PLANE GEOMETRY.-BOOK I.

5

two unequal lines from that point to the given line. Whi he line at the greater distance from the foot of the perpe

many equal straight lines can be drawn from a point to e?

m. If from a point in a perpendicular to a give
line, oblique lines are drawn to the given line,
oblique lines which meet the given line at equa
from the foot of the perpendicular are equal.
›blique lines which meet the given line at unequ
s from the foot of the perpendicular the mo
s the greater.

Any straight line, as AB;
endicular to AB, as PD;
point in PD, as C, from
lique lines, as CE, CF,
are drawn meeting AB so
DF, and DG is greater A

D

F

oduce CD to H, making DH = CD; draw EH and GH.

x. 2, CE + EH = 2 CE, and CG + GH = 2 CG.

131,

CE + EHCG + GH;

2 CE <2 CG, or CE < CG;

fore, etc.

CG > CE.

Q.E.

Cor. Only two equal straight lines can be drawn from a straight line; and of two unequal lines the greater cuts

Proposition XXXIV

134. Bisect any angle; from any point in the bisector draw lines per pendicular to the sides of the angle. How do the perpendiculars compare in length? How do the distances of the point from the sides or the angle compare?

Theorem. Every point in the bisector of an angle is equidistant from the sides of the angle.

Data: Any angle, as ABC, and any point in its bisector BD, as F.

To prove F equidistant from AB and CB.

Proof. Draw the perpendiculars FE and FG representing the distances of the point F from AB and CB respectively.

Ex. 82. The perpendicular let fall from the vertex to the base of a triangle divides the vertical angle into two angles. How does the difference of these angles compare with the difference of the base angles of the triangle ?

Ex. 83. ABC is a triangle. Angle A = 60°, angle B = 40°. The bisector of angle A is produced until it cuts the side BC. How many degrees are there in each angle thus formed?

Ex. 84. A perpendicular is let fall from one end of the base of an isosceles triangle upon the opposite side. How does the angle formed by the perpendicular and the base compare with the vertical angle ?

Ex. 85. If an angle of a triangle is equal to half the sum of the other two, what is the value of that angle?

Ex. 86. How does the sum of the lines from a point within a triangle to the vertices of the triangle compare with the sum of the sides of the triangle? With half that sum?

Proposition XXXV

135. Within an angle select any number of points that are each equidistant from its sides. Will the lines joining these points form a straight line? How will it divide the angle?

Theorem. Every point within an angle and equidistant from its sides lies in the bisector of the angle. (Converse of Prop. XXXIV.)

Data: Any angle, as ABC, and any point within the angle equidistant from AB and CB,

as F.

To prove F is in the bisector of the angle

ABC.

E

Proof. Through the point F draw BD; also draw the perpendiculars FE and FG representing the distances of the point F from AB and CB respectively.

Ex. 87. ABC is an isosceles triangle having a vertical angle of 30°. From each extremity of the base perpendiculars are drawn to the opposite sides. What angles are formed at the intersection of these perpendiculars? Ex. 88. The exterior angle at the vertex of an isosceles triangle is 110°. How many degrees are there in each angle of the triangle ?

Ex. 89. The exterior angle at the base of an isosceles triangle is 110°. How many degrees are there in each angle of the triangle?

Ex. 90. The angle C at the vertex of the isosceles triangle ABC is one fourth of the exterior angle at C. How many degrees are there in angle A? In the exterior angle at B?

Ex. 91. How does the angle formed by the bisectors of the base angles of an isosceles triangle compare with an exterior angle at the base ?

QUADRILATERALS

136. A portion of a plane bounded by four straight lines is called a Quadrilateral.

137. A quadrilateral which has no two sides. parallel is called a Trapezium.

138. A quadrilateral which has only two sides parallel is called a Trapezoid.

The parallel sides of a trapezoid are called its bases.

139. A trapezoid whose non-parallel sides are equal is called an Isosceles Trapezoid.

140. A quadrilateral whose opposite sides are parallel is called a Parallelogram.

141. A parallelogram whose angles are right angles is called a Rectangle.

142. A parallelogram whose angles are oblique angles is called a Rhomboid.

143. An equilateral rectangle is called a Square.

144. An equilateral rhomboid is called a Rhombus.

Diagonal

145. The straight lines which join the vertices of the opposite angles of a quadrilateral are called Diagonals.

146. The side upon which a figure is assumed to stand is called the Base.

The side upon which a trapezoid or a parallelogram is assumed to stand is called its lower base, and the side opposite is called its upper base.

147. The perpendicular distance between the bases of a trapezoid or of a parallelogram is called its Altitude.

Proposition XXXVI

148. 1. Draw a quadrilateral whose opposite sides are equal. What kind of a quadrilateral is it?

2. How do the opposite angles of a parallelogram compare in size?

Theorem. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

149. Cor. The opposite angles of a parallelogram are equal.

Ex. 92. If lines are drawn joining in succession the middle points of the sides of a square, what figure will be formed?

Ex. 93. To how many right angles is the sum of the angles of a parallelogram equal? To what is the sum of any two angles of a parallelogram, which are not opposite, equal?

Ex. 94. If medians are drawn from two vertices of a triangle and each is produced its own length, what kind of a line will join the extremities of the produced medians and the other vertex of the triangle?