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THEOREM 5. [Euc. I. 5.] The angles at the base of an isosceles

triangle are equal.

CoR. 1. If the equal sides of an isosceles triangle are pro-

duced, the exterior angles at the base are equal.

COR. 2. If a triangle is equilateral, it is also equiangular.

THEOREM 6. [Euc. I. 6.) If two angles of a triangle are equal

to one another, then the sides which are opposite to the equal

angles are equal to one another.

THEOREM 7. [Euc. I. 8.] If two triangles have the three sides

of the one equal to the three sides of the other, each to each,

they are equal in all respects.

THEOREM 8. [Fuc. I. 16.] If one side of a triangle is pro-

duced, then the exterior angle is greater than either of the

interior opposite angles.

COR. 1. Any two angles of a triangle are together less

than two right angles.

Cor. 2. Every triangle must have at least two acute

angles.

COR. 3. Only one perpendicular can be drawn to a

straight line from a given point outside it.

THEOREM 9. [Euc. I. 18.] If one side of a triangle is greater

than another, then the angle opposite to the greater side is

greater than the angle opposite to the less.

THEOREM 10. [Euc. I. 19.] If one angle of a triangle is

greater than another, then the side opposite to the greater

angle is greater than the side opposite to the less.

THEOREM 11. [Euc. I. 20.] Any two sides of a triangle are

together greater than the third side.

THEOREM 12. Of all straight lines from a given point to a

given straight line the perpendicular is the least.

COR. 1. If OC is the shortest straight line from O to the

straight line AB, then OC is perpendicular to AB.

CoR. 2. Two obliques OP, OQ, which cut AB at equal

distances from C the foot of the perpendicular, are equal.

COR. 3. Of two obliques OQ, OR, if OR cuts AB at the

greater distance from C the foot of the perpendicular, then

OR is greater than OQ.

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