Book I.  Proposition 15

Problems

Back to Proposition 15.

1.  PQ, RS are straight lines that intersect at T.  Name two pairs of
1.  vertical angles.

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Angles PTR, STQ, and angles RTQ, PTS.

2.   a)  State the hypothesis of Proposition 15.

Two straight lines intersect one another.

2.   b)  State the conclusion.

The vertical angles are equal.

2.   c)  Practice Proposition 15.

3.   Straight lines AC, DB intersect at E, at which both lines are bisected.

Prove that AB is equal to DC, and that angle A is equal to angle C.

By hypothesis, AE is equal to EC, and DE is equal to EB;
and angle DEC is equal to angle AEB, because they are vertical angles;
therefore the remaining side BA of triangle BAE is equal to the remaining side CD of triangle CDE  (S.A.S.).
And those angles are equal that are opposite the equal sides. Therefore angle A, opposite side BE, is equal to angle C, opposite the equal side ED.

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