Because ADB is an exterior angle of ▲ BCD,

▲ ADB is greater than ▲ C. But ADB = ▲ ABD, since AB

=

AD;

Much more, then, is ▲ ABC greater than C.

I. 16

I. 5

1. If two angles of a triangle be equal, the sides opposite them must also be equal.

2. A scalene triangle has all its angles unequal.

3. If one side of a triangle be less than another side, the angle opposite to it must be acute.

4. ABCD is a quadrilateral whose longest side is AD, and whose shortest is BC. Prove ABC greater than 4 ADC, and

5. Prove the proposition by producing AB to D, so that AD shall be equal to AC, and joining DC.

6. Prove the proposition from the following construction: Bisect A by AD, which meets BC at D; from AC cut off AE= AB, and join DE.

The greater angle of a triangle has the greater side opposite

Let ABC be a triangle having ▲ B greater than 4 C: it is required to prove AC greater than AB.

If AC be not greater than AB,

then AC must be

=

AB, or less than AB.
B = L C.

I. 5

If AC be less than AB, then ▲ B must be less than C. I. 18

But it is not;

... AC is not less than AB.

Hence AC must be greater than AB.

COR.—The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote.

From the given point, A, let there be drawn to the given straight line, BC, (1) the perpendicular AD, (2) AE and AF equally distant from the perpendicular, that is, so that DE = DF, (3) AG more remote than AE or AF: it is required to prove AD the least of these straight lines, and AG greater than AE or AF.

Because ADE is right, ... AED is acute;

I. 17

.. AG is greater than AE.

Hence also AG is greater than AF, and than AD.

I. 19

1. The hypotenuse of a right-angled triangle is greater than either of the other sides.

2. A diagonal of a square or of a rectangle is greater than any one of the sides.

3. In an obtuse-angled triangle the side opposite to the obtuse angle is greater than either of the other sides.

4. From A, one of the angular points of a square ABCD, a straight line is drawn to intersect BC and meet DC produced at E; prove that AE is greater than a diagonal of the square.

5. From a point outside not more than two equal straight lines can be drawn to a given straight line.

6. The circumference of a circle cannot cut a straight line in more than two points.

7. ABC is a triangle whose vertical angle A is bisected by a straight line which meets BC at D; prove that AB is greater than BD, and AC greater than CD.

The sum of any two sides of a triangle is greater than the

third side.

B

D

Let ABC be a triangle:

it is required to prove that the sum of any two of its sides is greater than the third side.

Produce BA to D, making AD = AC,

I. 3

Ꭰ

B

But BD

=

BA + AC;

.. BA + AC is greater than BC.

Now BA and AC are any two sides;

.. the sum of any two sides of a triangle is greater than the third side.

COR.-The difference of any two sides of a triangle is less than the third side.

For BA + AC is greater than BC.

Taking AC from each of these unequals,

there remains BA greater than BC – AC;

I. 20

I. Ax. 5

that is, the third side is greater than the difference between

the other two.

1. Prove the proposition by producing CA instead of BA.

4. In the first figure to I. 7, the sum of AD and BC is greater than

the sum of AC and BD.

5. A diameter of a circle is greater than any other straight line in the circle which is not a diameter.

6. Any side of a quadrilateral is less than the sum of the other three sides.

7. Any side of a polygon is less than the sum of the other sides. S. The sum of the distances of any point from the three angles of a triangle is greater than the semi-perimeter of the triangle. Discuss the three cases when the point is inside the triangle, when it is outside, and when it is on a side.

9. The semi-perimeter of a triangle is greater than any one side, and less than any two sides.

10. The sum of the two diagonals of any quadrilateral is greater than the sum of any pair of opposite sides.

11. The perimeter of a quadrilateral is greater than the sum and less than twice the sum of the two diagonals.

12. The sum of the diagonals of a quadrilateral is less than the sum of the four straight lines which can be drawn to the four angles from any other point except the intersection of the diagonals.

13. The sum of any two sides of a triangle is greater than twice the median* drawn to the third side, and the excess of this sum over the third side is less than twice the median.

14. The perimeter of a triangle is greater, and the semi-perimeter is less, than the sum of the three medians.

If from the ends of any side of a triangle there be drawn two straight lines to a point within the triangle, these straight lines shall be together less than the other two sides of the triangle, but shall contain a greater angle.

Let ABC be a triangle, and from B and C, the ends of BC, let BD, CD be drawn to any point D within the triangle:

it is required to prove (1) that BD + CD is less than AB + AC; (2) that ▲ BDC is greater than ▲ A.

* DEF.-A median line, or a median, is a straight line drawn from any vertex of a triangle to the middle point of the opposite side.