But BOD being an exterior angle, it is equal (by 80) to the sum of the interior and opposite angles, B and BAO; that is, But BOD is measured by the arc BD (by 174). Likewise measured by arc BD. arc DC, or ▲ BAC is Q.E.D. A 178. COR. 3. Every angle BAC, inscribed in a segment greater than a semicircle, is an acute angle; for it is measured by one-half the arc BDC, which is less than a quadrant. B D Every angle BDC, inscribed in a segment less than a semicircle, is an obtuse angle; for it is measured by one-half the arc BAC, which is greater than a quadrant. An angle formed by two chords which intersect within a circle, is measured by one-half the sum of the arcs intercepted between its sides and between its sides produced. That is, ▲ CAB is measured by one-half (BC+DE). See (80). B PROPOSITION X. THEOREM. C A D E 179. An angle formed by a tangent and a chord is measured by one-half its intercepted arc. Let AE be tangent to the circumference BCD at B, and let BC be a chord. To prove that ABC is measured by arc BC. At B erect a perpendicular; then (by 151) it will be a diameter, and the angle ABD is a right angle. Since a right angle is measured by a quadrant or one-half a semicircle, ABD is measured by arc BCD. And (by 175) ≤ CBD is measured by arc CD. But Therefore or ABC is measured by 1⁄2 arc BCD – 1⁄2 arc CD, (BCD - CD) = are BC. Q.E.D. 180. An angle formed by two secants, intersecting without the circumference, is measured by one-half the difference of the intercepted arcs. Let the angle BAC be formed by the secants AB and AC. To prove that the angle BAC is measured by one-half the arc BC minus one-half the arc DE. Join DC. Then (by 80) A D E B C By transposing, But and Therefore BDC is measured (by 175) by arc BC, ▲ BAC is measured by arc BC – 1⁄2 arc DE. EXERCISES. 1. An angle formed by a tangent and a secant is measured by onehalf the difference of the intercepted arcs. 2. The angle formed by two tangents is measured by one-half the difference of the intercepted arcs. 3. If a quadrilateral be inscribed in a circle, the sum of each pair of opposite angles is two right angles. = To prove that they intercept equal arcs; that is, arc BC = arc AF Join AB. But and ▲ BAC = ▲ ABF (by 62). BAC is measured by arc BC (175), Since the angles are equal, their measures are equal; that is, 1. Show that the above theorem is true if both lines are tangents, also when one is a chord and the other a tangent. CONSTRUCTION. Up to the present time it has been assumed that any needful line or combination of lines could be drawn, and the question has not arisen as to the possibility of drawing these lines with accuracy. In order to show that any required combination of lines, angles, or parts of lines or angles fulfilled the required conditions, principles were needed long before they could be demonstrated. Sufficient progress has now been made to render it possible to show that every assumed construction can be synthetically effected and proof furnished that each step is legitimate. The only instruments that can be employed in Elementary Geometry are the ruler and compasses. The former is used for drawing or producing straight lines, and the compasses for describing circles and for the transference of distances. The warrant for the use of these instruments is found in the three postulates already given (25). PROBLEMS IN CONSTRUCTION. PROPOSITION XIII. PROBLEM. 182. At a given point in a straight line to erect a perpendicular to that line. Let C be the given point in the line AB. To erect a perpendicular to AB at C. It is known (by 54) that every point that is equally distant from the extremities of a straight line is a perpendicular bisector of that line. Therefore it is simply necessary to make C the middle point of a portion of AB, by measuring off a distance CE, less than CB, and taking CD equal to CE. |