Geometry with Trigonometry

16 Preliminaries Ch. 1

3. With each wedge-angle∠BACwe associate a non-negative number, called its
   degree-measure, denoted by|∠BAC|◦, and for each straight-angleαwe take|α|◦=


4. 
   

A

B

C D

x

y

x+y

Figure 1.15. Addition of angle-measures.

A

B

D

C

x

y

180

A

B

C

H 1

k

Figure 1.16. Laying off an angle.

By observation, we note that ifA,B,Care non-collinear and[A,D is between
[A,B and[A,C,then|∠BAD|◦+|∠CAD|◦=|∠BAC|◦, while if[A,Band[A,Care
opposite andD∈AB,then|∠BAD|◦+|∠CAD|◦=180.

By observation, given any numberkwith 0≤k<180 and any half-line[A,B,on
each side of the lineABthere is a unique wedge-angle∠BACwith|∠BAC|◦=k.In
all cases|∠BAB|◦= 0 ,so that the degree-measure of each null angle is 0, while if
∠BACis not null then|∠BAC|◦> 0.

It follows from the foregoing, that if∠BADis any wedge-angle then
|∠BAD|◦<180, and that if∠BAD,∠CADaresupplementaryangles, then|∠CAD|◦
= 180 −|∠BAD|◦.

4. Given pointsBandCdistinct fromAsuch thatC∈[A,B, we can choose a point
   Dsuch that|∠BAD|◦is equal to half the degree-measure of the wedge or straight
   angle with support|BAC. Then for all pointsP=Aon the lineADwe have|∠BAP|◦=
   |∠PAC|◦. We callAPthemid-lineorbisectorof the support|BAC.