Ex. 135. If on the sides of an equilateral triangle, ABC, the points, D, E, F, are taken, so that AD = BE CF, and E, D, and Fare joined to the opposite vertices, then ▲ A'B'C' is equilateral. = Ex. 136. If the opposite sides of a hexagon are equal, and one pair of opposite sides are parallel, then the opposite angles are equal. Ex. 137. Two triangles are equal if one side and the altitudes upon the other sides of one triangle are equal respectively to one side and the altitudes upon the other sides of the other triangles. PROPOSITION XVIII. THEOREM 110. The line that joins the vertices of two isosceles triangles on the same base bisects the common base at right angles. Hyp. AABC and BCE are isosceles, and have the common base BC. To prove AE is the perpendicular-bisector of BC. .. ▲ BDA = rt. (being one-half of a st. <). .. AE is the perpendicular-bisector of BC. (Hyp.) Q.E.D. 111. COR. 1. Two points equidistant from the ends of a line determine the perpendicular-bisector to that line. 112. COR. 2. Every point equidistant from the ends of a line lies in the perpendicular-bisector to that line. Ex. 138. If the four sides of a quadrilateral are equal, the diagonals bisect each other. CONSTRUCTIONS PROPOSITION XIX. PROBLEM 113. To bisect a given straight line. 사 B Given a straight line AB. Required to bisect AB. Construction. From A and B as centers, with equal radii greater than AB, describe arcs intersecting at C and E. Join CE. Then the line CE bisects AB at D. Q.E.F. Proof. Two points equidistant from the ends of a line determine the perpendicular-bisector to that line. SCHOLIUM. CE is the perpendicular-bisector of AB. (111) Ex. 139. Divide a given line into four equal parts. PROPOSITION XX. PROBLEM 114. To bisect a given angle. Given. Z CAB. Required. To bisect / CAB. Construction. From A as a center, with any radius, as AB, describe an arc cutting the sides of the A at B and C. From B and C as centers, with equal radii greater than one-half the distance from B to C, describe two arcs intersecting at D. Join AD. AD bisects / CAB. Q.E.F. HINT. - What is the usual means of proving the equality of angles? Ex. 141. Divide an angle into four equal parts. Ex. 142. To bisect a straight angle. Ex. 143. Construct an angle of 90°; of 135°. Ex. 144. Draw the three bisectors of a triangle. PROPOSITION XXI. PROBLEM 115. At a given point in a given straight line, to erect a perpendicular to that line. Α Given. Point O in line AB. D B Required. A perpendicular to the line AB at O. Construction. From O as a center, with any radius OC, describe an arc intersecting AB in C and D. From C and D as centers, with equal radii greater than OC, describe two arcs intersecting at E. Join OE. OE is the perpendicular to the line AB at O. [The proof is left to the student.] Q.E.F. Ex. 145. Construct the complement of a given angle. What kind of an angle must the given angle be? Ex. 146. By what angle is the supplement of an angle greater than its complement? Ex. 147. Construct the supplement of a given angle. Ex. 148. To bisect a given reflex angle. Ex. 149. Construct an angle of 67° 30'. Ex. 150. Construct an angle of 60°. Ex. 151. Construct an angle of 30°. Ex. 152. Construct an angle of 120°; of 75°. Ex. 153. Construct the three perpendicular bisectors of the three sides of any given triangle. 116. From a point without a straight line, to let fall a perpendicular upon that line. B D Given. A straight line BC, and a point A without the line. Required. A perpendicular from the point A to the line BC. Construction. From A as a center, with a radius sufficiently great, describe an arc cutting BC in C' and D'. From D' and C" as centers, with equal radii greater than D'C', describe two arcs intersecting at O. Draw 40 intersecting BC in D. AD is a perpendicular from the point A to the line BC. Q.E.F. [The proof is left to the student.] (111) Ex. 154. Construct the three altitudes of an acute triangle; of an obtuse triangle. Ex. 155. From a given point without a line, to draw a line forming with the given line an angle equal to half a right angle. Ex. 156. From a given point without a line, to draw a line forming with the given line an angle of 60°. How many such lines can be drawn? Ex. 157. From a given point without a line, to draw a line forming with the given line an angle of 30°. |