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A B is parallel to CE (a).
by Euc. I. 29, (6) angle BAC is equal to the alternate
angle ACE. Again, A B is parallel to CE (a). BD falls upon them.
by Euc. I. 29, (1) angle ABC is equal to exterior angle
ACE in (b), and ECD in (c),
make up the whole angle A CD. .. the exterior angle ACD is equal to the two interior opposite angles B A C, A BC.
Q. E. D.
THEOREM (Euclid 1. 32, Second Part).
Repeat.-The enunciation of Euc. I. 13 and Axioms 1 and 2a. General Enunciation. The three interior angles of every triangle
are equal to two right angles.
A B C.
that the three angles
right angles. Construction.
Produce B C to D (see next page).
(Conclusion of last theorem.) Add angle A CB to each of these equals.
Then, by Axiom 2a, (6) The two angles A CD, ACB are equal to the three angles BAC, ABC, ACB.
But, by Euc. I. 13, Angles A CD, ACB are equal to two right angles.
by Axiom 1, angles BAC, ACB, A B C are equal to two right angles.
Q. E. D.
triangle, having A B equal to
A C, A D equal to A B. Required.–To prove that angle BCD is a right angle (D AC is the exterior angle of triangle ABC: show that ADC is an isosceles triangle : and use Euc. I. 32, Second Part.)
THEOREM (Euclid I. 33). Repeat.-The enunciations of Euc. I. 4; I. 27; I. 29; and Axiom 2a. General Enunciation. The straight lines which join the extremi
ties of two equal and parallel straight lines towards the same parts, are them
selves equal and parallel. Particular Enunciation. Given.-(a) The straight lines A B, CD, equal
(6) The straight line AC joining the extremities
A and C, and (c) The straight line B D joining the extremities
B and D. Required.—To prove that A C and B D are
equal and parallel.
(NOTE.-A C is said to join the extremities towards the same parts.
CB joins the extremities towards opposite parts.) Proof.
A B is parallel to CD (Given a).
.. by Euc. I. 29, (d) the angle ABC is equal to the alternate angle BCD.
Again, AB is equal to CD (Given a),
.'. by Axiom 2a, (e) the two sides A B, BC, are equal to the two sides DC, CB, each to each. So that in the two triangles ABC, DCB, we have two sides equal to two sides (e), and the included angles equal (d).
by Euc. I. 4, we have (f) the base AC equal to the base BD, and (8) the angle ACB equal to the angle D BC.
So, the straight line B C meets A C, BC,
AC is parallel to BC, and
AC, B D equal.
Required.—To prove that angles C AB, BDC
are together equal to two right angles. (Draw B E parallel to A C. Foin C B. Euc. I. 33, 1. 5, 1. 8, 1. 13, 1. 32.)
DEFINITIONS. A quadrilateral is a four-sided figure. A square has four equal sides and four right angles.
A rectangle has four right angles, and its opposite sides equal; but its adjacent sides unequal.
A parallelogram is a four-sided figure having its opposite sides parallel.
(A square and a rectangle are parallelograms, so are Figures A and B.)
A is a rhombus, B is a rhomboid.
A trapezium is any four-sided figure that is not a square, a rectangle, or a parallelogram, as C.
A diameter of a parallelogram is the straight line which joins any two of its opposite angles.