82. Theorem. A theorem is a statement to be proved. 83. Problem. A problem is a construction to be made so that it shall satisfy certain given conditions. Either a theorem or a problem may be called a proposition. 84. Problem. To construct an equilateral triangle when a side is given. On the line DF, construct the equilateral triangle DEF. § 25 With the first proposition in his book, Euclid used the method shown in Fig. 2 to construct an equilateral triangle. 85. Problem. To construct an isosceles triangle when the base and one of the equal sides are given. (Fig. 3.) 86. Theorem. In an isosceles triangle, the angles opposite the equal sides are equal. Given the isosceles triangle ABD B C D 87. Theorem. An equilateral triangle is also equiangular. Show that an equilateral triangle is a special form of an isosceles triangle. 88. Theorem. Each angle of an equilateral triangle is equal to 60°. EXERCISES B 1. Show how the isosceles triangle ABC is constructed if the vertex angle B is given. 2. Construct an isosceles triangle having the vertex angle a right angle. The vertex angle an obtuse angle. 3. Draw an isosceles triangle, using only a straightedge. 4. Mention some concrete illustrations of the isosceles triangle; e.g. a tripod. 5. If any angle of an isosceles triangle is equal to 60°, the triangle equiangular. 6. Construct an angle of 60° without using the protractor. 7. In the isosceles triangle HKE, ZK=30°. How many degrees in Zx 8. Prove that DB divides the square ABCD into two equal isoscele triangles. 9. In the square HKEM, prove that HE = KM. 10. Triangle ABC is isosceles. Prove that x = Lz. 11. Triangle ABC is isosceles. AE= DC. Prove that AEBD is isosceles 12. The two same base AC. isosceles triangles ABC and ADC are constructed upon the Prove that BD bisects LABC. 13. Triangle DEF is isosceles. H is the middle point of DF DK FL. Prove that x = Ly. 14. Triangle ABE is isosceles. The lines BC and BD are drawn, making Zz = Zv. Prove that ZxZy. 15. Prove that the following method is correct for bisecting an angle with a steel square. Take BA=BC. Place the square so that DA = DC. Will the line BD bisect ZABC? Why? B 89. Theorem. If two angles of a triangle are equal the sides opposite the equal angles are equal and the triangle is isosceles. Given ABD having ZA = ZD. To prove that AB= DB, and that AABD is isosceles. Proof. Draw BC L AD. B ZA=ZD. Why? ZBCA = ZBCD. (Being right angles, since BC was constructed 1 AD.) (If two angles of one triangle are equal respectively to two angles of another, the third angles are equal.) 90. Theorem. An equiangular triangle is also equilateral. B EXERCISES E Iden. $19 § 21 § 78 АДАА FIG. 1 H M D F 1. In Fig. 1, Zx = Lz. 2. In Fig. 2, Zv=Zw. 3. In Fig. 3, r = Ls. Prove that AABC is isosceles. Prove that ADEF is isosceles. 4. In Fig. 4, AHOM is isosceles. KO=EO. Prove that KH =EM. 5. In the isosceles triangle ABC, AE and CD are drawn, making 4x=4z. Prove that AE = CD. 6. Two straight lines drawn from a point in a perpendicular to a given line, cutting off on the given line equal lengths from the foot of the perpendicular, are equal, and make equal angles with the perpendicular. A4 91. Theorem. The sum of the acute angles of a right triangle is equal to 90°, or one right angle. § 30 92. Theorem. If two right triangles have one acute angle of one equal to one acute angle of the other, the other acute angles, also, are equal. § 32 93. Theorem. Each acute angle of an isosceles right triangle is equal to 45°. Why? 94. Prove that rt.AABC= rt. AFDE, if one side BC and an adjacent acute angle C, are equal respectively to the corresponding side DE, and the corresponding acute angle E. C E B D $ 19 K M 95. Prove that rt.AHEK=rt.^ONM, if one side KE, and the opposite acute angle H, are equal respectively to the corresponding side MN and the opposite acute angle 0. (Use § 32 and § 19.) H EN 96. Prove that rt.ACDE=rt.AKFH, if the hypotenuse CE and an acute angle C, are equal respectively to the hypotenuse KH and the acute angle K. (Proof similar to § 95.) E H D F K 97. From § 94, § 95, and § 96, the following must be true: If two right triangles have a side and an acute angle of one equal respectively to the corresponding side and the corresponding acute angle of the other, the triangles are equal. 98. Theorem. If two triangles have a side and any two angles of one equal respectively to the corresponding side and the two corresponding angles of the other, the triangles are equal. § 32 100. Prove that rt.AABC=rt. FED, if the hypotenuse AC and side BC are equal respectively to the hypotenuse FD and side ED. Proof. Place the triangles so that the two equal sides, BC and ED, take the position HK, the vertices A and F H falling on opposite sides of HK. Then ZKHF and ZKHA are right angles. AHF is a straight line. Given § 51 (If two adjacent angles are supplementary, their exterior sides 101. From § 99 and § 100, the following must be true: If two right triangles have two sides of one equal respectively to two corresponding sides of the other, the triangles are equal. 102. In drawing lines according to K points of the compass, follow the directions of geographical maps. Vertical lines indicate north and south. Horizontal lines indicate east and west. For example, B is east of 0. A is west of 0. H is south of O. N is north of O. K is north of A. C is east of D. A D N H B |