Solutions

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(a) AB, BC, CD, AC, and AD. These segments may also be named by interchanging the letters; thus, BA, CB, DC, CA, and DA are also correct. (b) AB, AC, and AD (c) BD (d) D (e) C

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Fig. 1-3

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1.3 Finding lengths and points of line segments See Fig. 1-4. (a) State the lengths of AB, AC, and AF. (b) Name two midpoints. (c) Name two bisectors. (d) Name all congruent segments.

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(a) AB 3 7 10; AC 5 5 10 20; AF 5 5 10.

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Fig. 1-4

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(b) E is midpoint of AF; F is midpoint of AC. (c) DE is bisector of AF; BF is bisector of AC. (d) AB, AF, and FC (all have length 10); AE and EF (both have length 5).

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1.4 Circles

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A circle is the set of all points in a plane that are the same distance from the center. The symbol for circle is s (; for circles, s. Thus, (O stands for the circle whose center is O. The circumference of a circle is the distance around the circle. It contains 360 degrees (360 ). A radius is a segment joining the center of a circle to a point on the circle (see Fig. 1-5). From the definition of a circle, it follows that the radii of a circle are congruent. Thus, OA, OB, and OC of Fig. 1-5 are radii of (O and OA > OB > OC.

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CHAPTER 1 Lines, Angles, and Triangles

Fig. 1-5

Fig. 1-6

A chord is a segment joining any two points on a circle. Thus, AB and AC are chords of (O. A diameter is a chord through the center of the circle; it is the longest chord and is twice the length of a radius. AC is a diameter of (O. An arc is a continuous part of a circle. The symbol for arc is , so that AB stands for arc AB. An arc of measure 1 is 1/360th of a circumference. A semicircle is an arc measuring one-half of the circumference of a circle and thus contains 180 . A diameter divides a circle into two semicircles. For example, diameter AC cuts (O of Fig. 1-5 into two semicircles. A central angle is an angle formed by two radii. Thus, the angle between radii OB and OC is a central angle. A central angle measuring 1 cuts off an arc of 1 ; thus, if the central angle between OE and OF in Fig. 1-6 is 1 , then EF measures 1 . Congruent circles are circles having congruent radii. Thus, if OE > O G, then ( O > ( O .

SOLVED PROBLEMS

1.4 Finding lines and arcs in a circle In Fig. 1-7 find (a) OC and AB; (b) the number of degrees in AD; (c) the number of degrees in BC.

Fig. 1-7

Solutions

(a) Radius OC radius OD 12. Diameter AB 24. 100 70 80 . 110 .

(b) Since semicircle ADB contains 180 , AD contains 180 (c) Since semicircle ACB contains 180 , BC contains 180

1.5 Angles

An angle is the figure formed by two rays with a common end point. The rays are the sides of the angle, while the end point is itsSvertex. The symbol for angle is / or ]; the plural is . S Thus, AB and AC are the sides of the angle shown in Fig. 1-8(a), and A is its vertex.

CHAPTER 1 Lines, Angles, and Triangles

1.5A Naming an Angle

An angle may be named in any of the following ways: 1. With the vertex letter, if there is only one angle having this vertex, as /B in Fig. 1-8(b). 2. With a small letter or a number placed between the sides of the angle and near the vertex, as /a or /1 in Fig. 1-8(c). 3. With three capital letters, such that the vertex letter is between two others, one from each side of the angle. In Fig. 1-8(d), /E may be named /DEG or /GED.

Fig. 1-8