Geometry: An Interactive Journey to Mastery

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In )LJXUH, O is the center of the circle.
Given: PS is the diameter.
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Prove: Arcs PQ and QR are congruent.
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i. Draw an additional radius and label angles x 1 , x 2 , x 3 , and x 4 , as
shown in )LJXUH.
ii. x 1 # x 2 because they are corresponding angles for parallel lines
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iii. x 2 # x 3 because they are base angles of an isosceles triangle
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iv. x 3 # x 4 because they are alternate interior angles for parallel lines.
v. Thus, x 1 # x 4.
vi. The two arcs in consideration are congruent because x 1
is the measure of arc PQ and x 4 is the measure of arc QR.
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Two circles of radii 7 and 2 have centers 12 units apart.
Find the length x of the common tangent shown in )LJXUH.
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Draw in two radii to make a quadrilateral containing two right angles
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Draw a dotted line to make a rectangle within this quadrilateral.
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The Pythagorean theorem gives x^2 + 5^2 = 12^2 , yielding x 119.

P

Q

R S

O

Figure 19.5

Figure 19.6

P

Q

R S

xx 41 O

x (^3) x 2
(^732)
x
Figure 19.7

7 3 2 2

7 x

Figure 19.8

12 2

52 x

x

Figure 19.9