base and the adjacent one as the altitude. The rectangle having two given segments as its base and altitude is called the rectangle on these segments.

Notation. The symbol □ stands for the word rectangle and for parallelogram. C

Rectangles and parallelograms are commonly indicated by naming a pair of their opposite vertices.

COMPARISON OF AREAS--RECTANGLES,
PARALLELOGRAMS, TRIANGLES.

SECTION I.

139°. Theorem.—1. Rectangles with equal bases and equal altitudes are equal.

2. Equal rectangles with equal bases have equal altitudes. 3. Equal rectangles with equal altitudes have equal bases.

I. In the s BD and FH, if

AD=EH,

AB=EF,

OBD=0FH.

and

then

BP

A

F

E

Proof.--Place E at A and EH along AD. Then, as ¿FEH=_BAD=7, EF will lie along AB. And because EH=AD and EF = AB, therefore H falls at D and F at B, and the two □s are congruent and therefore equal. q.e.d.

2. If BD=FH and AD=EH, then AB=EF.

Proof.—If EF is not equal to AB, let AB be > EF. Make AP EF and complete the

Then

but

D

G

PD.
☐PD=FH, by the first part,
OBD=FH,

H

(hyp.)

E

AB and EF cannot be unequal, or

AB=EF.

3. If BD=FH and AB=EF, then AD=EH.

Proof. Let AB and EF be taken as bases and AD and EH as altitudes (138°), and the theorem follows from the second part. q.e.d.

A

OPD=BD, which is not true,

Cor. In any rectangle we have the three parts, base, altitude, and area. If any two of these are given the third is given also.

140°. Theorem.—A parallelogram is equal to the rectangle

B

F

Con its base and altitude.

AC is a

and DF is the altitude.

Then AC on AD and DF.

Proof.-Complete the ADFE by drawing AE to CB

AAEB ADFC, ... AE=DF, AB=DC,
LEAB=LFDC;

and

.. ▲DFC may be transferred to the position AEB, ABCD becomes the AEFD,

AC=□ on AD and DF.

produced.

Then

q.e.d.

D

whereof AD is the base

and

q.e.d.

Cor. 1. Parallelograms with equal bases and equal altitudes are equal. For they are equal to the same rectangle.

Cor. 2. Equal parallelograms with equal bases have equal altitudes, and equal parallelograms with equal altitudes have equal bases.

Cor. 3. If equal parallelograms be upon the same side of the same base, their sides opposite the common base are in line.

141°. Theorem.—A triangle is equal to one-half the rectangle on its base and altitude.

ABC is a triangle of which AC is the base and BE the altitude.

B

D

Then ABC on AC and BE.

Proof.-Complete the ABDC, of which AB and AC are adjacent sides. A AABC=ADCB,

Then

...

▲ABC=OAD=□ on AC and BE. (140°) q.e.d.

Cor. 1. A triangle is equal to one-half the parallelogram having the same base and altitude.

Cor. 2. Triangles with equal bases and equal altitudes are equal. For they are equal to one-half of the same rectangle.

E

Cor. 3. A median of a triangle bisects the area. For the median bisects the base.

Cor. 4. Equal triangles with equal bases have equal altitudes, and equal triangles with equal altitudes have equal bases.

Cor. 5. If equal triangles be upon the same side of the same base, the line through their vertices is parallel to their common base.

142°. Theorem.—If two triangles are upon opposite sides of the same base

1. When the triangles are equal, the base bisects the segment joining their vertices;

2. When the base bisects the segment joining their vertices, the triangles are equal. (Converse of 1.)

B

ABC and ADC are two triangles upon opposite sides of the common base AC. 1. If AABC AADC,

then

BH=HD.

Proof. Let BE and DF be altitudes, Then .

A

2. If BH = HD, then ABC=^ADC.

E

▲ABC=ADC, .. BE=DF,
EBH=AFDH, and BH=HD.

HF

q.e.d.

and

Proof. Since BH=HD, :: ΔΑΒΗ=ΔΑΦΗ, ДСВН=ДСDH. ... adding, AABC AADC.

143. Def.-By the sum or difference of two closed figures is meant the sum or difference of the areas of the figures.

If a rectangle be equal to the sum of two other rectangles its area may be so superimposed upon the others as to cover both.

144°. Theorem.—If two rectangles have equal altitudes, their sum is equal to the rectangle on their common altitude and the sum of their bases.

B

D

F

X

==

Proof.-Let the □s X and Y, having equal altitudes, be so placed as to have E their altitudes in common at CD, and so may not overlap the other.

<BDC=LCDF=¬,

BDF is a line.

(38°, Cor. 2)

Similarly

ACE is a line.

But BD is || to AC, and BA is || to DC || to FE; therefore AF is the on the altitude AB and the sum of the bases AC and CE; and the

AF=¬AD÷□CF.

q.e.d.

A

that one Then

(141°, Cor. 3)
q.e.d.

C

Y

Cor. 1. If two triangles have equal altitudes, their sum is equal to the triangle having the same altitude and having a base equal to the sum of the bases of the two triangles.

Cor. 2. If two triangles have equal altitudes, their sum is equal to one-half the rectangle on their common altitude and the sum of their bases.

Cor. 3. If any number of triangles have equal altitudes, their sum is equal to one-half the rectangle on their common altitude and the sum of their bases.

In any of the above, "base" and "altitude" are interchangeable.

145°. Theorem.-Two lines parallel to the sides of a parallelogram and intersecting upon a diagonal divide the parallelogram into four parallelograms such that the two through which the diagonal does not pass are equal to one another.

In the ABCD, EF is || to AD and GH is to BA, and these intersect at O on the diagonal AC.

Then

7BO=OD.

and

but

and

..

... the

Proof.-ABC=▲ADC, and ▲AEO=^AHO,

AOGC=AOFC;

BO=ABC-AAEO-AOGC,
OD=AADC-AAHO-AOFC.

7B0=0D.

7BF=

GD.

=

Ex. Let ABCD be a trapezoid. (84°, Def.) In line with AD make DE=BC, and in line with BC make CF AD.

A

Cor. I.

Cor. 2. If B0=OD, O is on the diagonal AC. (Converse of the theorem.)

For if ( is not on the diagonal, let the diagonal cut EF in O'. Then BO'= O'D. (145°)

But

E

A

G

B

B

BO' is <BO, and

O'D is >

OD;

7BO is >□OD, which is contrary to the hypothesis; gonal cuts EF in O.

(141°, Cor. 1)

с

D

q.e.d.

Then BF AE and BFEA is a

But the trapezoid CE can be superimposed on the trapezoid DB, since the sides are respectively equal, and

<F=A, and LE = B, etc.

trapezoid BDBE,

or, a trapezoid is equal to one-half the rectangle on its altitude and the sum of its bases.

E