146. If two parallel lines be crossed by a transversal, the bisectors of the interior angles form a rectangle.

Sug. Consult 88, III., 92, 93, and 64.

147. The lines which bisect the angles of a rhomboid form a rectangle.

148. If two parallelograms have two sides and their included angle of one equal respectively to two sides and their included angle of the other, the two parallelograms are equal in every respect.

149. If a line be drawn parallel to one side of a triangle, and bisecting another side, this line will bisect the third side of the triangle, and be equal to one-half the side parallel to it. 150. Converse of 149.

151. The median of a trapezoid is parallel to its bases and equal to one-half their sum.

Sug. Through one extremity of the median draw a line parallel to one leg, and extend the shorter base to meet it.

ADVANCE THEOREMS.

152. If from any point in the base of an isosceles triangle lines are drawn parallel to the equal sides, the perimeter of the parallelogram thus formed will be equal to the sum of the two equal sides of the triangle.

153. If the legs of a trapezoid are equal, the angles which they make with either base are equal.

154. Converse of 153.

155. If the angles at one base of a trapezoid are equal, the angles at the other base are also equal.

156. The line which is parallel to the bases of a trapezoid and bisects one leg is a median.

157. The line joining the vertex of the right angle to the middle point of the hypothenuse in a right triangle is equal to one-half the hypothenuse.

158. The lines which join the middle points of the sides of a triangle divide the triangle into four equal triangles.

159. The three medians of a triangle meet in one point.

Sug. From the point of intersection of the two medians draw a line to the third vertex. From the middle point of this line draw another to the point midway between one of the other vertices and the point of intersection of the two medians. By extending the first line and connecting certain points a parallelogram may be formed.

160. The lines which join the middle points of the sides of any quadrilateral form a parallelogram.

Sug. Draw the diagonals, then consult 150.

161. The lines which join the middle points of the sides of a rhombus form a rectangle.

162. The lines which join the middle points of the sides of a square form a square.

163. The lines which join the middle points of the sides. of a rectangle form a rhombus.

164. The lines which join the middle points of the sides of an isosceles trapezoid form a rhombus.

165. The median and diagonals of a trapezoid intersect at the same point.

Sug. Consult 151 and 149.

166. The diagonals of an isosceles trapezoid are equal.

167. Converse of 166.

168. If a trapezoid be isosceles, the opposite angles are supplementary.

169. The line which joins the middle points of the diagonals of a trapezoid equals one-half the difference of the bases.

170. The two perpendiculars from the extremities of the base to the equal sides of an isosceles triangle are equal.

171. The medians drawn to the equal sides of an isosceles triangle are equal.

172. The bisectors of the angles at the base of an isosceles triangle are equal.

173. The two perpendiculars from the middle point of the base of an isosceles triangle to the equal sides are equal.

174. State and prove the converse of each of 170, 171, 172, 173.

175. If one of the equal sides of an isosceles triangle be extended at the vertex, making the extension equal to the side, the line joining the end of the extension with the nearer extremity of the base is perpendicular to the base.

176. If one angle of an isosceles triangle be 60°, the triangle is equilateral.

177. If from the middle points of two opposite sides of a parallelogram lines be drawn to the vertices of the angles opposite, these lines will trisect the diagonal that joins the other two vertices.

178. If the two base angles of a triangle are bisected, and through the point of intersection of these bisectors a line be drawn parallel to the base and terminating in the sides, this line is equal to the sum of the two segments of the sides between this parallel and the base.

179. The bisectors of the vertical angle of a triangle and the angles formed by extending the sides below the base meet in a point which is equally distant from the base and the extensions of the sides.

180. If one of the acute angles of a right triangle is double the other, the hypothenuse is double the shorter leg.

181. If from any two points selected at random in the base of an isosceles triangle perpendiculars be drawn to the equal sides, the sum of the perpendiculars from one point equals the sum of the perpendiculars from the other point.

182. If from any point selected at random in an equilateral triangle perpendiculars be drawn to the sides, the sum of these perpendiculars is constant, and equal to the altitude of the triangle.

CIRCLES.

183. A circle is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within called the centre.

The bounding line is called the circumference.

Any portion of the circumference is called an arc.

A radius (plural radii) is any line from centre to circumference.

A diameter is any line passing through the centre and terminating both ways in the circumference.

How do the radius and diameter of the same circle compare in magnitude?

How do the radii of a circle compare in magnitude? The diameters?

A semicircumference is one-half the circumference.

A sector is that part of a circle included between an arc and the radii drawn to its extremities.

A quadrant is a sector which is one-fourth the circle.

A quadrant arc is one-fourth the circumference.

A chord is any straight line whose extremities are in the circumference.

A segment is that portion of the circle included between an arc and the chord which joins its extremities.

Every chord, therefore, must divide the circumference into two arcs and the circle into two segments.

If the arcs are unequal, they are designated as major and minor arcs, and the segments as major and minor segments. The chord is said to subtend the arc, and the arc is said to be subtended by the chord.

Whenever a chord and its subtended arc are mentioned, the minor arc is meant unless it is otherwise specified.

If two circles have the same centre, they are said to be concentric.

A central angle is an angle formed by two radii.

An inscribed angle is an angle formed by two chords, with its vertex in the circumference.

When would an angle be said to be inscribed in a segment? A tangent is a straight line that touches the circumference of a circle, but on being extended does not intersect it; i.e. the tangent and the circumference have one point, and only one, in common. This point is called the point of contact, or point of tangency.

Two circumferences are tangent to each other when they have one point in common but do not intersect; i.e. when they touch each other.

If one of two tangent circumferences lies within the other, they are said to be tangent internally; if it lies without, they are said to be tangent externally.

A secant is a straight line that intersects a circumference in two points lying partly within and partly without the circle; e.g. a chord extended in either direction becomes a secant.

The term circle is also sometimes used to designate a circumference.

Construct diagrams illustrating the above magnitudes and their relations.

184. A diameter is greater than any other chord.

Sug. Draw the diameter AD, and let HB be any other chord. Join CH and CB. Consult 97.