Proposition 24. Theorem. 110. Two right-angled triangles are equal when the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other. Hyp. Let ABC, DEF be two rt. As having To prove hyp. AC DF, and BC= EF. = Δ ΑΒΓ = Δ DEF. E Proof. Apply the ▲ ABC to the A DEF so that BC shall coincide with its equal EF. Then, since B E a rt. 2, = .. BA will lie along ED, and pt. A will fall somewhere on ED. But the equal oblique lines CA and FD cut off on ED equal distances from the foot of the EF (61). .. pt. A will fall at D. .. the two as coincide in all their parts, and are equal. Q.E.D. EXERCISES. 1. If BC, the base of an isosceles triangle, ABC, is produced to any point D, show that AD is greater than either of the equal sides. 2. Prove that the sum of the distances of any point from the three vertices of a triangle is greater than half its perimeter. 3. If one of the acute angles of a right triangle is 40° 14' 48', what is the value of the other acute angle? Proposition 25. Theorem. 111. In an isosceles triangle the angles opposite the equal sides are equal. Hyp. Let ABC be an isosceles A having Proof. Draw the line CD from the vertex C to the middle pt. D of the A CD= CD, (being common to both ▲s,) .. ▲ ADC = ▲ BDC. Two As are equal if the three sides are equal each to each (108). .. ZA Z B, = being opposite equal sides (109). Q.E.D. 112. COR. 1. Since ▲ ADC = ▲ BDC, we have ACD = BCD, and ADC = BDC. Therefore, the straight line which joins the vertex to the middle point of the base of an isosceles triangle, is at right angles to the base, and bisects the vertical angle. Hence, also, the bisector of the vertical angle of an isosceles triangle bisects the base at right angles. 113. COR. 2. The perpendicular from the vertex to the base of an isosceles triangle bisects the base and the angle at the vertex. An equilateral triangle is also equiangular. Proposition 26. Theorem. 114. Conversely, if two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isoceles. Hyp. Let ABC be a ▲ having having a side and the two adjacent angles equal, each to each (105). ... AC = BC, being homologous sides of equal ▲s (109). Q.E.D. 115. COR. An equiangular triangle is also equilateral. EXERCISES. 1. ABC is an isosceles triangle, with AB bisectors of the angles B and C meet at O. CO = BO. AC. The Prove that 2. ABC is a triangle; BA is produced to D so that AD = AC, and DC is joined. Prove that BCD > ▲ BDC. 3. The angle C is twice as large as either of the angles A and B: how many degrees are there in each angle? Proposition 27. Theorem. 116. If one side of a triangle be greater than another, the angle opposite the greater side is greater than the angle opposite the less. B Hyp. Let ABC be a ▲ having the side But ext. ≤CDB > opposite int. ≤ DAB of ▲ ADB. (98) = 1. Prove Prop. 27 by producing CB to E, making CE CA. 2. ABC is a triangle in which OB, OC bisect the angles B, C, respectively; show that, if AB is greater than AC, then OB is greater than OC. 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 ZABD BAD. A = AD = BD, being opposite equal Zs (114). in ▲ BCD, BD + DC>BC. Either side of a ▲ is < the sum of the other two (96). In every NOTE.-These two propositions may be stated briefly as follows: 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. |