122.

Proposition 33.

Two straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel.

Let AB, CD represent two finite straight lines which are of equal lengths and parallel to each other, and let AC, BD join their extremities towards the same parts;

it is required to prove

that AC, BD are (i) of equal lengths and (ii) parallel to each other.

Join CB.

Because AB, CD are parallel

and the angles ABC, DCB are alternate angles,

therefore they are equal.

Now consider the triangles ABC, DCB;

[Prop. 29.]

because two sides AB, BC and the angle ABC between them of the triangle ABC are equal respectively

to the two sides DC, CB and the angle DCB between them of the triangle DCB;

therefore the triangles are equal in all respects; [Prop. 4.] so that the side AC is equal to the corresponding side DB, [I.] and the angle ACB is equal to the corresponding angle DBC.

Now the angles ACB, DBC are alternate angles with reference to the two lines AC, BD; and because they are equal, therefore AC is parallel to BD.

Wherefore, two straight lines which join, etc. Q.E.D.

[II.]

123. NOTE. The expression towards the same parts, should be noticed; it seems to express clearly enough what is meant.

The lines AD, BC which join the extremities of AB, CD towards opposite parts are called diagonals of the quadrilateral ABCD.

EXAMPLES XXXVIII.

1. Prove that the lines which join the extremities of two equal and parallel straight lines towards opposite parts bisect each other.

2. If the straight lines which join the extremities of two equal straight lines towards opposite parts are equal, then the lines which join their extremities towards the same parts make equal angles with these two pairs of equal straight lines.

3. Prove that in Question 2 the third pair of lines are parallel.

4. If the lines which join the extremities of two equal straight lines towards the same parts are equal, they are also parallel.

124. DEF. A four-sided figure whose opposite sides are parallel is called a parallelogram.

Thus, in the figure on page 121, if AD, BC represent two parallel straight lines, and also AB, DC then ABCD represents a parallelogram.

125. DEF. The line joining two opposite angular points of a rectilineal figure having an even number of sides is called a diagonal.

Thus, in the figure on page 121, BD is a diagonal of ABCD; AC is also a diagonal.

DEF.

When a diagonal divides the figure into two equal parts it is called a diameter.

126. A parallelogram has many properties such as
The opposite sides of a parallelogram are equal.
The opposite angles of a parallelogram are equal.
Each diagonal bisects its area.

(Thus each diagonal is a diameter.)

Its diameters bisect each other.

127. A four-sided figure of no definite shape is called a quadrilateral.

Proposition 34.

128. The opposite sides and angles of a parallelogram are equal and a diagonal bisects its area.

Let ABCD represent a parallelogram,

it is required to prove

(i) that the side AB, is equal to CD,

(ii) that AD is equal to BC,

(iii) that the angle BCD is equal to DAB,

(iv) that the angle ABC is equal to CDA,

(v) that a diagonal BD bisects the area of the parallelogram.

B

Now, because AB, DC are parallel, therefore

the alternate angles ABD, BDC are equal; [Prop. 29.]
and because AD and BC are parallel, therefore
the alternate angles ADB, CBD are equal. [Prop. 29.]
Now consider the triangles ABD, CDB;

because the angles ABD, ADB and the side BD adjacent, in the triangle ABD, are equal respectively

to the angles CDB, DBC and the side BD adjacent,
in the triangle DCB,

therefore the triangles are equal in all respects; [Prop. 26, I.]
so that AB is equal to DC (i), AD is equal to BC (ii),
the angle BAD is equal to DCB (iii),

and the areas of the triangles are equal (v).
Also, because the angles ABD, CDB are equal

and the angles CBD, ADB are equal,
therefore the whole angles ABC, CDA are equal (iv).
Wherefore, the opposite sides and angles, etc.

Q.E.D.

129. DEF.

A parallelogram of which two adjacent

sides are equal is called a rhombus.

A

B

130. DEF. A parallelogram of which two adjacent sides are equal and one angle a right angle is called a square.

131. DEF. A parallelogram of which one angle is a right angle is a rectangle.

Example i. A quadrilateral whose opposite angles are equal is a parallelogram.

B

Let ABCD represent a quadrilateral such that the angles ABC, ADC are equal and the angles DAB, BCD are equal,

then ABCD is a parallelogram.

The four interior angles of a quadrilateral figure together make up four right angles. [Prop. 32, Cor. I.] Therefore the four angles at A, at B, at C, at D together make up four right angles.