Geometry with Trigonometry

Sec. 3.5 Degree-measure of angles 41

AXIOM A 5 .Degree-measure||◦of angles has the following properties:-

(i)In all cases|α|◦≥0;

(ii)Ifαis a straight-angle, then|α|◦= 180;

(iii)If∠BAC is a wedge-angle and the point D=A lies in the interior region

IR(|BAC),then

|∠BAD|◦+|∠DAC|◦=|∠BAC|◦,

while if|BACis a straight angle-support and D∈AB, then

|∠BAD|◦+|∠DAC|◦=180;

(iv) If B=A, ifH 1 is a closed half-plane with edge AB and if the half-lines[A,C

and[A,DinH 1 are such that|∠BAC|◦=|∠BAD|◦,then[A,D=[A,C;

(v)If B=A, ifH 1 is a closed half-plane with edge AB and if 0 <k< 180 ,then

there is a half-line[A,CinH 1 such that|∠BAC|◦=k.|

A

B

D

C

x

y

x+y

Figure 3.6. Addition of angle-measures.

A

B

D

C

x

y

180

COMMENT. The properties and proofs for degree-measure are quite like those for
distance, with the role of interior regions analogous to that of segments. We note that
A 5 (i) is like A 4 (i), A 5 (iii) is like A 4 (iii), A 5 (iv) is like the uniqueness part of A 4 (iv)
and A 5 (v) is like the existence part of A 4 (iv). Wedge-angles∠BADand∠DACsuch
as those in the second part of A 5 (iii) are said to besupplementary

3.5.2 Derived properties of degree-measure

Definition. For a wedge-angle∠BAC, if we take a pointB 1 =Aso thatA∈[B,B 1 ]
and a pointC 1 =Aso thatA∈[C,C 1 ],then∠B 1 AC 1 is called thevertically opposite
angleof∠BAC.