Book I, Proposition 5


1.   a)  State the hypothesis of Proposition 5.

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A triangle is isosceles.

1.   b)  State the conclusion.

The angles at the base are equal.

2.   ABC is a triangle with AB equal to AC; the base BC is bisected (cut

3.   into two equal parts) at D; and EB is equal to FC.  Prove that ED is
3.   equal to FD.

Since, by hypothesis, EB is equal to FC, and BD equal to DC,
then the two sides EB, BD are equal to the two sides FC, CD respectively;
and since AB is equal to AC, triangle ABC is isosceles, and therefore the angles at the base are equal: angle B is equal to angle C.
Therefore two sides and the included angle of triangle EBD are equal respectively to two sides and the included angle of triangle FCD;
therefore, the remaining side will equal the remaining side (S.A.S.); side ED is equal to side FD.

3.   If two isosceles triangles are on opposite sides of the same base, then
3.   the straight line that joins their vertices bisects the base at right
3.   angles.

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