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THEOREM 1. [Euc. I. 13.] The adjacent angles which one
straight line makes with another straight line on one side
of it are together equal to two right angles.
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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
to four right angles.
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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
another, the vertically opposite angles are equal.
Triangles.
DEFINITIONS
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
are equal in all respects.
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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. [Euc. 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.
angles.
COR. 3.
Every triangle must have at least two acute
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.
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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.
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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.
Parallels.
PLAYFAIR'S AXIOM
THEOREM 13. [Euc. I. 27 and 28.] If a straight line cuts two
other straight lines so as to make (i) the alternate angles
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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.
PARALLELS ILLUSTRATED BY ROTATION.
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HYPOTHETICAL CON
STRUCTION
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THEOREM 15. [Euc. I. 30.] Straight lines which are parallel
to the same straight line are parallel to one another.
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Triangles continued.
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
reentrant angle, are produced in order, then all the exterior
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 rightangled 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
base of the other.
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THEOREM 20. [Euc. I. 33.] The straight lines which join the
extremities of two equal and parallel straight lines towards
the same parts are themselves equal and parallel.
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COR. 1. If one angle of a parallelogram is a right angle, all its angles are right angles.
THEOREM 21. [Euc. I. 34.] The opposite sides and angles of a
parallelogram are equal to one another, and each diagonal
bisects the parallelogram.
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COR. 2. All the sides of a square are equal; and all its
angles are right angles.
COR. 3. The diagonals of a parallelogram bisect one
another.
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
are also equal.
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.
DIAGONAL SCALES
Practical Geometry. Problems.
INTRODUCTION. NECESSARY INSTRUMENTS
PROBLEMS ON LINES AND ANGLES.
PROBLEM 1. To bisect a given angle.
PROBLEM 2. To bisect a given straight line.
PROBLEM 3. To draw a straight line perpendicular to a given
straight line at a given point in it.
PROBLEM 4. To draw a straight line perpendicular to a given
straight line from a given external point.
PROBLEM 5. At a given point in a given straight line to make
an angle equal to a given angle.
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PROBLEM 6. Through a given point to draw a straight line
parallel to a given straight line.
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PROBLEM 7. To divide a given straight line into any number
of equal parts.
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THE CONSTRUCTION OF TRIANGLES.
PROBLEM 8. To draw a triangle, having given the lengths of
the three sides.
PROBLEM 9. To construct a triangle having given two sides
and an angle opposite to one of them.
PROBLEM 10. To construct a rightangled triangle having given
the hypotenuse and one side.
THE CONSTRUCTION OF QUADRILATERALS.
PROBLEM 11. To construct a quadrilateral, given the lengths
of the four sides, and one angle.
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PROBLEM 12. To construct a parallelogram having given two
adjacent sides and the included angle.
PROBLEM 13. To construct a square on a given side.
Loci.
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
equal to one another.
PROBLEM 15. To find the locus of a point P which moves so
that its perpendicular distances from two given straight lines
AB, CD are equal to one another.
INTERSECTION OF LOCI
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.
III. The medians of a triangle are concurrent.
COR.
The three medians of a triangle cut one another at a
point of trisection, the greater segment in each being towards
the angular point.
Areas.
PART II.
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THEOREM 23. AREA OF A RECTANGLE.
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THEOREM 24.
[Euc. I. 35.] Parallelograms on the same base
and between the same parallels are equal in area.
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AREA OF A PARALLELOGRAM

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THEOREM 25. AREA OF A TRIANGLE.
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THEOREM 26. [Euc. I. 37.] Triangles on the same base and
between the same parallels (hence, of the same altitude) are
equal in area.
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.
(ii) ANY QUADRILATERAL.
AREA OF ANY RECTILINEAL FIGURE 
THEOREM 29. [Euc. I. 47. PYTHAGORAS'S THEOREM.] In a
rightangled triangle the square described on the hypotenuse
is equal to the sum of the squares described on the other two
sides.
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