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THEOREM 1. [Euc. I. 13.] The adjacent angles which one
straight line makes with another straight line on one side
COR. 1. If two straight lines cut one another, the four
angles so formed are together equal to four right angles.
COR. 2. When any number of straight lines meet at a
point, the sum of the consecutive angles so formed is equal
COR. 3. (i) Supplements of the same angle are equal.
(ii) Complements of the same angle are equal.
THEOREM 2. [Euc. I. 14.] If, at a point in a straight line,
two other straight lines, on opposite sides of it, make the
adjacent angles together equal to two right angles, then
these two straight lines are in one and the same straight line.
THEOREM 3. [Euc. I. 15.] If two straight lines cut one
THE COMPARISON OF TWO TRIANGLES
THEOREM 4. [Euc. I. 4.] If two triangles have two sides
of the one equal to two sides of the other, each to each, and
the angles included by those sides equal, then the triangles
THEOREM 5. [Euc. I. 5.] The angles at the base of an isosceles
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. [Euc. I. 16.] If one side of a triangle is pro-
duced, then the exterior angle is greater than either of the
COR. 1. Any two angles of a triangle are together less
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
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
THEOREM 12. Of all straight lines from a given point to a
COR. 1. If OC is the shortest straight line from O to the
COR. 2. Two obliques OP, OQ, which cut AB at equal
COR. 3. Of two obliques OQ, OR, if OR cuts AB at the
greater distance from C the foot of the perpendicular, then
THEOREM 13. [Euc. I. 27 and 28.] If a straight line cuts two
equal, or (ii) an exterior angle equal to the interior opposite
angle on the same side of the cutting line, or (iii) the interior
angles on the same side equal to two right angles; then in
each case the two straight lines are parallel.
THEOREM 14. [Euc. I. 29.] If a straight line cuts two parallel
lines, it makes (i) the alternate angles equal to one another;
(ii) the exterior angle equal to the interior opposite angle on
the same side of the cutting line; (iii) the two interior angles
on the same side together equal to two right angles.
THEOREM 15. [Euc. I. 30.] Straight lines which are parallel
THEOREM 16. [Euc. I. 32.] The three angles of a triangle
are together equal to two right angles.
COR. 1. All the interior angles of any rectilineal figure,
together with four right angles, are equal to twice as many
right angles as the figure has sides.
COR. 2. If the sides of a rectilineal figure, which has no
re-entrant angle, are produced in order, then all the exterior
angles so formed are together equal to four right angles.
THEOREM 17. [Euc. I. 26.] If two triangles have two angles
of one equal to two angles of the other, each to each, and any
side of the first equal to the corresponding side of the other,
the triangles are equal in all respects.
ON THE IDENTICAL EQUALITY OF TRIANGLES
THEOREM 18. Two right-angled triangles which have their
hypotenuses equal, and one side of one equal to one side of
the other, are equal in all respects.
THEOREM 19. [Euc. I. 24.] If two triangles have two sides of
the one equal to two sides of the other, each to each, but the
angle included by the two sides of one greater than the angle
included by the corresponding sides of the other; then the
base of that which has the greater angle is greater than the
THEOREM 20. [Euc. I. 33.] The straight lines which join the
extremities of two equal and parallel straight lines towards
COR. 1. If one angle of a parallelogram is a right angle, all
THEOREM 21. [Euc. I. 34.] The opposite sides and angles of a
parallelogram are equal to one another, and each diagonal
COR. 3. The diagonals of a parallelogram bisect one
THEOREM 22. If there are three or more parallel straight lines,
and the intercepts made by them on any transversal are equal,
then the corresponding intercepts on any other transversal
COR. In a triangle ABC, if a set of lines Pp, Qq, Rr,...,
drawn parallel to the base, divide one side AB into equal parts,
they also divide the other side AC into equal parts.
INTRODUCTION. NECESSARY INSTRUMENTS
PROBLEM 4. To draw a straight line perpendicular to a given
THE CONSTRUCTION OF TRIANGLES.
PROBLEM 8. To draw a triangle, having given the lengths of
PROBLEM 12. To construct a parallelogram having given two
adjacent sides and the included angle.
PROBLEM 14. To find the locus of a point P which moves so
that its distances from two fixed points A and B are always
PROBLEM 15. To find the locus of a point P which moves so
that its perpendicular distances from two given straight lines
THE CONCURRENCE OF STRAIGHT LINES IN A TRIANGLE.
I. The perpendiculars drawn to the sides of a triangle from
their middle points are concurrent.
II. The bisectors of the angles of a triangle are concurrent.
The three medians of a triangle cut one another at a
point of trisection, the greater segment in each being towards
THEOREM 26. [Euc. I. 37.] Triangles on the same base and
between the same parallels (hence, of the same altitude) are
THEOREM 27. [Euc. I. 39.] If two triangles are equal in area,
and stand on the same base and on the same side of it, they
are between the same parallels.
THEOREM 28. AREA OF (i) A TRAPEZIUM.
AREA OF ANY RECTILINEAL FIGURE -
THEOREM 29. [Euc. I. 47. PYTHAGORAS'S THEOREM.] In a
right-angled triangle the square described on the hypotenuse
is equal to the sum of the squares described on the other two