1.2. Perpendicular Lines, Kiselev's Geometry, Book I. Planimetry, Givental Alexander.

 Exercises.

Prove:

40. If at a point O of the line AB (in the above figure) two congruent angles AOD and BOC are built on the opposite sides of AB, then their sides OD and OC form a straight line.

Proof: Assume that OD and OC do not form a straight line.

Then ∠AOC, ∠AOD, and ∠AOC, ∠BOC are not supplementary pairs, i.e. the sums ∠AOC + ∠AOD and ∠AOC + ∠BOC cannot be equal. In other words ∠AOD and ∠BOC cannot be equal. We have reached a contradiction.

41. If from the point O (in the same above figure) rays OA, OB, OC, OD are constructed in such a way that ∠AOC = ∠DOB and ∠AOD = ∠COB, then OB is the continuation of OA, and OD is the continuation of OC.

Proof: Now ∠AOC = ∠DOB and ∠AOD = ∠COB.

We know that ∠AOC + ∠DOB +∠AOD + ∠COB = 360°, because the sum forms a full angle. This implies that 2∠AOC + 2∠AOD = 360°. In other words, ∠AOC + ∠AOD = 180°, i.e. ∠AOC and ∠AOD are supplementary. Thus, OD is the continuation of  OC.

In a similar way we can prove that OB is the continuation of OA. Both cases are now proved.

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