Geometry with Trigonometry

Sec. 5.2 Parallelograms 63

asAC=BDwould implyB∈AC. SimilarlyC∈IR(|BAD)soTis on[B,D]. Thus

[A,C]∩[B,D]=0 so as in 2.4.4 a convex quadrilateral/ [A,B,C,D]can be defined, and

in this case it is called aparallelogram. The terminology of 2.4.4 then applies.

A

B C

D

T

Figure 5.3. A parallelogram.

A

B C

D

T

A rectangle.

Definition.If[A,B,C,D]is a parallelogram in whichAB⊥AD, then, asAB‖
CD, by 5.1.1 we haveAD⊥CD. Thus if two adjacent side-lines of a parallelogram
are perpendicular, each pair of adjacent side-lines are perpendicular; we call such a
parallelogram arectangle.
Parallelograms have the following properties:-

(i)Opposite sides of a parallelogram have equal lengths.

(ii) The point of intersection of the diagonals of a parallelogram is the mid-point

of each diagonal.

Proof.

(i) With the notation above for a parallelogram, the triangles with vertices

{A,B,D}and{C,D,B}are congruent in the correspondence(A,B,D)→(C,D,B)by

the ASA principle. First note that|B,D|=|D,B|. Secondly note thatAB‖CDandA

andCare on opposite sides ofBDso that∠ABDand∠CDBare alternate angles, and

hence|∠ABD|◦=|∠CDB|◦. FinallyAD‖BC,andAandCare on opposite sides of

BD,sothat∠ADBand∠CBDare alternate angles and hence|∠ADB|◦=|∠CBD|◦.

It follows that|A,B|=|C,D|,|A,D|=|B,C|.

(ii) LetTbe the point of intersection of the diagonals. Then the triangles[A,B,T],

[C,D,T]are congruent by the ASA principle, as

|A,B|=|C,D|,|∠ABT|◦=|∠CDT|◦,|∠BAT|◦=|∠DCT|◦.

It follows that|A,T|=|C,T|,|B,T|=|D,T|.

5.2.2 Sum of measures of wedge-angles of a triangle

If A,B,C are non-collinear points, then

|∠CAB|◦+|∠ABC|◦+|∠BCA|◦= 180.