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(c) A right triangle if its angle measures are in the ratio 2:3:5 (d) An obtuse triangle if one angle measures 64 and the larger of the other two measures 10 less than five times the measure of the smaller 4.17. (a) Find the sum of the measures of the interior angles (in straight angles) of a polygon of 9 sides; of 32 sides. (4.15) (b) Find the sum of the measures of the interior angles (in degrees) of a polygon of 11 sides; of 32 sides; of 1002 sides. (c) Find the number of sides a polygon has if the sum of the measures of the interior angles is 28 straight angles; 20 right angles; 4500 ; 36,000 . 4.18. (a) Find the measure of each exterior angle of a regular polygon having 18 sides; 20 sides; 40 sides. (b) Find the measure of each interior angle of a regular polygon having 18 sides; 20 sides; 40 sides. (c) Find the number of sides a regular polygon has if each exterior angle measures 120 ; 40 ; 18 ; 2 . (d) Find the number of sides a regular polygon has if each interior angle measures 60 , 150 ; 170 , 175 ; 179 . 4.19. (a) Find each interior angle of a quadrilateral if its interior angles are represented by x 5, x 20, 2x 45, and 2x 30. (4.17) (b) Find the measure of each interior angle of a quadrilateral if the measures of its exterior angles are in the ratio of 1:2:3:3. 4.20. 4.21. In each part of Fig. 4-71, select congruent triangles and state the reason for the congruency. (4.18) (4.16)



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In each part of Fig. 4-72, two triangles can be proved congruent. Make a diagram showing the congruent parts of both triangles, and state the reason for the congruency. (4.19)

Fig. 4-71

Fig. 4-72

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Provide the proofs requested in Fig. 4-73.

3

(4.20)

Fig. 4-73

Prove each of the following:

(4.21)





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(a) If the perpendiculars to two sides of a triangle from the midpoint of the third side are congruent, then the triangle is isosceles. (b) Perpendiculars from a point in the bisector of an angle to the sides of the angle are congruent. (c) If the altitudes to two sides of a triangle are congruent, then the triangle is isosceles. (d) Two right triangles are congruent if the hypotenuse and an acute angle of one are congruent to the corresponding parts of the other.

A trapezoid is a quadrilateral having two, and only two, parallel sides. The bases of the trapezoid are its parallel sides; the legs are its nonparallel sides. The median of the trapezoid is the segment joining the midpoints of its legs. Thus in trapezoid ABCD in Fig. 5-1, the bases are AD and BC, and the legs are AB and CD. If M and N are midpoints, then MN is the median of the trapezoid.

Fig. 5-1

In This :

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An isosceles trapezoid is a trapezoid whose legs are congruent. Thus in isosceles trapezoid ABCD in Fig. 5-2 AB > CD. The base angles of a trapezoid are the angles at the ends of its longer base: /A and /D are the base angles of isosceles trapezoid ABCD.

Fig. 5-2

PRINCIPLE 1:

Fig. 5-3

PRINCIPLE 2:

Fig. 5-4

Summary

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(a) Since AD i BC, (2x Also, y 70 5) (x 110. 20 or x 10. 50 180 or y 130. 5) 180; then 3x 180 and x 60. 180 or y

(b) Since /A > /D, 5x 3x 20, so that 2x Since BC i AD, y (3x 20) 180, so y (c) Since BC i AD, 3x Since /D > /A, y 2x 180 or x 72. 36. 2x or y

Proof of a trapezoid principle stated in words Prove that the base angles of an isosceles trapezoid are congruent.

Isosceles trapezoid ABCD (BC i AD, AB > CD) To Prove: /A > /D Plan: Draw $ to base from B and C. Prove nI > II.

PROOF:

Given:

Statements 1. Draw BE ' AD and CF ' AD. 2. BC i AD, AB > CD 3. BE > CF 4. /1 > /2 5. nI > nII 6. /A > /D

In the preceding chapters, we discussed the na ture of business transactions and the manner in which they are analyzed and classi ed. The pri mary emphasis was the why rather than the how of accounting operations; we aimed at an understanding of the reason for making the entry in a particular way. We showed the effects of transactions by making entries in T accounts. However, these entries do not provide the necessary data for a particular transaction, nor do they provide a chronological record of transactions.

Reasons 1. A ' may be drawn to a line from an outside point. 2. Given 3. Parallel lines are everywhere equidistant. Definition of congruent segments. 4. Perpendiculars form rt. . All rt. are congruent. 5. Hy-leg s 6. Corresponding parts of congruent n are congruent.

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