Preparing for the Regents Examination Geometry, AK

22 Label arcs with degree measures that sum
to 360, as with x, 2x, 3x, 4x. Then find the
values of the inscribed angles.
23 Parallel lines cut congruent arcs x and y.
Therefore,
2x 2y 360
x y 180
Each inscribed angle is equal to one-half the
intercepted arc: 90.
13-4 Tangents and Secants
(pages 363–365)
1 a
b
c
d
2 a 12 b 3
3 a 3.25 b 1
4 a disjoint b externally tangent
c intersecting twice
d internally tangent
e disjoint internally
f concentric
5 a Sketch of two circles externally disjoint
b Sketch of two intersecting circles, not
tangent
c Sketch of externally tangent circles
d Sketch of internally tangent circles
6 a 1 b 3
c 2 d 4
7 a 160 b 130 c 90
d 40 e 180 x
8 a 60 b 45 c 35
d 51 e 67.5
180 n
9 a _ b 90 2n c
n
2 _
2
d 45 n e __
180 m n
2
10 16
11 80
12 48
13 234
14 a 18 b 72 c 29.25
15 a 11, 17, 18 b no sides equal
16 a 12, 16, 20
b Satisfies Pythagorean theorem:
12 2 16 2 20 2
17 a 10, 15, 15 b two sides are equal
18 BE 4, EC 5, CF 5, AF 6, AC 11,
AB 10
19 AB 20, CB 20, RB 10
20 AB 24, DR 14, DB 32
21 OB 26, DB 36, RB 16
22 OB 20, AB 2 √ 91 , CB 2 √ 91
23 OB 22, CB 8 √ 6 , AB 8 √ 6
24 AB 20, CB 20, OC 15, OB 25
25 Tangent segments from the same point are
congruent. Base angles are equal, each measuring
60. Therefore, APB is equilateral.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
26 1. Let E be the point of intersection of −−
AB
and −−−
CD .
2. −−
EB and −−
ED are tangent segments to
circle P.
3. −−
EB −−
ED (Two tangent segments
drawn from an external
point are congruent.)
4. −−
EA and −−
EC are tangent segments to
circle O.
5. −−
EA −−
EC
6. −−
EA −−
EB ( Addition postulate)
−−
EC −−−
CD
or −−
AB −−−
CD
13-4 Tangents and Secants 81