PROPOSITION XXIX. THEOREM.

If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

Let the st. line EF fall on the parallel st. lines AB, CD.

Then must

I.

II.

LEGB= corresponding ↳ GHD.

III. ▲ s BGH, GHD together =two rt. Ls.

I. If LAGH be not = GHD, let AGH be greater than

GHD.

Add to each 4 BGH.

Then 48 AGH, BGH are together greater than 4s GHD, BGH together.

Now 48 AGH, BGH together = two rt. <s;

:. 48 GHD, BGH are together less than two rt.

:. AB and CD will meet if produced towards B, D.

But they cannot meet, they are parallel ;

:. LAGH is not greater than ▲ GHD.

Similarly it may be shewn that

LAGH is not less than GHD;

I. 13.

<s;

Post. 6.

.. adding to each ▲ BGH,

▲ s BGH, GHD together= ≤s BGH, EGB together.

But s BGH, EGB together=two rt. 48;

... 28 BGH, GHD together= two rt. 4 s.

I. 13.

Q. E. D.

EXERCISES.

1. If through a point, equidistant from two parallel straight lines, two straight lines be drawn cutting the parallel straight lines; they will intercept equal portions of those lines.

2. If a straight line be drawn, bisecting one of the angles of a triangle, to meet the opposite side; the straight lines drawn from the point of section, parallel to the other sides and terminated by those sides, will be equal.

3. If any straight line joining two parallel straight lines be bisected, any other straight line, drawn through the point of bisection to meet the two lines, will be bisected in that point.

4

S. E.

PROPOSITION XXX. THEOREM.

Straight lines which are parallel to the same straight line are parallel to one another.

Let the st. lines AB, CD be each || to EF.

Then must AB be || to CD.

Draw the st. line GH, cutting AB, CD, EF in the pts. O, P, Q. 0,

The following Theorems are important. They admit of easy proof, and are therefore left as Exercises for the student.

1. If two straight lines be parallel to two other straight lines, each to each, the first pair make the same angles with one another as the second.

2. If two straight lines be perpendicular to two other straight lines, each to each, the first pair make the same angles with one another as the second.

PROPOSITION XXXI. PROBLEM.

To draw a straight line through a given point parallel to a given straight line.

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= L ADC.

Produce EA to F. Then EF shall be || to BC.

I. 23.

For AD, meeting EF and BC, makes the alternate angles equal, that is, EAD= L ADC,

.. EF is to BC.

.. a st. line has been drawn through A || to BC.

I. 27.

Q. E. D.

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.

B

N

E

D

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

Then will

I. LACD = 48 ABC, BAC together.

II.

LS ABC, BAC, ACB together=two rt, Ls.

..

4 s ECD, ACE together= 48 ABC, BAC together;

LACD=LS ABC, BAC together.

..

to each of these equals add ▲ ACB ;

then 28 ABC, BAC, ACB together= 48 ACD, ACB together,

... 28 ABC, BAC, ACB together=two rt. 4 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.