Ex. 158. Prove that the diagonals of a rhombus are perpendicular to each other. Ex. 159. Prove that the diagonals of a rhombus bisect the angles. Note. Supplementary Exercises 55–57, p. 278, can be studied now. 143. A Square is a parallelogram having two adjacent sides equal and one angle a right angle. It can be proved and it is important to remember that All the angles of a square are right angles and all the sides are equal. Note. The square is a special rectangle and also a special rhombus. Hence every theorem true about a rectangic or a rhombus is true about a square. (See Note, § 141.) 144. Many artistic designs are made on a network of squares as illustrated below. Ex. 160. Make a list of facts about the square which may be inferred from known facts about the parallelogram, the rectangle, and the rhombus. Ex. 161. How large are the angles into which a diagonal of a square divides its angles? Ex. 162. Construct a square whose diagonals shall be 2 in. in length. Prove that the lines drawn from the ends of one side of a square to the mid-points of the two adjacent sides are equal. Ex. 164. Prove that if the diagonals of a quadrilateral are perpendicular to and bisect each other, the figure is a rhombus. Ex. 165. If E, F, G, and H are points on the sides, AB, BC, CD, and AD respectively of square ABCD, such that AE = BF = CG = DH, prove that EFGH is a square. Suggestions. 2 E 4 B 1. Try to prove EFGH is a, having two adj. sides equal, and having one (24) a right angle. (§ 143.) Note. Supplementary Exercises 58-60, p. 278, can be studied now. TRAPEZOIDS 145. A Trapezoid is a quadrilateral which has one and only one pair of parallel sides; AB and CD are called the non-parallel sides. The parallel sides of a trapezoid are called the Bases. B C D The perpendicular distance between the bases is called the Altitude. The line joining the mid-points of the non-parallel sides is called the Median of the trapezoid. 146. An Isosceles Trapezoid is a trapezoid the non-parallel sides of which are equal. Ex. 166. Construct the trapezoid having lower base of 4 in., one of its non-parallel sides 2 in., the angle between these two sides being 60°, and the upper base being 1.5 in. Ex. 167. If the angles at the ends of one base of a trapezoid are equal, the angles at the ends of the other base are also equal. Ex. 168. If a trapezoid is isosceles, the lower base angles are equal. (If AB = CD, prove ▲ A = 2D. Draw BE || CD. Compare AEB with 2D and ▲ A.) B Ex. 169. If one pair of base angles of a trapezoid are equal, the trapezoid is isosceles. Ex. 170. Prove that the diagonals of an isosceles trapezoid are equal. Ex. 171. Prove that the opposite angles of an isosceles trapezoid are supplementary. Note. Supplementary Exercises 61-63, p. 278, can be studied now. 147. If three or more parallels intercept equal lengths on one transversal, they intercept equal lengths on all transversals. Plan. Try to prove BD, DF, and FH homologous sides of cong. A. Proof. 1. Draw BI, DJ, FK parallel to AG. 2. ... BI || DJ FK 3. Also BI = AC, DJ = CE, and FK = EG. 4. But 5. ACCE EG. = .. BI = DJ = FK. [Complete the proof by proving A BDI, DJF, are congruent, and then proving that BD = DF $ 91 Why? Hyp. Ax. 1, § 51 and FHK FH.] 148. Cor. 1. If a line bisects one side of a triangle, and is parallel to a second side, it bisects the third side also. 149. Cor. 2. If a line is parallel to the bases of a trapezoid and bisects one of the non-parallel sides, it bisects the other also. Note. Supplementary Exercises 64–65, p. 278, can be studied now. Required to divide AB into five equal parts. Construction. 1. Draw line AC, making a convenient with AB 2. Upon AC, lay off AD = DE = EF= FG: 4. Through D, E, F, G, and H, draw lines parallel to HB, meeting AB at X, Y, Z, and W. Statement. AX=XY= YZ = ZW: WB. Proof. 1. Assume RS through A parallel to HB. 2. 3. .. RS DX EY FZ|| GW || HB. .. AX = XY= YZ = ZW=WB. Why? Note. Supplementary Exercises 66–67, p. 279, can be studied now. 151. If a line joins the mid-points of two sides of a triangle, it is parallel to the third side and equal to one half of it. Hypothesis. D is the mid-point of AB, and E is the midpoint of AC in ▲ ABC. Conclusion. DE || BC; DE = 1⁄2 BC. Plan. Extend DE its own length to F. Try to prove FE = BC, and FE || BC. To do this, try to prove FECB is a . Proof. 1. Extend DE to F, making DF=DE. Draw BF. Give the proof. Why? Why? Why? Why? Why? Also FE = BC, and .. DE = 1⁄2 BC. Why? .. BFEC is a parallelogram. .. FE or DE is parallel to BC. Note. This theorem is very important. 152. The proof of Proposition XXIX illustrates another valuable device for proving theorems. Principle III. To prove that one segment is double another, either double the shorter and prove the result equal to the longer, or halve the longer and prove the result equal to the shorter. The first of these plans is followed in the proof of Proposition XXIX; the second plan will be used in Proposition XL. |