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.

A

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?

Proposition XXXVII

150. 1. Draw a quadrilateral having two of its sides equal and par allel to each other. What kind of a quadrilateral is it?

2. Draw two parallel lines and two parallel transversals. How do the segments of the transversals between the parallel lines compare in length? Theorem. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

Data: Any quadrilateral, as ABCD,

in which two of the sides, as AB and DC, are equal and parallel.

To prove ABCD a parallelogram.

Proof. Draw AC.

In the AABC and ADC,

D

A

B

151. Cor. Parallel lines intercepted between parallel lines are equal, and parallel lines are everywhere equally distant.

Ex. 95. If the sides of a parallelogram are bisected and these middle points joined in succession, what figure is formed by the connecting lines? Ex. 96. If the four interior angles formed by a transversal crossing two parallel lines are bisected and the bisectors produced until they meet, what figure will be formed?

Ex. 97. If a line is drawn through the vertices of two isosceles triangles on the same base, how does it divide the base?

Ex. 98. If two equal straight lines are drawn from a point to a line, how do the angles formed with the given line compare?

Ex. 99. If lines are drawn from the vertex of an isosceles triangle to points in the base equally distant from its extremities, how do they compare in length?

Proposition XXXVIII

152. 1. Draw a parallelogram and either diagonal. How do the triangles thus formed compare in size?

2. How do the opposite sides of the parallelogram compare in length? Theorem. The diagonal of a parallelogram divides the figure into two equal triangles.

Data: Any parallelogram, as ABCD, and one of its diagonals, as AC.

To prove triangles ABC and ADC equal.

Proof. To be given by the student.

D

B

153. Cor. The opposite sides of a parallelogram are equal.

Proposition XXXIX

154. Draw a parallelogram and its diagonals. How do the segments of each diagonal compare in length?

Theorem. The diagonals of a parailelogram bisect each other.

Data: Any parallelogram, as ABCD, and its diagonals, AC and BD, intersecting at E.

To prove AE = CE and BE = DE.

Proof. In the AABE and CDE,

D

E

B

Ex. 100. If the diagonals of a quadrilateral are equal and bisect each other, what kind of a figure is the quadrilateral?

Ex. 101. From a figure representing a parallelogram and its diagonals, select four pairs of equal triangles.