Book I.]

PROPOSITION XXXII.

53

Ex. 3. If the side BC of the triangle ABC be produced to D, and AE be drawn bisecting the angle BAC and meeting BC in E; shew that the angles ABD, ACD are together double of the angle AED.

Ex. 4. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet; shew that they will contain an angle equal to an exterior angle at the base of the triangle.

Ex. 5. If the straight line bisecting the external angle of a triangle be parallel to the base; prove that the triangle is isosceles.

The following Corollaries to Prop. 32 were first given in Simson's Edition of Euclid.

COR. 1. The sum of the interior angles of any rectilinear figure together with four right angles is equal to twice as many right angles as the figure has sides.

B

Let ABCDE be any rectilinear figure.

Take any pt. F within the figure, and from F draw the st. lines FA, FB, FC, FD, FE to the angular pts. of the figure Then there are formed as many s as the figure has sides.

The threes in each of these As together=two rt. 4 s.

..all the s in these As together twice as many right 4s as there are As, that is, twice as many rights as the figure has sides.

Now angles of all the Ass at A, B, C, D, E and s at F,

that is,

and...

= 48 of the figure and s at F,

= 8 of the figure and four rt. 4 s. I. 15. Cor. 2. ..s of the figure and four rt. s=twice as many rt. 48 as the figure has sides.

COR. 2. The exterior angles of any convex rectilinear figure, made by producing each of its sides in succession, are together equal to four right angles.

Every interior angle, as ABC, and its adjacent exterior angle, as ABD, together are two rt. 4 s.

D B

..all the intr. 4s together with all the extr. <s
=twice as many rt. s as the figure has sides.
But all the intr. 4s together with four rt. LS
=twice as many rt. 4s as the figure has sides.
all the intr. 4s together with all the extr. 4 s
=all the intr. 4s together with four rt. 4 s.

.. all the extr. 4 s=four rt. 4 s.

NOTE. The latter of these corollaries refers only to convex figures, that is, figures in which every interior angle is less than two right angles. When a figure contains an angle greater

than two right angles, as the angle marked by the dotted line in the diagram, this is called a reflex angle. See p. 149.

Ex. 1. The exterior angles of a quadrilateral made by producing the sides successively are together equal to the interior gles.

Ex. 2. Prove that the interior angles of a hexagon are equal to eight right angles.

Ex. 3. Shew that the angle of an equiangular pentagon is g of a right angle.

Ex. 4. How many sides has the rectilinear figure, the sum of whose interior angles is double that of its exterior angles? Ex. 5. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles?

PROPOSITION XXXIII. THEOREM.

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

Let the equal and || st. lines AB, CD be joined towards the same parts by the st. lines AC, BD.

.::AB=CD, and BC is common, and ▲ ABC= ▲ DCB,

..AC=BD, and ▲ ACB= 2 DBC.

Then BC, meeting AC and BD,

I. 29.

I. 4.

makes the alternate

s ACB, DBC equal,

.. AC is to BD.

Q. E. D.

Macalunama Ecorrises on Sections I. and II.

1 Evo exterior angles of a triangle be bisected by straight Ines which meet in 0; prove that the perpendiculars from 9 m the sides, or the sides produced, of the triangle are

3. The sectors of the three angles of a triangle meet in one point.

4. The perpendiculas to the three sides of a triangle drawn fum the mille points of the sides meet in one point.

5. The ance between the bisector of the angle BAC of the miangle 130 and the perpendicular from A on BC, is equal ahut the fference between the angles at B and C.

6. E me sirin The AD bisect the angle at A of the mangle 430, and BDF be drawn perpendicular to AD, and meeting 12 x 40 produced, in E; shew that BD is equal

7. Divide a right-angled triangle into two isosceles triangies.

8. 43. Care we given straigha Enes. Through a point 3 berveen them faw a straight line GEH, such that the inpercepted portion Eshal be bisected in E.

A The vertical sarle 0 of a triangle OPQ is a right, acute, ze jòcuse savie, scuring as OR, the line bisecting PQ, is INDT I INs than the half of PQ.

SECTION III.

On the Equality of Rectilinear Figures in respect of Area.

THE amount of space enclosed by a Figure is called the Area of that figure.

Euclid calls two figures equal when they enclose the same amount of space. They may be dissimilar in shape, but if the areas contained within the boundaries of the figures be the same, then he calls the figures equal. He regards a triangle, for example, as a figure having sides and angles and area, and he proves in this section that two triangles may have equality of area, though the sides and angles of each may be unequal.

Coincidence of their boundaries is a test of the equality of all geometrical magnitudes, as we explained in Note 1, page 14.

In the case of lines and angles it is the only test: in the case of figures it is a test, but not the only test; as we shall shew in this Section.

The sign, standing between the symbols denoting two figures, must be read is equal in area to.

Before we proceed to prove the Propositions included in this Section, we must complete the list of Definitions required in Book I., continuing the numbers prefixed to the definitions in page 6.