Ex. 107. If an arm of an isosceles triangle is produced by its own length through the vertex, and the end of the prolongation is joined to the nearest end of the base, the line joining is perpendicular to the base. Ex. 108. The bisectors of the base angles of an isosceles triangle are equal. PROPOSITION XV. THEOREM 103. If two angles of a triangle are equal, the sides opposite these angles are equal. HINT.Let AD be the bisector of angle BAD, and prove the equality of the triangles. 104. COR. An equiangular triangle is also equilateral. 105. REMARK. The above two propositions may also be used to prove the equality of lines and angles. Ex. 109. The bisectors of the base angles of an isosceles triangle form, if they meet, an isosceles triangle. Ex. 110. If at the ends of the base of an isosceles triangle perpendiculars are erected upon the arms, another isosceles triangle is formed. Ex. 111. A line parallel to the base of an isosceles triangle, and intersecting the two arms, cuts off another isosceles triangle. Ex. 112. If two exterior angles, formed by producing one side of a triangle at both ends, are equal, the other sides are equal. Ex. 113. If one angle of an isosceles triangle is 60°, the triangle is equilateral. Ex. 114. If the bisector of an exterior angle of a triangle is parallel to one side, the triangle is isosceles. 106. Two triangles are equal if three sides of the one are respectively equal to three sides of the other. (S. 3. 38. 3. 3.) B Hyp. In AABC and A'B'C', To prove AB = A'B', BC = B'C', AC = A'C'. ▲ ABC ▲ A'B'C'. Proof. Apply ▲ ABC to ▲ A'B'C' so that its greatest side BC coincides with B'C', and A and A' lie on opposite sides of B'C'. Ex. 117. If the opposite sides of a quadrilateral are equal, the opposite angles are equal. Ex. 118. The median to the base of an isosceles triangle is perpendicular to the base. Ex. 119. If a quadrilateral has two pairs of equal adjacent sides, the angles included by the unequal sides are equal. Ex. 120. The medians to the arms of an isosceles triangle are equal. Ex. 121. If the opposite sides of a quadrilateral are equal, the sides are parallel. Ex. 122. If in the sides of an equilateral ▲ ABC, the points D, E, and F be taken so Ex. 123. If the sides of an equilateral ▲ ABC are produced so that Ex. 124. If the opposite sides of a quadrilateral are equal, and a line be drawn through the midpoint of the diagonal and terminating in two sides, this line is bisected. Ex. 125. Two triangles are equal if two sides and the median to one of these sides are equal respectively to two sides and the homologous median of the other. Ex. 126. Two isosceles triangles are equal if one angle and the altitude upon one arm of one triangle are equal respectively to one angle and the altitude upon one arm of the other. * Ex. 127. If the opposite sides of a quadrilateral are equal, the diagonals bisect each other. Ex. 128. Two triangles are equal if the base, an adjacent angle, and the altitude upon the base of one triangle are equal respectively to the base, an adjacent angle, and the altitude of the other. B A C Ex. 129. In the annexed diagram if AB = AC, and AD bisects / BAC, then LDBE LDCE. PROPOSITION XVII. THEOREM 108. Two right triangles are equal if the hypotenuse and an arm of the one are respectively equal to the hypotenuse and an arm of the other. (hy. arm-hy. arm.) 44 Hyp. In the right AABC and A'B'C', To prove Β' AB= A'B'; AC = A'C'. ▲ ABC=▲ A'B'C'. Proof. Apply AA'B'C' to AABC, so that A'C' coincides with AC, and B' falls remote from B. BCB' is a straight line, (4 BCB = 2 rt. 4). AB = AB'. :. LB = LB', (base of an isos. ▲). (s. a. a. s. a. a.). .. ΔΑΒΟ = AA'B'C'. (Hyp.) Q.E.D. 109. DEF. A circumference is a curved line, all of whose points are equidistant from a point within called the center, as ABC, the center being D. A circle is the portion of a plane bounded by a circumference, and is usually read "the circle D" or "the O ABC." A B D A radius is any straight line drawn from the center to the circumference. An arc is any portion of a circumfer ence. Ex. 130. If two altitudes of a triangle are equal, the corresponding sides are equal, and the triangle is isosceles. Ex. 131. Two triangles are equal if two sides and the altitude upon one of them of one triangle are respectively equal to two sides and the homologous altitude of the other. Ex. 132. If the perpendiculars from the midpoint of one side of a triangle upon the two other sides are equal, the triangle is isosceles. Ex. 133. Two triangles are equal if the base, the median, and altitude to the base of one triangle are equal respectively to the base, the median, and altitude to the base of the other. Ex. 134. If two circles intersect in A and B, a line joining their centers bisects AB in E. D E B |