117. If a line bisect an angle, any point selected at random in this bisector will be equally distant from the sides of the angle.

Sug. Consult 72, Remark (a), and 106.

118. If a line bisect an angle, any point selected at random outside this bisector is unequally distant from the sides of the angle.

Sug. If K be the point, then what is the relation between HS and HB? Why?

What is the relation between BK and KH + HS? Why? What is the relation of KS to KH+HS? Why?

What is its relation, then, to BK?

What is the relation between KS and KP?

What, then, must be the relation between KP and BK?

119. If a point be equally distant from the sides of an angle, the line joining it with the vertex will bisect the angle.

120. If the angles at the base of an isosceles triangle be bisected, the bisectors will form, with the base, an isosceles triangle.

121. If the angles at the base of an isosceles triangle be double the vertical angle, a line bisecting either of the former will divide the triangle into two isosceles triangles.

122. If two angles of a triangle be unequal, the side opposite the greater of these two angles is longer than the side opposite the lesser.

Post. Let ABC be any triangle with <B>ZA.

We are to prove that the side AC, opposite the greater angle B, is longer than the side CB, opposite the lesser angle A.

Cons. Let BD be drawn so

A

D

C

B

as to cut off a portion of the larger angle B, making it equal to angle 4, so that they will both be in the same triangle as DAB.

What relation between AD and DB? Why?

What relation between BC and CD+DB? Why?

(The pupil should finish it without difficulty.)

123. Converse of 122.

Sug. Three possible relations between the two angles. Prove the impossibility of two of them; or, a method similar to that in 122 may be employed.

123 (a). If two triangles have the two sides of one equal respectively to two sides of the other, but the included angles unequal, then the third side of that triangle having the greater included angle is longer than the third side of the other.

Sug. Apply the two triangles so that two of the equal sides

shall coincide, as is represented in the above diagram.

Will the other pair of equal sides coincide? Why?
Join the other two vertices.

What kind of a triangle is HKB'?

What relation, then, between angles HKB' and HB'K? What must be the relation, therefore, between the angles DKB' and C'B'K?

Consult Theorem 122.

Or, instead of joining B' and K, the angle KA'B' may be bisected, and the point where this bisector cuts DK joined with B'. Then consult 101 and 97.

124. Converse of 123.

Sug. There are only three possible relations between those angles. Prove the impossibility of two of them.

ADVANCE THEOREMS.

125. The bisectors of the three angles of a triangle meet in one point.

Sug. From the point of intersection of two of the bisectors draw a line to the other vertex, and also perpendiculars to the three sides. Prove the former a bisector by means of equality of triangles.

126. The perpendiculars which bisect the three sides of a triangle meet in a point.

Sug. From the point of intersection of two of the perpendiculars draw a line to the middle point of the other side. Prove this line perpendicular, by consulting 74 and 103.

127. The perpendiculars from the three vertices of a triangle to the opposite sides meet in one point.

Sug. Through each vertex draw a line parallel to the opposite side. Then consult 102 and 127.

128. If two angles of an equilateral triangle be bisected, and lines be drawn through the point of intersection parallel to the sides, the sides will be trisected.

QUADRILATERALS.

129. If two lines in the same plane be crossed by two transversals, a figure of four sides may be formed, which is called a quadrilateral. Hence a quadrilateral may be defined as a plane figure bounded by four straight lines. A quadrilateral is also called a tetragon.

If each of the two pairs of lines is parallel, the quadrilateral thus formed is called a parallelogram.

Define, then, a parallelogram.

If a parallelogram have all its sides equal, it is called a rhombus.

If its angles are right angles, it is called a rectangle.

If it is both a rhombus and a rectangle, it is called a square.

Give all the names applicable to a square.

If a quadrilateral have only two of its sides parallel, it is called a trapezoid.

If no two of its sides are parallel, it is called a trapezium. A parallelogram whose angles are oblique and adjacent sides unequal is sometimes called a rhomboid.

A rectangle whose adjacent sides are unequal is sometimes called an oblong.

The line which joins two opposite vertices of a quadrilateral is called its diagonal.

The side upon which a parallelogram is conceived to rest, and the side opposite the latter, are termed, respectively, the lower and upper bases.

The parallel sides of a trapezoid are always considered as its bases, the other two sides its legs, while the line bisecting the legs is called the median.

The altitude of either a parallelogram or trapezoid is the perpendicular distance between its bases.

The pupil should construct figures to represent all the abovementioned quantities.

The sum of the four sides of a quadrilateral is called its perimeter.

If the legs of a trapezoid are equal, it is called an isosceles trapezoid.

130. Either diagonal divides the parallelogram into two equal triangles.

Sug. Consult 87 and 102.

Query. Is the converse of this theorem true?

131. The sum of the four angles of a quadrilateral equals four right angles.

132. The opposite sides of a parallelogram are equal.

132 (a). Two parallel lines are everywhere equally distant. 133. Converse of 132.

Sug. Draw one diagonal, then consult 45 and 34.

134. The opposite angles of a parallelogram are equal. 135. Converse of 134.

Sug. Consult 131 and 91.

136. If two sides of a quadrilateral are equal and parallel, the quadrilateral is a parallelogram.

137. The two diagonals of a parallelogram bisect each other. 138. Converse of 137.

139. If one angle of a parallelogram be a right angle, the other three angles are right angles, and the parallelogram is therefore a rectangle.

140. The diagonals of a rectangle are equal.

141. Converse of 140.

142. The diagonals of a rhombus are perpendicular to each other.

143. Converse of 142.

144. The diagonals of a rhombus bisect the angles.

145. Converse of 144.