BD IIAE

R

Ex.2.

2 Given: RS == LS;

SP

^{bisects L TSL.}

Prove:

SP

^II

RL.

182

T B

Ex. 3.

3. Given: DB

^{bisects LADC;}

D Upper base C

Median ---

Lower base

'B Fig. 6.4.

184

(b) FUNDAMENTALS OF COLLEGE GEOMETRY

L~D

^(a)

(d)

(c)

(e) (f)

Fig. 6.1. Polygons.

Definitions: A polygon is a quadrilateral iff it has four sides; it is a pent gon iff it has five sides, a hexagon iff it has six sides, an octagon iff it has eighl sides, a decagon iff it has ten sides, and an n-gon iff it has n sides. .

Definitions: A polygon is equilateral iff all its sides are congruent.-c polygon is equiangular iff all its angles are congruent. A polygon is a regult polygon iff it is both equilateral and equiangular.

Definitions: The sum of the measures of the sides of a polygon is called tli perimeter of the polygon. The perimeter will always be a positive numbe'

A diagonal of a polygon is a segment whose endpoints are nonadjace venice:, uf Lhe pUlygoll. The side UPUll which Lhe pulygon appcaL5-tQ is called the base of the polygon. In Fig. 6.2, ABCDE is a polygon of

D

1\

/ \

/ \

/ \

I \

I \

--/

I _\ \

-- --

^{,/ ,/}

/ '"

~dO(\~

-

^\--

-

^/

I v~ ~

-

^{\ /' /'}

__r-

^{\ //}

Ef::--

^{/ )/}

"""'

I

,/ \

-J... ^{// \}

I """' //

\

I /"

\

1,/ \

./ Base""'" """' \

c

A B

Fig. 6.2. Diagonals and base of a polygon.

---

POLYGONS

-

PARALLELOGRAMS

185

Fig.6.3 Exterior angles of a polygon.

sides; A, E, C, D, E are vertices of the polygon; LA, LB, LC, LD, LE are the angles of the polygon. There are two diagonals drawn from each vertex of the figure.

Definition: An exterior angle of a polygon is an angle that is adjacent to and supplementary to an angle of the polygon (Fig. 6.3).

6.3. Quadrilaterals. Unlike the triangle, the quadrilateral is not a rigid figure. The quadrilateral may assume many different shapes. Some quadrilaterals with special properties are referred to by particular names.

We will define a few of them.

Definitions: A quadrilateral is a trapezoid (symbol D) iff it has one and only one pair of parallel sides (Fig. 6.4). The parallel sides are the b~~es (Upper and lower) of the trapezoid. The nonparallel sides arc the legs. TiIe altitude of a trapezoid is a segment, as DE, which is perpendicular to one of the bases and whose endpoints are elements of the lines of which the bases are subsets. Often the word altitude is used to mean the distance between the bases. The median is the line segment connecting the midpoints of the non- parallel sides. An isosceles trapezoid is one the legs of which are congruent (Fig. 6.5). A pair of angles which share a base is called base angles.

A

Fig. 6.5.

Q

L

R

T

~ ^s

------

Definitions: A quadrilateral is a D

parallelogram (symbol D) iff the pairs of opposite sides are parallel. Any side of the parallelogram may be called the base, as AB of Fig. 6.6. The distance between two parallel lines is the perpen- dicular distance from any point on one

of the lines to the other line. An Fig. 6.6.

altitude of a parallelogram is the segment perpendicular to a side of the parallelogram and whose endpoints are in that side and the opposite side or the line of which the opposite side is a subset. DE of Fig. 6.6 is an altitude of 0 ABCD. Here, again, "altitude" is often referred to as the distance be- tween the two parallel sides. A parallelogram has two altitudes.

A rhombus is an equilateral parallelogram (Fig. 6.7).

A rectangle (symbol D) is a parallelogram that has a right angle (Fig. 6.7).

A rectangle is a square iff it has four congruent sides. Thus it is an equi- lateral rectangle (Fig. 6.7).

D C

0

^B

A

Base E

Q T

G

Hp

E

Rectangle

A

^R

s

Rhombus Fig. 6.7. Square

Theorem 6.1

6.4. All angles of a rectangle are right angles.

Given: ABCD is a rectangle with LA a right angle.

Prove: LB, LC, and LD are right angles.

(The proof of this theorem is left to the student. Hint: Use Theorem 5.15.)

D --C

A B

Theorem 6.1.

Exercises (A)

Indicate which of the following statements are always true and which are not always true.

.

n

- ---

c ^-

PARALLELOGRAMS

187

1. The sides of polygons are segments.

2. The opposite sides of a trapezoid are parallel.

3. Every quadrilateral has two diagonals.

4. Some trapezoids are equiangular.

5. All rectangles are equiangular.

6. The set of parallelograms are subsets of rectangles.

7. An octagon has eight angles.

8. An octagon has five diagonals.

9. The set of diagonals of a given triangle is a null set.

10. The diagonals of a polygon need not be coplanar.

11. Every polygon has at least three angles.

12. If a polygon does not have five sides it is not a pentagon.

13. A rhombus is a regular polygon.

]4. Each exterior angle of a polygon is supplementary to its adjacent angle of the polygon.

15. Only five exterior angles can be formed from a given pentagon.

]6. A square is a rectangle.

]7. A square is a rhombus.

]8. A square is a parallelogram.

]9. A rectangle is a square.

20. A rectangle is a rhombus.

2]. A rectangle is a parallelogram.

22. A quadrilateral is a polygon.

23. A quadrilateral is a trapezoid.

24. A quadrilateral is a rectangle.

25. A polygon is a quadrilateral.

Exercises (B)

]. Draw a convex quadrilateral and a diagonal from one vertex. Determine the sum of the measures of the four angles of the quadrilateral.

2. Draw a convex pentagon and as many diagonals as possible from one of its vertices. (a) How many triangles are formed? (b) What will be the Sum of the Pleasures of the angles of the pentagon?

3. Repeat problem 2 for a hexagon.

4. Repeat problem 2 for an octagon.

5. Using problems 2-5 as a guide, what would be the sum of the measures

.

of the angles of a polygon of 102 sides?

6. What is the measure of each angle of a regular pentagon?

7. What is the measure of each angle of a regular hexagon?

8. What is the measure of each exterior angle of a regular octagon?

9. What is the measure of each angle of a regular decagon?

All sides Opposssite Diagonals bisect Opposite Diagonals are are sides are each the .1 of

.1 are

Rclationshi ps

- - ^II other polygon - - ^.1

Parallelogram Rectangle Rhombus Square Trapezoid Isosceles

trapezoid 188 FUNDAMENTALS OF COLLEGE GEOMETRY

10. Using the set of polygons as the Universal set, draw a Venn diagram' relating polygons, rhombuses, quadrilaterals, and parallelograms.

11. Using the set of quadrilaterals as the Universal set, draw a Venn diagram relating quadrilaterals, squares, parallelograms, rhombuses, and trape- zoids.

12. Using the set of parallelograms as the Universal set, draw a Venn diagram relating parallelograms, squares, rectangles, and rhombuses.

Theorem 6.2

6.5. The opposite sides and the opposite angles of a parallelogram are congruent.

D

s

"",,"'"

//

y,

/-

"",-""

-_/ /-

X

-- /-

-

^r

Given: DAB CD.

Conclusion: AB ₌₌

DC; AD

⁼⁼

BC;

LA

==

LC; LB

⁼⁼

LD.

A

Theorem 6.2.

Proof

ST ATEMENTS REASONS

1. ABCD is a D.

2. Draw the diagonalAC.

3. AB

II

DC; AD

II

Be.

4. Lx

== Ly; Lr ₌₌

Ls.

5. AC

==

AC.

6. LABC

₌₌

LCDA.

7. AB

==

DC;AD

⁼⁼

BC.

8. LB

== LD.

9. LA ₌₌ LC.

6.6. Corollary:

triangles.

6.7. Corollary:

mentary.

6.8. Corollary: Segments of a pair of parallel lines cut off by a second pair of parallel lines are congruent.

6.9. Corollary: Two parallel lines are everywhere equidistant.

6.10. Corollary: The diagonals of a rectangle are congruent.

Either diagonal divides a parallelogram into two congruent Any two adjacent angles of a parallelogram are supple-j

POL YGONS - PARALLELOGRAMS 189 Theorem 6.3

6.11. The diagonals of a parallelogram bisect each other.

Given: DABCD with diagonals intersecting atE.

Conclusion: AC and BD bisect each other.

D C

~ ~

A B

Proof _{Theorem 6.3.}

STATEMENTS REASONS

I. ABCD is aD.

2. AB II

De.

3. Lz

== Ly, Lr

== Ls.

4. AB ₌₌ DC.

5. LABE

== LCDE.

6. AE

==

EC, and BE

==

DE.

1. Given.

2. Definition of a D.

3. Theorem 5.13.

4. Theorem 6.2.

5. A.S.A.

6. Corresponding sides of

₌₌

&, are

7. AC and BD bisect each other.

7. Definition of bisector.

6.12. Corollary:

other.

The diagonals of a rhombus are perpendicular to each Exercises (A)

Copy the chart below. Then put check marks (x) whenever the polygon

has lile il}dicat~(lrela~911~Jl_U:> n n n_n

--

xercises (B) 190

1. Given: ABCD is a 0;

DR 1.. AC; BT 1.. AC.

Prove: DR

==

BT.

2. Given: QRST is a 0;

RM

⁼⁼

NT.

Prove: QM

==

SN.

3. Given: ABCD is a 0;

DE 1.. AB; CF 1.. AB produced.

Prove: DE

⁼⁼

CF.

4. Given: ABCD is an isosceles trapezoid with AD

⁼⁼

Be.

Prove: LA

==

LB.

(Hint: Draw CE

II DA. )

5. Given: AB

==

CD;

AD

==

BC.

Prove: ABCD is aD.

D C

A~

EX.i.

~ T

~

R

Ex. 2.

ALl _E

C

B ZL- ^F

Ex.3.

D

Ai

C

E DB

Ex.4.

D

A

...-,/'

----

-

--- ...-..-

.-

...- ./ ..-

-

..-

----

Ex.5.

C

6. Given: RS == QT;

RS

_II

QT.

Prove: QRSTis aD.

7. Given: ABCD is aD;

DE bisects LD;

BF bisects LB.

Prove: DE

II

BF.

8. Prove that the diagonal QS of rhombus QRST bisects LQ and LS.

POLYGONS - PARALLELOGRAMS 191

Q T

R

-

---...- ---...-

.-

---

- -

.-- ...- .--

I

S Ex. 6.

D F C

~

^{A E B}

Ex. 7.

T

Qn

s

Ex. 8.

9. Prove that if the base angles of a trapezoid are congruent, the trapezoid is isosceles.

]0. Prove that if the diagonals of a parallelogram are perpendicular to each other, the parallelogram is a rhombus.

] ]. Prove that if the diagonals of a parallelogram are congruent, it is a rectangle.

]2. Prove that the bisectors of two consecutive angles of a parallelogram are perpendicular to each other.

Theorem 6.4

6.13. If the opposite sides of a quad-

~ilateral are congruent, the quadrilateral IS a parallelogram.

Given: Quadrilateral ABCD with AB

₌₌

CD; AD

_==BC.

Prove: ABCD is aD.

D C

A

y---

~~~~~~~~~~

s ,

r....--- x

B Theorem 6.4.

--

192

FUNDAMENTALS OF COLLEGE GEOMETRY Proof

STATEMENTS REASONS

1. AB ₌₌

CD; AD

==

BC.

2. Draw diagonal Ae.

3. AC

==

AC.

4. LABC

⁼⁼

LCDA.

5. Lx

⁼⁼

^{Ly; Lr}

==

Ls.

1. Given.

2. Postulate 2.

3. Reflexive property of congruence.

4. S.S.S.

5. Corresponding parts of

== &.

are 6. AB

II

CD; AD

II

BC.

7. :. ABCD is a D.

6. Theorem 5.11.

7. Definition of D.

Theorem 6.5

6.14. If two sides of a quadrilateral are congruent and parallel, the quadrilateral is a parallelogram.

!

""-/

//// J

//- / /

x

. I

Given: Quadrilateral ABCD with I

AB

==

CD; AB

^II

CD.

Conclusion: ABCD is a D.

A

Theorem 6.5.

STATEMENTS REASONS

The proof is left to the student.

Theorem 6.6

6.15. If the diagonals of a quadrilateral bisect each 0 C

-

ther, the quadrilateral is a parallelogram.

[;jjQ

D

y

s!

Given: Quadrilateral ABCD with AC E

and BD bisecting each other at E. x

Conclusion: ABCD is a D. ArB _>'

Proof

Theorem 6.6.

----.--- ---

STATEMENTS

POL YGONS

-

PARALLELOGRAMS

193

REASONS

1. A C and BD bisect each other at E.

2. AE

==

CE;BE

==

DE.

3. Lx

==

Ly.

4. LA BE

==

LCDE.

5. AB

==

CD.

6. Lr ==

Ls.

7. AB

"

CD.

8. ABCD is a D.

1. Given.

2. Definition of bisector.

3. Vertical angles are congruent.

4. S.A.S.

5. Corresponding parts of

_{== &.}

are congruent.

6. Same as 4.

7. Theorem 5.11.

8. Theorem 6.5.

Theorem 6.7

6.16. If three or more parallel lines cut off congruent segments on one trans-

versal, they cut off congruent segments on every transversal.

Given: Parallel lines t, m, and n cut by transversals rand s;AB

₌₌

BC.

Conclusion: DE

_==EF.

Prn°f"

STATEMENTS

1. Through D and E draw DG

II

r

and EH

II

r.

2. DC

_II

EH.

3. AD IllfE

" CF.

4. :ADGB andBEHC are m.

5. AB

₌₌

DG andBC

==

EH.

6. AB

==

Be.

7. DG

==

EH.

8. La

₌₌ L{3 and L"y ==

LB.

9. LDCE

== LEHF.

10. LDGE ₌₌

LEHF.

II. DE

== EF.

--

m

n

T.

REASONS

I. Postulate 18; Theorem 5.7.

2. Theorem 5.8.

3. Given.

4. Definition of D.

5. Theorem 6.2.

6. Given.

7. Theorem 3.5 and transitive pro- perty of'congruence.

8. Theorem 5.14.

9. ~ 5.28.

10. A.S.A.

11. § 4.28.

Exercises

1. Given: ABCD is aD;

M is midpoint of AD;

N is midpoint of Be.

Prove: MBND is a D.

2. Given: QR

II

ST;

Lx

== Ly.

Prove: QRST is a D.

3. Given: ABCD is aD;

AM

==

CN.

Prove: MBND is aD.

4. Given: QRSTisaD;

QL bisects LTQR;

SM bisects LRST.

QLSMisaD.

Prove:

5. Given: 0 ABCD with diagonals in tersecting at E.

Prove: E bisects FG.

f::::IC

A B

Ex. I.

S T

\L\

^{R Q}

Ex. 2.

D

c

A&Jl

Ex. 3.

T~

S R

Ex.4.

~C

A G B

Ex. 5.

6. Given: LMNP is aD.

PR 1- LN;

MS1-LN.

Prove: RMSP is aD.

7. Given: ,6.ABC with D midpoint of AC; E midpoint of BC; DE

_==EF.

Prove: ABFD is aD.

8. In Ex. 7, prove rnDE = tmAB.

POL YGONS - PARALLELOGRAMS 19.f1

P N

[S<;J

L M

Ex. 6.

C

D~E F

AI ~I _B

Ex.\.7,8.

9. Prove that two parallelograms are congruent if two sides and the included angle of one are congruent respectively to two sides and the included angle of the other.

10. Given: 0 QRSTwith AQ

==

SC;

RB

==

DT.

Prove: ABCD is aD.

11. Given: Trapezoid ABCD with AB

_II

DC;

AD

₌₌

De.

Prove: AC bisects LA.

R ~' _B S

Ex.10.

AMB

Ex.ll.

FUNDAMENTALS OF COLLEGE GEOMETRY

196

^POLYGONS

-

PARALLELOGRAMS

197

12. Prove that line segments drawn from A and B of LABC to the op- posite sides cannot bisect each other.

(Hint: Use indirect method by assuming AS and RB bisect each other; then ABSR is aD, ete.)

c

~

A B

L

^{G //}

^H/

^//

^K

B L-_-

A ^/

/ / / /

EL _D

Ex. 12.

13. Prove that a quadrilateral is a rhombus if the diagonals bisect each other and are perpendicular to each other.

14. Prove that if from the point where the bisector of an angle of a triangle meets the opposite side parallels to the other sides are drawn a rhombus is formed.

6.17. Direction of rays. Two rays have the same direction if and only if either they are parallel and are contained in the same closed half-plane determined by the line through their endpoints or if one ray is a subset of the other (Fig. : 6.8).

Theorem 6.8.

Given: LABC and LDEF with

'l3t

II

EP

and with same direction;