160. The vertexes of all the angles of a given magnitude whose sides pass through two fixed points, lie on a circle that passes through the two fixed points and any one of the vertexes. In Fig. 99, let APB be an angle of the given magnitude and A and B the fixed points. Through A, B, and P, pass a circle. Now, any angle, as AP, B or AP, B, whose sides pass through A and B and whose vertex lies on the arc APB is (Art. 158) equal to the given angle APB. Again, any angle, as A P' B, whose sides pass through A and B and whose vertex lies without the arc APB is less than the angle APB. For if AP is produced to meet B P' at Q, the angle APB being an exterior angle of the triangle BPQ, is equal to PQB PBQ and is therefore greater than PQB, and, as PQB FIG. 99 is greater than A P' B (since P Q B = A P' B + QA P'), it follows that APB is greater than A P' B. In like manner it can be shown that any angle, as A P" B, whose sides pass through A and B and whose vertex lies within the arc A PB, is greater than the given angle A PB. 161. An angle formed by a tangent, as TM, Fig. 100, and a chord, as TP, is measured by one-half the intercepted arc TEP. Draw the diameter TOA. Then MTA is a right angle and is, therefore, measured by one-half the semi-circumference TEPA. The angle PTA is measured by one-half the arc PA. Hence, the angle MTP, -M equal to MTA minus PTA, is measured by one-half the difference between the semicircumference and PA; that is, by one FIG. 100 half the arc TE P. EXAMPLES FOR PRACTICE A 1. Prove that the angle BPA, Fig. 101, formed by two secants intersecting without the circumference is measured by one-half the difference of the intercepted arcs A B and CD; that is, by (AB – C D). SUGGESTION.-Join C and B. Then angle BCA is an exterior angle of the triangle BCP, and angle BPC is equal to angle A CB minus angle DBC. Show that the angle APB, Fig. 102, formed by two tangents PT and PT' is measured by one-half the difference of the intercepted arcs TQ T' and TRT'. SUGGESTION.-Join T and T'. Then ATT' is an exterior angle of triangle TTP, while ATT' and PT' Tare angles formed by a tangent and a chord. B R FIG. 102 162. The angle of intersection of two tangents is the angle formed by one tangent with the prolongation of the P S FIG. 103 other tangent. Thus, the angle APT', Fig. 103, is the angle of intersection of the two tangents TP and PT'. 163. The angle of intersection of two tangents is equal to the central angle whose sides pass through the points of tangency. In Fig. 103, join TT'. Then, the angle APT' is equal to the sum of the equal angles PTT' and PTT. But each of these angles is made by a tangent and a chord and is, therefore, measured by one-half of the arc TS T'. Hence, the angle APT' is measured by the arc TS T'. The central angle O is also measured by this arc; therefore, the angle O is equal to the angle APT'. 164. The opposite angles of an inscribed quadrilateral are supplementary; that is, their sum is equal to two right angles or 180°. In Fig. 104, the angle B is measured by one-half the arc ADC, and the opposite angle D is measured by onehalf the arc ABC. The sum of the arcs ADC and ABC is a circumference, or 360°. Hence, the sum of the angles ADC and ABC is measured by one-half of 360°, or 180°. 165. If the opposite angles of a quadrilateral are supplementary, the quadrilateral can be inscribed in a circle. FIG. 104 EXAMPLE 1.-What is the number of degrees in each angle of an equilateral triangle? SOLUTION.-The sum of the three angles of the triangle is two right angles, or 180°. Since the three angles are equal, each angle is one180° third of 180°, or = 60°. Ans. 3 EXAMPLE 2.-The unequal angle of an isosceles triangle is 75° 32′ 10′′; what is the magnitude of each of the equal angles? SOLUTION.-Since the sum of the three angles is 180°, the sum of the two equal angles is 180° minus the other angle, or 180° – 75° 32′ 10′′ 104° 27′ 50′′, and each of them is one-half of this sum, or (104° 27′ 50′′) ÷ 2 52° 13′ 55′′. Ans. = = EXAMPLE 3.-The exterior angle of a triangle is 124° 3′ 40′′, and one of the opposite-interior angles is 60°; find the other two angles of the triangle. SOLUTION.-Let the given exterior angle be denoted by A, the given interior angle by B, the other opposite interior angle by C, and the third angle of the triangle by A'. (Let the student draw the triangle and mark these angles.) Then, A 124° 3′ 40′′ – 60° = 64° 3′ 40′′. Ans. = 180°. A = A' = 180° - 124° 3′ 40′′ = BC; whence, C = A – B Also, A + A' = 180°; whence, 55° 56′ 20. Ans. EXAMPLES FOR PRACTICE 1. Show that the only parallelogram that can be inscribed in a circle is a rectangle. E Ꭲ B 2. Show that if from a point A, Fig. 105, on the arc of a circle a chord A B and a tangent A T are drawn, the perpendiculars DC and D E drawn to them from the middle point D of the subtended arc are equal. 3. The angle of intersection of two tangents is 100°; find the number of degrees in each angle formed by the tangents and the chord through the points of contact. Ans. 50° 4. One of the acute angles of a right triangle is 50°; what is the magnitude of the other acute angle? Ans. 40° 5. Each of the equal angles of an isosceles triangle is 45°; show that the triangle is right-angled. 6. Two angles of a triangle are 37° 41′ 33′′ and 86° 51′ 2′′; what is the value of the other angle? Ans. 55° 27′ 22′′ GEOMETRY (PART 2) PROPORTION DEFINITIONS AND GENERAL PRINCIPLES 1. A proportion is an equality of ratios or of fractions. 4 Thus, the fractions and , being equal, form a proportion. 8 10' form a proportion, which may be written in any of the following forms: = a с a b d ': b = cd, a b c d. When writ ten in either of the last two forms, the proportion is read a is to b as c is to d. 2. Properties of Proportions.-The first and the fourth term of a proportion are called the extremes; the second and the third, the means. Thus, in the proportion a: b = cd, the extremes are a and d, and the means, b and c. This 3. If any four quantities are in proportion, the product of the extremes is equal to the product of the means. principle follows at once from the definition of a proportion, as will be explained presently. portion, then, by the definition, a If a, b, c, and d, are in pro b d This equation may be treated the same as any other algebraic equation. Both members of the equation may be COPYRIGHTED BY INTERNATIONAL TEXTBOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON |