PROPOSITION XXIX. THEOREM. 113. A straight line which bisects the angle at the vertex af an isosceles triangle divides the triangle into two equal triangles, is perpendicular to the base, and bisects the base. Let the line C E bisect the Z A C B of the isosceles A ACB. We are to prove I. A AC E= A BCE; II. line CEI to AB; III. A E= BE. AC=BC, CE=CE, Hyp. Iden. Cons. ..A ACE=A BCE, § 106 (having two sides and the included L of the one equal respectively to two sides and the included L of the other). Also, II. ZCEA=CEB, (being homologous ts of equal A). .. C E is I to A B, (a straight line meeting another, making the adjacent s equal, is I to that line). A E= EB, Q. E. D. PROPOSITION XXX. THEOREM. 114. If two angles of a triangle be equal, the sides opposite the equal angles are equal, and the triangle is isosceles. -- In the triangle A BC, let the x B= L C. Draw A D I to BC. In the rt. A A D B and A DC, Iden. § 111 (having a side and an acute Z of the one equal respectively to a side and an acute Zof the other). .. A B = AC, Q. E. D. Ex. Show that an equiangular triangle is also equilateral. PROPOSITION XXXI. THEOREM. 115. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first will be greater than the third side of the second. CÁ In the A ABC and A B E, let A B= A B, BC=BE; but Z A B C > LABE. We are to prove AC > A E. Place the A so that A B of the one shall coincide with A B of the other. Draw B F so as to bisect Z EBC. Draw E F. Hyp. Iden. ZEBF= 2 C BF, Cons. .. the A EBF and C BF are equal, § 106 (having two sides and the included L of one equal respectively to two sides and the included 2 of the other). . EF= FC, (being homologous sides of equal ). $ 96 (the sum of two sides of a A is greater than the third side). Substitute for F E its equal FC. Then A F + FC > A E; or, Q. E. D. PROPOSITION XXXII. THEOREM. 116. CONVERSELY: If two sides of a triangle be equal respectively to two sides of another, but the third side of the first triangle be greater than the third side of the second, then the angle opposite the third side of the first triangle is greater than the angle opposite the third side of the second. SC BL If In the ABC and A' B'C', let AB= A' B', AC = A'C'; but BC > B'C'. We are to prove ZA> <A'. ZA=ZA', $ 106 (having two sides and the included 2 of the one equal respectively to two sides and the included 2 of the other), BC= B'C', A < A', $ 115 (if two sides of a A be equal respectively to two sides of another A, but the included Z of the first be greater than the included Z of the second, the third side of the first will be greater than the third side of the second.) But both these conclusions are contrary to the hypothesis ; ..Z A does not equal Z A', and is not less than 2 A'. ..Z A > Z A. And if Q. E. D PROPOSITION XXXIII. THEOREM. 117. Of two sides of a triangle, that is the greater which is opposite the greater angle. In the triangle A B C let angle A C B be greater than angle B. We are to prove A B > A C. Draw C E so as to make ZBCE= Z B. Then § 114 § 96 (the sum of two sides of a A is greater than the third side). Substitute for EC its equal E B. Then A E + EB > AC, or Q. E. D. Ex. A B C and A B D are two triangles on the same base A B, and on the same side of it, the vertex of each triangle being without the other. If A C equal A D, show that BC cannot equal B D. |