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PROPOSITION XXXI. PROBLEM. To draw a straight line through a given point parallel to a given straight line.

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Let A be the given pt. and BC the given st. line.

It is required to draw through A a st. line || to BC.

In BC take any pt. D, and join AD.

Make . DAE=;ADC.

I. 23.

Produce EA to F. Then EF shall be 'l to BC.

For : AD, meeting EF and BC, makes the alternate angles equal, that is, - EAD= . ADC, .. EF is || to BC.

I. 27. .. a st. line has been drawn through A ll to BC.

Q. E. F.

Ex. 1. From a given point draw a straight line, to make an angle with a given straight line that shall be equal to a given angle.

Ex. 2. Through a given point A draw a straight line ABC, meeting two parallel straight lines in B and C, so that BC may be equal to a given straight line.

PROPOSITION XXXII. THEOREM.

If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles.

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Let ABC be a A, and let one of its sides, BC, be produced to D. Then will

1. LACD= 28 ABC, BAC together. II. 28 ABC, BAC, ACB together=two rt. 2 8. From C draw CE || to AB.

I. 31. Then I. :: BD meets the ||s EC, AB, .. extr. ECD=intr. LABC.

I. 29. And :: AC meets the 1s EC, AB, ... ACE=alternate . BAC.

I. 29. 1.68 ECD, ACE together= 28 ABC, BAC together ;

... ACD= 28 ABC, BAC together. And II. ; 28 ABC, BAO together= L ACD,

to each of these equals add . ACB; then 28 ABC, BAC, ACB together= 28 ACD, ACB together, .: 68 ABC, BAC, ACB together=two rt. 2 s. I. 13.

Q. E. D. Ex. 1. In an acute-angled triangle, any two angles are greater than the third.

Ex. 2. The straight line, which bisects the external vertical angle of an isosceles triangle is parallel to the base.

Ex. 3. If the side BC of the triangle ABC be produced to D, and AE be drawn bisecting the angle BAC and meeting BC in E ; shew that the angles ABD, ACD are together

; double of the angle AED.

Ex, 4. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet ; shew that they will contain an angle equal to an exterior angle at the base of the triangle.

Ex. 5. If the straight line bisecting the external angle of a triangle be parallel to the base ; prove that the triangle is isosceles.

The following Corollaries to Prop. 32 were first given in Simson's Edition of Euclid.

COR. 1. The sum of the interior angles of any rectilinear figure together with four right angles is equal to twice as many right angles as the figure has sides.

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A

Let ABCDE be any rectilinear figure.

Take any pt. F within the figure, and from F draw the st. lines FA, FB, FC, FD, FE to the angular pts. of the figure

Then there are formed as many 28 as the figure has sides.

The three zs in each of these as together=two rt. 2 8.

..all the zs in these as together=twice as many right 28 as there are as, that is, twice as many right 28 as the figure has sides.

Now angles of all the as= ls at A, B, C, D, E and as

at F,

that is, = 2 8 of the figure and 2 s at F, and ...

= 2 8 of the figure and four rt. 2 s. I. 15. Cor. 2. .: 28 of the figure and four rt. Zs=twice as many rt. 28 as the figure has sides.

Cor. 2. The exterior angles of any convex rectilinear figure, made by producing each of its sides in succession, are together equal to four right angles.

Every interior angle, as ABC, and its adjacent exterior angle, as ABD, together are=two rt. 2 8.

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.. all the intr. 28 together with all the extr. 28

=twice as many rt. Z s as the figure has sides. But all the intr. 28 together with four rt. 28

=twice as many rt. 28 as the figure has sides. .. all the intr. 28 together with all the extr. 28 =all the intr. 28 together with four rt. 28.

.. all the extr. Ls=four rt. 28. NOTE. The latter of these corollaries refers only to convex figures, that is, figures in which every interior angle is less than two right angles. When a figure contains an angle greater

a

than two right angles, as the angle marked by the dotted line in the diagram, this is called a reflex angle. See p. 149.

Ex. 1. The exterior angles of a quadrilateral made by producing the sides successively are together equal to the interior angles.

Ex. 2. Prove that the interior angles of a hexagon are equal to eight right angles.

Ex. 3. Shew that the angle of an equiangular pentagon is g of a right angle.

Ex. 4. How many sides has the rectilinear figure, the sum of whose interior angles is double that of its exterior angles ?

Ex. 5. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles ?

PROPOSITION XXXIII. THEOREM.

The straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel.

Let the equal and || st. lines AB, CD be joined towards the same parts by the st. lines AC, BD. Then must AC and BD be equal and Il.

Join BC.
Then .: AB is || to CD,
.. LABC=alternate DCB.

I. 29.
Then in As ABC, BCD,
:: AB=CD, and BC is common, and 2 ABC= 2 DCB,

.. AC=BD, and LACB= 2 DBC. I. 4. Then :: BC, meeting AC and BD, makes the alternate 2s ACB, DBC equal, .. AC is ll to BD.

Q. E. D.

=L

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