*Ex. 251. If in the diagram opposite, A, B, and C are right angles, ≤D is also a right angle. *Ex. 252. If in the diagram opposite, AC FD, AF CD, and FB|| EC, prove that 108. Two angles whose corresponding sides are parallel and extend in the same direction from their vertices are equal. HINT. Produce if necessary BC and B'A' until they meet at D. 109. COR. Angles whose corresponding sides are parallel are either equal or supplementary. Thus, in the diagram opposite, if the correspond ing sides are parallel, 1 and 3 equal A, and 2 and 4 are sup. of ZA. 2 3 4 Ex. 253. If COLAO, and DOL OB, then 21 = 42. Ex. 254. Two angles are either equal or supplementary if the sides of one are respectively perpendicular to the sides of the other (as 1 and 2, or 1 and 3, or 1 and 4, or 1 and 5). HINT. Draw 26 whose sides are parallel to MN and PQ respectively. 2 TRIANGLES-PART II PROPOSITION XXII. THEOREM 110. The sum of the angles of a triangle is equal to a straight angle. Given ▲ ABC. To prove A B ZA+ZB+<C= a st. <. Proof 111. COR. 1. In a triangle there can be at most one obtuse angle or one right angle. 112. COR. 2. The acute angles of a right triangle are complementary. 113. COR. 3. If two triangles have two angles of the one respectively equal to two angles of the other, the third angles are equal. 114. COR. 4. Two triangles are congruent if two angles and the side opposite one of them are equal respectively to two angles and the homologous side of the other (s. a. a. = s. a. a.). 115. COR. 5. From a point without a line there can be only one perpendicular to that line. 116. COR. 6. Each angle of an equiangular triangle is equal to sixty degrees. Ex. 255. In the annexed diagram find the value of (a) ZA, if ZB = 60°, and ≤ C = 50°. = 3 B\2 (g) ZA+ZB if Z C = m°. Ex. 256. If C is the vertex angle of isosceles triangle ABC, find Ex. 257. How many degrees are there in each angle of an isosceles right triangle? Ex. 258. If two angles of a triangle are 60° and 40° respectively, what is the angle formed by the bisectors of these angles? Ex. 259. If two angles of a triangle are given, construct the third. Ex. 260. If a base angle of an isosceles triangle is given, construct the vertex angle. Ex. 261. If the vertex angle of an isosceles triangle is given, construct a base angle. Ex. 265. The bisectors of two interior angles on the same side of a transversal to two parallel lines are perpendicular to each other. Ex. 266. If CB and ED are drawn respectively perpendicular upon the sides of angle A, prove 21 = 22. Ex. 267. If CD is the altitude upon the hypotenuse of right triangle ABC, prove that = LB. ACD Ex. 268. If CD and AE are altitudes of triangle ABC, prove that 12. Ex. 269. Find the sum of the four angles of a quadrilateral. Ex. 270. If two angles of a triangle are equal, the bisector of the third angle divides the figure into two equal triangles. Ex. 271. Two right triangles are equal if the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other. Ex. 272. The altitudes upon the arms of an isosceles triangle are equal. Ex. 273. The lines drawn from the mid-point of the base of an isosceles triangle perpendicular to the arms are equal. PROPOSITION XXIII. THEOREM 117. An exterior angle of a triangle is equal to the sum of the two remote interior angles. 118. COR. An exterior angle of a triangle diminished by a remote interior angle equal the other remote interior angle. Ex. 274. Given two angles of a triangle, construct an angle equal to the remote exterior angle. Ex. 275. Given an exterior angle and a remote interior angle of a triangle. Construct an angle equal to the other remote interior angle. Ex. 276. If two angles of a triangle are equal, the bisector of the remote exterior angle is parallel to the opposite side of the triangle. Ex. 277. If at each vertex of a triangle, one exterior angle is drawn, the sum of the three angles is equal to four right angles. Ex. 278. The sum of two exterior angles of a triangle diminished by the third interior angle is equal to two right angles. * Ex. 279. If the sum of two exterior angles of a triangle is equal to three right angles, the triangle is a right one. * Ex. 280. The bisectors of two exterior angles of a triangle include an angle equal to half the third exterior angle. * Ex. 281. The sum of two interior angles of a quadrilateral is equal to the sum of the two nonadjacent exterior angles. (I.e, ZA+Z B = 21+ 22.) B |