Proposition 28. Theorem. 117. Conversely, if one angle of a triangle be greater than another, the side opposite the greater angle is greater than the side opposite the less. Hyp. Let ABC be a ▲ having But ABD = BAD. A AD = BD, being opposite equal 28 (114). in ▲ BCD, BD + DC>BC. Either side of a ▲ is < the sum of the other two (96). And, since AD = BD, ... BD + DC = AC. ... AC> BC. Q. E.D. NOTE. These two propositions may be stated briefly as follows: In every triangle, the greater side is opposite the greater angle, and conversely, the greater angle is opposite the greater side. 118. COR. The hypotenuse is the greatest side of a right-angled triangle. EXERCISES. 1. In a ▲ ABC, if AC is not greater than AB, show that any straight line drawn through the vertex A and terminated by the base BC, is less than AB. 2. Any two sides of a triangle are together greater than twice the straight line drawn from the vertex to the middle point of the third side. Proposition 29. Theorem. 119. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle in the first triangle greater than the included angle in the second, then the third side of the first triangle is greater than the third side of the second. Hyp. Let ABC, DEF be two As, having Proof. Apply the ▲ ABC to the ▲ DEF so that AB shall coincide with DE, and the pt. C fall at H. Join FH. Then, since DH = DF, .'. /DHF = /DFH. But DHF >< EHF. The whole is greater than any of its parts (Ax. 8). ... also DFH >< EHF. (Hyp.) (111) But <EFH >< DFH. (Ax. 8) Much more ... < EFH >< EHF. ... EH > EF. But The greater side is opposite the greater angle (117). EH = BC. ... BC > EF. Q. E.D. Proposition 30. Theorem. 120. Conversely, if two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first triangle greater than the third side of the second, then the included angle of the first triangle is greater than the included angle of the second. Hyp. Let ABC, DEF be two ▲s ABDE, ACDF, BC > EF. having but To prove Proof. If A is not > D, then either <A> <D. (1) A = <D, (2) ZA ZD. AA or (1) If then ZA = 2D, BC= EF, having two sides and the included equal, each to each (104). But this is contrary to the hypothesis. Hence, since A can neither be nor <≤ D, NOTE.-Prop. 30 is here proved by an indirect method. This method is sometimes called the reductio ad absurdum. It consists in assuming that the conclusion to be proved is not true, and showing that this assumption leads to an absurdity, or to a result inconsistent with the hypothesis. QUADRILATERALS. 121. A quadrilateral is a plane figure bounded by four straight lines, which are called its sides. The straight lines which join opposite angles of a quadrilateral are called the diagonals. 122. A trapezium is a quadrilateral which has no two of its sides parallel. 123. A trapezoid is a quadrilateral which has two of its sides parallel. The parallel sides of a trapezoid are called the bases, and the perpendicular distance between them is called the altitude. The line joining the middle points of the non-parallel sides is called the middle parallel* of the trapezoid. 124. A parallelogram is a quadrilateral which has its opposite sides parallel. The bases of a parallelogram are the side on which it stands and the opposite side. The perpendicular distance between the bases is called the altitude. 125. A rectangle is a parallelogram whose angles are right angles.t 126. A square is a rectangle whose sides are all equal.‡ 127. A rhomboid is a parallelogram whose angles are oblique and whose adjacent sides are unequal. 128. A rhombus, or lozenge, is a parallelogram whose sides are all equal. § *Called also the median. + Called also a right-angled parallelogram. § Called also equilateral rhomboid. Proposition 31. Theorem. 129. In every parallelogram, the opposite sides are equal, and the opposite angles are equal. having a side and the two adjacent s equal, each to each (105), 130. COR. 1. A diagonal of a parallelogram divides it into two equal triangles. 131. COR. 2. Two parallels included between two other parallels are equal. |