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(b) By Pr. 8, x 45 Since mjABC 60 and mjCBD 45 y 60 45 105
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4.13 Applying ratios to angle-measure sums Find the measure of each angle (a) Of a triangle if its angle measures are in the ratio of 3:4:5 [Fig. 3-49(a)] (b) Of a quadrilateral if its angle measures are in the ratio of 3:4:5:6 [(b)] (c) Of a right triangle if the ratio of the measures of its acute angles is 2:3 [(c)]
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Fig. 4-49
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CHAPTER 4 Parallel Lines, Distances, and Angle Sums
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(a) Let 3x, 4x, and 5x represent the measures of the angles. Then 12x Now 3x 45, 4x 60, and 5x 75. Ans. 45 , 60 , 75 180 by Principle 1, so that x 15.
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(b) Let 3x, 4x, 5x, and 6x represent the measures of the angles. Then 18x x 20. Now 3x 60, 4x 80, and so forth. Ans. 60 , 80 , 100 , 120 (c) Let 2x and 3x represent the measures of the acute angles. Then 5x Now 2x 36 and 3x 54. Ans. 36 , 54 , 90
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360 by Principle 3, so that 18.
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90 by Principle 7 so that x
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4.14 Using algebra to prove angle-measure-sum problems (a) Prove that if the measure of one angle of a triangle equals the sum of the measures of the other two, then the triangle is a right triangle. (b) Prove that if the opposite angles of a quadrilateral are congruent, then its opposite sides are parallel.
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(a) Given: ^ ABC, mjC mj A mj B To Prove: ^ ABC is a right triangle. Plan: Prove mjC 90 ALGEBRAIC PROOF: number of degrees in j A number of degrees in jB Then a number of degrees in jC a b (a b) 180 (Pr. 1) 2a 2b 180 a b 90 Since mjC 90 , ^ ABC is a rt. ^ . Let a b b (b) Given: Quadrilateral ABCD jA > jC, jB > jD To Prove: AB i CD, BC i AD s Plan: Prove int. j on same side of transversal are supplementary.
ALGEBRAIC PROOF: number of degrees in j A and jC, number of degrees in jB and jD. 2a 2b 360 (Pr. 3) a b 180 Since jA and jB are supplementary, BC i AD. Since jA and jD are supplementary, AB i CD. Let a b
4.4 Sum of the Measures of the Angles of a Polygon
A polygon is a closed plane figure bounded by straight line segments as sides. An n-gon is a polygon of n sides. Thus, a polygon of 20 sides is a 20-gon.
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
Fig. 4-50
A regular polygon is an equilateral and equiangular polygon. Thus, a regular pentagon is a polygon having 5 congruent angles and 5 congruent sides (Fig. 4-50). A square is a regular polygon of 4 sides.
Names of Polygons According to the Number of Sides Number of Sides 3 4 5 6 7 Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Number of Sides 8 9 10 12 n Polygon Octagon Nonagon Decagon Dodecagon n-gon
4.4A Sum of the Measures of the Interior Angles of a Polygon
By drawing diagonals from any vertex to each of the other vertices, as in Fig. 4-51, a polygon of 7 sides is divisible into 5 triangles. Note that each triangle has one side of the polygon, except the first and last triangles which have two such sides. In general, this process will divide a polygon of n sides into n 2 triangles; that is, the number of such triangles is always two less than the number of sides of the polygon. The sum of the measures of the interior angles of the polygon equals the sum of the measures of the interior angles of the triangles. Hence: Sum of measures of interior angles of a polygon of n sides (n 2)180
Fig. 4-51
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
4.4B Sum of the Measures of the Exterior Angles of a Polygon
The exterior angles of a polygon can be reproduced together, so that they have the same vertex. To do this, draw lines parallel to the sides of the polygon from a point, as shown in Fig. 4-52. If this is done, it can be seen that regardless of the number of sides, the sum of the measures of the exterior angle equals 360 . Hence: Sum of measures of exterior angles of a polygon of n sides 360
Fig. 4-52
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