Geometry with Trigonometry

Sec. 10.10 Angles between lines 169

10.10.3Duo-angles ..............................

Whenl 1 ,l 2 are distinct lines, intersecting atZ 1 , we call the pairs

({l 1 ,l 2 },D 1 ),({l 1 ,l 2 },D 2 ),

duo-angles, with armsl 1 ,l 2 ;inthisD 1 ,D 2 are the duo-sectors of 10.10.2. We denote
these duo-angles byαd,βd, respectively. We call the bisectorl 3 theindicatorofαd,
and the bisectorl 4 the indicator ofβd.Wedefinethedegree-magnitudes of these
duo-anglesby

|αd|◦=|∠Z 2 Z 1 Z 4 |◦=|∠Z 3 Z 1 Z 5 |◦,|βd|◦=|∠Z 2 Z 1 Z 5 |◦=|∠Z 3 Z 1 Z 4 |◦.

Ifl 1 ⊥l 2 we have that|αd|◦=|βd|◦=90, and we call theseright duo-angles.
Whenl 1 =l 2 we takeαd=({l 1 ,l 2 },l 1 )to be a duo-angle with armsl 1 ,l 1 ,and
callitanull duo-angle. Its indicator isl 1 , and we define its degree-measure to be 0.
We do not define a straight duo-angle. Thus the measure of a duo-angleγdalways
satisfies 0≤|γd|◦<180.
Whenl 1 =l 2 we define

sinαd=sin(∠Z 2 Z 1 Z 4 )=sin(∠Z 3 Z 1 Z 5 ),

cosαd=cos(∠Z 2 Z 1 Z 4 )=cos(∠Z 3 Z 1 Z 5 ),

sinβd=sin(∠Z 2 Z 1 Z 5 )=sin(∠Z 3 Z 1 Z 4 ),

cosβd=cos(∠Z 2 Z 1 Z 5 )=cos(∠Z 3 Z 1 Z 4 ).

For a right duo-angle these have the values 1 and 0, respectively.
Whenl 1 andl 2 are not perpendicular, we can define as well tanαd=cossinααdd,

tanβd=cossinββdd.
Ifαdis a null duo-angle we define sinαd= 0 ,cosαd= 1 ,tanαd=0.

10.10.4 Duo-angles in standard position ...................

O

I

J

H 1

H 2

H 3

H 4 D 1

D 2

D 1

D 2

l

m

Figure 10.13.

O

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J

H 1

H 2

H 4 H 3

D 1

D 2

D 1

D 2

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