

K

E

Geometry 5.5Polygon Properties Practice QuizDetermine the measures of each missing angle below:The nonagon in the figure is regular.

1. a= ______

2. b= ______

3. c= ______

4. d= ______

5. e= ______

6. Two of the angles in a convex kite measure 100o and 45o. What is the LARGESTpossible angle measure in the kite?

6. ______

Determine the measures or length for each: (not to scale)

7. L= ______

8. ES= ______

9. NES= ______

10. LS= ______

11. AN= ______

12. FBK= ______

13. FS= ______

14. Use the given information to find the length of AE:14. ______

Name________________________ Period _____

a

c

d

b e

B

A

E

L

NK

S

Given:NE = EK FAN = 84o

GSE = 32o

F

G

6cm

8cm

Given:Perimeter of AKE = 33cmPerimeter of PAKE = 42cm

P

AR

K

E

~

12cm

GeometrySolve each:

1. In regular octagon ABCDEFGH, what is the measure of angle ACH?

_______

2. In parallelogram ABCD, Diagonal AC is twice the length of diagonal BD.The perimeter of triangle ABC is 21 and the perimeter of triangle BCDis 17cm, what is the perimeter of parallelogram ABCD?

_______

3. Convex kite QRST has vertex angles Q and S, where Q is three times themeasure of S. Angle QRT is 30 degrees, find the measure of angle SRT.

_______

4. The sides of a regular polygon are 2cm long and the interior angles of thepolygon measure 171 degrees. What is the perimeter of the polygon?

_______

5. The midpoints of quadrilateral ABCD are connected to form quadrilateral WXYZ.If diagonals AC=10cm and BD=8cm, what is the perimeter of quadrilateral WXYZ?

_______

6. What is the smallest angle which can be created by connecting three vertices ofa regular 36-gon?

_______

7. The long diagonal of a rhombus is 3 times the length of the short diagonal.The perimeter of the rhombus is 40cm, what is its area?(Hard, think it through!)

_______

Challenge: How many regular polygons have interior angle measures of integral(integer) measure?

_______

More Difficult Polygons PracticeName________________________ Period _____

GeometrySolve each:1. In regular nonagon ABCDEFGHI, what is the measure of angle AFB?

_______

2. Rhombus ABCD has diagonal lengths of 10 and 24.What is the perimeter of the rhombus?

_______

3. Concave kite FGHJ has a vertex angle H measuring 10 degrees.Interior angle F is three times the measure of angle G. What is the measureof the non-vertex angles?

_______

4. In the section of regular polygon below, the measure of angle ACB is 5 degrees.What is the perimeter of the polygon?

_______

5. The midpoints of quadrilateral WXYZ are connected to form rhombus PQRS.If the perimeter of PQRS = 18cm, what is the length of diagonal WY?

_______

6. One of the angles in a convex kite measures 174 degrees.One vertex angle is two degrees larger than the other.What is the smallest possible angle measure in the kite?

_______

7. The diagonals of parallelogram ABCD intersect at X. If AX = 25, and AB = 7,and angle ABD is a right angle. What is the area of the parallelogram?

_______

More Dificult Polygons PracticeName________________________ Period _____

AB

C5cm

Geometry 5.6Special ParallelogramsSpecial Parallelograms:Rhombuses are equilateral ___________.Rectangles are ________ parallelograms.Squares are __________ and __________ parallelograms.

Construct rhombus ABCD on your paper with diagonals AC and BD.What can you infer about the diagonals of a rhombus?

Are they perpendicular?Do they bisect each other?Are they congruent?Do they bisect the angles?Explain the proof for each time youanswered ‘yes’ above.

Construct rectangle ABCD on your paper with diagonals AC and BD.What can you infer about the diagonals of a rectangle?

Are they perpendicular?Do they bisect each other?Are they congruent?Do they bisect the angles?Explain the proof for each time youanswered ‘yes’ above.

Construct square ABCD on your paper with diagonals AC and BD.What can you infer about the diagonals of a square?

Are they perpendicular?Do they bisect each other?Are they congruent?Do they bisect the angles?Explain the proof for each time youanswered ‘yes’ above.

A E

D

C

B

A

E

D C

B

A

E

D C

B

Geometry 5.7Proofs PracticeProve:Write a two-column or flowchart proof for each of the following:

1. If the diagonals of a quadrilateral bisect each other, then thequadrilateral is a parallelogram.

Challenge. If the diagonals of a quadrilateral are perpendicular, congru-ent, and bisect each other, then the quadrilateral is a square.

hint: Use “Congruent parts of congruentsegments” in your proof.

Prove:Write a two-column or flowchart proof for each of the following:

1. Prove that if two sides of a quadrilateral are congruent and parallel, thenthe other two sides are congruent.

Challenge. If the diagonals of a trapezoid are congruent, then the trap-ezoid is isosceles.

hint: Construct the parallelline shown in the diagram.

B U

MP

Y

H

R

I

G

T

H

E L

IU

M

F

I

L

E

T

GeometryWrite a Complete Proof for each statement and diagram below. You may usea two-column or flow-chart proof. Only use the information included in thestatements and diagrams.

1. Prove that if opposite sides of a quadrilateral are congruent,then opposite sides are parallel.

2. Prove that the opposite sides of a rectangle are congruent.

5.7Proofs PracticeName________________________ Period _____

F

H

I

E

T

B

K

A E

R

GeometryWrite a Complete Proof for each statement and diagram below. You may usea two-column or flow-chart proof. Only use the information included in thestatements and diagrams.

3. Prove that opposite angles in a parallelogram are congruent.

4. Prove that if the diagonals of a quadrilateral are perpendicularand one bisects the other, then the quadrilateral is a kite.

5.7Proofs PracticeName________________________ Period _____

S

T

P

E

I

T E

S

K

GeometryAnswer the following:

1. The measure of an interior angle in a regular octagon is ___.1. ________

2. If the exterior angles in a regular polygon measure 72 degrees,how many sides does the figure have?

2. ________

3. The sum of the interor angles in a polygon is equal to the sum of theexterior angles. How many sides does the polygon have?

3. ________

4. In an isosceles trapezoid, two of the angles measure 56 degrees.What are the measures of the other two angles?

4. ________

5. The two vertex angles in a kite measure 55 and 170degrees. What are the measures of the nonvertex angles?

5. ________

6. The perimeter of the triangle formed by the midsegments oftriangle ABC is 14 inches. If AB = 6 and BC = 12, find thelength of AC.

6. ________

7. Trapezoid EFGH has bases measuring x-10 and 3x. If the midsegmentis 19 inches long, what is the length of the short base?

7. ________

8. Triangle ABC has midpoint D on segment AB and E on segment AC.If the perimeter of triangle ADE is 20 inches, what is the perimeterof triangle ABC?

8. ________

9. In parallelogram ABCD, angle A measures (x+5)o and angle Cmeasures (2x-16)o. Find the measure of angle B.

9. ________

10. In parallelogram QRST, the diagonals intersect at X.The perimeter of QRST is 26.The perimeter of RST is 20.Find the length of RX.

10. ________

5.7Quadrilaterals Practice TestName________________________ Period _____

Geometry

Write a two-column proof to prove that if opposite sides of a quadrilateralare congruent and parallel, then the remaining sides are also parallel.

note:Use only what is givenin the diagram.

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

5.7Quadrilaterals Practice TestName________________________ Period _____

Find each missing angle:The pentagon and hexagon are regular.

11. a=_______

12. b=_______

13. c=_______

14. d=_______

15. e=_______

a cbd

e

S

T

P

E

GeometryWhat is the measure of an interior angle in a regular pentagon?

The exterior angles in a regular polygon measure 18 degrees.How many sides does it have?

One vertex angle in a kite measures 57 degrees and one nonvertexangle measures 125 degrees. What are the measures of the otherangles?

An isosceles triangle has a perimeter of 42 inches, and has mid-points A, B, and C. If BC = 3 inches, what is the length of AB?

The midsegment of a trapezoid measures (x+3) inches. The longbase of the trapezoid measures (2x+3) inches. What is the lengthof the short base?

Midsegment GH of triangle XYZ is parallel to side YZ. If GH is 7inches long, GX is 6 inches, and XZ is 16 inches, what is the perim-eter of triangle XYZ?

Rectangle WAVE has diagonals which intersect at N. WAVE has aperimeter of 34 inches. If the perimeter of triangle AVE is 30inches, what is the length of AN?

Isosceles trapezoid QRST has midsegment AB which is 10 incheslong. The perimeter of ARSB is 40 inches and the perimeter ofQABT is 46 inches. What is the length of RS?

One of the exterior angles in a rhombus is 37 degrees. What arethe measures of the four interior angles?

In parallelogram GRAM, angle G is (3x-5) degrees and angle R is(x+25) degrees. What is the measure of angle A in degrees?

(give the actual angle measure in degrees)

In parallelogram MANY the diagonals intersect at X.If MX = (2x+7) cm and NX = (3x-5) cm, how manycentimeters long is MN? (give the actual length)

5.7Quadrilaterals Test Review

Geometry1. Find each missing angle measure below:Note: All figures that appear regular are regular.

a ______

b ______

c ______

d ______

e ______

2. Two angles of a convex kite measure 120 and 90 degrees.What is the largest possible measure of the other angle(s)?

2. ________3. Isosceles triangle ABC has a perimeter of 18 inches, with congruent

sides AB and BC. AC = 4 inches. Triangle XYZ is formed by themidsegments of triangle ABC. What are the lengths of all threesides of triangle XYZ?

3. ____ ____ ____

4. A trapezoid has midsegment JK. One of the bases of the trapezoidis 12 inches longer than the other. JK is 11 inches long.What is the length of the short base?

4. ________

5. In parallelogram ABCD, angle A measures (x+24)o and angle Bmeasures (2x-60)o. Find the measure of angle D.

5. ________6. In parallelogram EXTR the diagonals intersect at A. If EA is (7x-6) cm

long and AT is (2x+4) cm long, what is the length of segmentET in centimeters?

6. ________7. In triangle ART, midsegment KB is parallel to side AT. If angle

KBT is 75 degrees, what is the measure of angle ATB?

7. ________

RePolygon ReviewName________________________ Period _____

ab c

d

e

Geometry RePolygon ReviewName________________________ Period _____

8. Find each missing angle measure below:Note: The nonagon is regular.

a ______

b ______

c ______

d ______

e ______

9. Prove that the diagonals of a rhombus are perpendicular.Note: only use the following given information:Opposite sides are parallel.All sides are congruent.

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________

____________________________ ____________________________________Add more lines on separate paper if needed.

a

b

c

d

e

B N

R AD

10. Given:Perimeter of MSA = 39Perimeter of MSG = 44Perimeter of MGA = 63

MR = _____

GR

M A

S