4. Show that the three angles of any triangle in Fig. 1 are equal respectively to the three angles of each of the other triangles. Connect this exercise with § 28. 5. In Fig. 2, BA || GF, and BC | DE. Prove that 4B and y are supplementary, and that ZB=Lz. 6. In Fig. 3, which shows a common form of a "scissors roof truss," AC-EC, and BC=DC. Prove that <CAD=ZCEB, and that AD=EB, A0=EO, and that OC bisects LACE. 7. In Fig. 4, ZBAD=ZCDA, and AB=DC. Prove that AAED is isosceles. 8. AB | CD, FK bisects ZEFB, EK bisects ZDEF. Prove FK 1 EK. 9. If lines AB and CD are parallel, what is the value of x? 10. In Fig. 7, AB || CD. At what angle does the transversal intersect the parallel lines? (Fig. 6.) *11. Draw the designs of the tile floor borders. They are Greek designs, depending upon perpendicular and parallel lines. 142. Exterior Angles.. An angle formed by one side of a figure and an adjacent side produced is an exterior angle of the figure. B Thus ZBCF is an exterior angle of AABC. C In AABC, ZA and B are the opposite interior angles in relation to the exterior angle BCF. 143. Theorem. The sum of the angles of a triangle is equal 144. Theorem. An exterior angle of a triangle is equal to the sum of the opposite interior angles, and is therefore greater than either of them. 145. Theorem. In a triangle, there can be only one right angle, or one obtuse angle. Why? 1. Prove the theorem of § 143, using the above figures. 2. Show how the sum of the angles of a triangle may be found to be equal to a straight angle by cutting out a paper triangle, and tearing it into three pieces, as in Fig. 1. Place the three pieces as in Fig. 2. 3. How many degrees in <r+Zs+Zt? 4. <z=110°. ZB-ZA=40°. How many degrees in each angle of the triangle? 5. Triangle DEF is isosceles. KH and KC bisect the two exterior angles. Prove that ADKF is isosceles. 6. Show how the distance from B to D may be found by the accompanying figure. How many degrees in ZD? Sailors use this method of determining distances, which is known as "doubling the angle on the bow." 7. When each base angle of an isosceles triangle is twice the vertex angle, the bisector of either base angle divides the triangle into two isosceles triangles. SUGGESTION. Show that <C=36°; Zx=≤C; Zz=ZE. 8. Triangle DBE is isosceles. BC bisects LABE. Prove BC | DE. 9. Triangle BDE is a right triangle, BA =BD, and EC=ED. Show that ZADC=135°. Also show that <r+2s = 270°. 10. In the isosceles triangle AKB, AH=BE. Prove that ZABH = ZBAE. *11. The latitude of a place may be determined by observing the North Star. HK is the equator. D is the North Pole. OD 1 HK. AC 1 AO. AB and OD both point to the North Star and are considered to be parallel on account of the great distance of the star. To the observer at A, Zx is the angle of elevation of the North Star. Show that Zz = Ly=Zx. When the observer measures Zx he determines the latitude at A, which is equal to Zz. A H B K QUADRILATERALS 146. Closed Figure. If a portion of a plane surface is entirely separated from the remaining portion by a line or lines, a closed figure is formed. 147. Quadrilateral. A closed figure formed by four straight lines is a quadrilateral. 148. Quadrilaterals Classified. A trapezium is a quadrilateral having no two sides parallel. A trapezoid is a quadrilateral having two sides and only two sides parallel. A parallelogram is a quadrilateral having its opposite sides parallel. TRAPEZIUM TRAPEZOID 149. Parallelograms Classified. PARALLELOGRAM A rhomboid is a parallelogram having no right angles. A rhombus is a rhomboid having all of its sides equal. A rectangle is a parallelogram having all of its angles right angles. A square is a rectangle having all of its sides equal. 150. An isosceles trapezoid is a trapezoid in which the non-parallel sides are equal. 151. The median of a trapezoid is the line joining the middle points of the non-parallel sides. The bases of a trapezoid are the parallel sides. Thus HK is the median of the trapezoid ABCD. BC and AD are the bases. A D 152. The bases of a parallelogram are the side on which it is supposed to rest and the opposite side. They are called the lower base and upper base. Either pair of parallel sides may be considered as the bases. 153. Diagonal. A diagonal of a figure is a straight line joining any two vertices not consecutive. Thus BD is a diagonal of the quadrilateral ABCD. A B 154. Theorem. A diagonal divides a parallelogram into two 155. Theorem. In any parallelogram, the opposite sides are equal, and the opposite angles are equal. 156. Theorem. The diagonals of a parallelogram bisect each other. Given AC and BD, the diagonals of To prove A0=OC and BO=OD. Show that ABOC=^AOD. A C B x D 157. Theorem. Any two consecutive angles of a parallelo |