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PROPOSITION XVII. THEOREM.

Any two angles of a triangle are together less than two right angles.

GIVEN the triangle ABC;

IT IS REQUIRED TO PROVE that any two of its angles are together less than two right angles.

B

Produce BC to D.

Because ACD is an exterior angle of the triangle ABC;
therefore the angle ACD is greater than the angle ABC.

Post. 2.

Constr.

I. 16.

Because the angle ACD is greater than the angle ABC, Proved.
let the angle ACB be added to each;

therefore the angles ACD and ACB are together greater than
the angles ABC and ACB.

Because the angles ACD and ACB are together equal to two

right angles,

and the angles ACD and ACB are together greater than the

angles ABC and ACB;

W.

I. 13.

Proved.

t.

therefore the angles ABC and ACB are together less than two

right angles.

In the same manner it may be shown that the angles BAC and ACB, as also the angles CAB and ABC are together less than two right angles.

Therefore, any two angles &c.

Q. E. D.

Postulate 2. A terminated straight line may be produced to any length in a straight line.

Proposition XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Proposition XIII. The angles which one straight line makes with another straight line on one side of it, are together equal to two right angles.

[blocks in formation]

The greater side of every triangle has the greater angle opposite to it.

In the triangle ABC;

GIVEN that AC is greater than AB;

IT IS REQUIRED TO PROVE that the angle ABC is greater than the angle ACB.

[blocks in formation]

Because ADB is the exterior angle of the triangle BDC;
therefore the angle ADB is greater than the angle ACB.

Because AD is equal to AB;

therefore the angle ADB is equal to the angle ABD.

Because the angle ADB is greater than the angle ACB,
and the angle ABD is equal to the angle ADB ;
therefore the angle ABD is greater than the angle ACB.

Because the angle ABC is greater than the angle ABD,
and the angle ABD is greater than the angle ACB;
therefore the angle ABC is greater than the angle ACB.

Therefore, the greater side &c.

Нур.

I. 3.

Post. 1.

Constr.

I. 16.

Constr.

I. 5.

Proved.
Proved.

t.

Axiom 9.
Proved.

V.

Q. E. D.

Proposition III. From the greater of two given straight lines a part may be cut off equal to the less.

Postulate 1. A straight line may be drawn from any one point to any other point.

Proposition XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Proposition V. The angles at the base of an isosceles triangle are equal.

Axiom 9. The whole is greater than its part.

[blocks in formation]

The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

In the triangle ABC;

GIVEN that the angle ABC is greater than the angle ACB;
IT IS REQUIRED TO PROVE that AC is greater than AB.

B

Because, if AC were equal to AB, the angle ABC would

be equal to the angle ACB,

and the angle ABC is greater than the angle ACB;
therefore AC is not equal to AB.

Because, if AC were less than AB, the angle ABC would
be less than the angle ACB,

and the angle ABC is greater than the angle ACB;
therefore AC is not less than AB.

Because AC is not equal to or less than AB;

therefore AC is greater than AB.

I. 5.

Hyp.

G.

I. 18.
Hyp.

G.

Proved.

0.

Therefore, the greater angle &c.

Q. E. D.

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