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EXERCISES.

Ex. (1). The lines which bisect the angles at the base of an isosceles triangle, and meet the opposite sides, are equal.

Let ABC be an isosceles triangle.

Data. AB AC, and the angles at B and C bisected by BD, CE.

Proof. In the triangles ACE, ABD we have

AC=AB,

angle at A common,

E

(Hyp.)

and angle ACE=angle ABD. (Hyp. and Th. 6.))

Therefore the base CE=the base BD.

B

A

(Th. 7.) Q. E. D.

Ex. (2). The bisectors of the three angles of a triangle will meet in one point.

Let ABC be a triangle, and let the bisectors of the angles ABC, ACB be BO, CO, meeting in O; then the Theorem will be proved if we can shew that AO is the bisector of the angle BAC.

Let perpendiculars OP, OQ, OR be drawn to

the three sides BC, CA, AB.

Proof. In the triangles OQC, OPC we have

OQC=OPC, (Constr.))

OCQ=OCP, (Hyp.)

OC common.

(Hyp.)

Therefore OQ=OP by Theorem 17.

Similarly from the triangles OPB, ORB, it follows that OP=OR ; therefore OR=0Q;

and therefore the right-angled triangles OQA, ORA have the hypotenuse and one side of the one equal to the hypotenuse and one side of the other, and are therefore equal in all respects by Theorem 20, Cor. 1.

Therefore the angle OAQ=the angle OAR, that is, OA is the bisector of the angle BAC.

EXERCISES FOR SOLUTION.

I. OA and OB are any two equal lines, and AB is joined; shew that AB makes equal angles with OA and ов.

2. If the bisectors of the equal angles B, C of an isosceles triangle meet in O, shew that OBC is also an isosceles triangle.

3. The line drawn to bisect the vertical angle of an isosceles triangle will also bisect the base, and be perpendicular to it.

4. The lines joining the middle points of the sides of an isosceles triangle to the opposite extremities of the base will be equal to one another.

5. The line drawn from the vertex of an isosceles triangle to bisect the base will cut it at right angles, and bisect the vertical angle.

6. Prove that the lines which bisect the sides of a triangle and are perpendicular to them meet in one point.

7. The perpendiculars let fall from the extremities of the base of an isosceles triangle upon the opposite sides will be equal, and will make equal angles with the base.

8. The perpendicular let fall from the vertex of an isosceles triangle to the base, will bisect the base and the vertical angle.

9. If two exterior angles of a triangle be bisected by straight lines which meet in O, prove that the perpendiculars, from O on the sides or sides produced of the triangle are equal to one another.

EXERCISES ON THEOREMS OF INEQUALITY.

The line that joins the vertex to the middle point of the base of a triangle is less than half the sum of the two sides.

Let D be the middle point of AC,

then is BD less than half the sum of AB, BC.

Proof. Produce BD to B', making DB'=DB. Join AB'.

Then since the two triangles BDC, B'DA have two sides BD, DC and the included angle BDC of the one respectively equal to the two sides B'D, DA and the included angle B'DA of the other, therefore (Theorem 5) the base BC the base AB';

[blocks in formation]

B

.. AB+BC>B'B, which is twice BD,

that is, BD is less than half the sum of BC and BA.

B

(Th. 12.)

I.

EXERCISES FOR SOLUTION.

Prove that any one side of a four-sided figure is less than the sum of the other three sides.

2. Prove that the sum of the lines which join the opposite angles of any four-sided figure is together greater than the sum of either pair of opposite sides of the figure.

3. Prove that the sum of the diagonals of a quadrilateral figure is less than the sum of the four lines which can be drawn to the angles from any other point than the intersection of the diagonals.

4. O is any point within the triangle ABC; prove that OA+OB+ OC are less than the sum and greater than half the sum of AB + BC + CA.

5. Prove that the sum of the four sides of a quadrilateral figure is greater than the sum and less than twice the sum of the diagonals.

6. If ABC is a triangle in which AB is greater than AC, and D is the middle point of BC, and AB is joined, prove that the angle ADB is an obtuse angle.

7. Prove that the sum of the three sides of a triangle is greater than the sum of the three medians.

NOTE. The median of a triangle is the line that joins any angle to the middle point of the opposite side.

8. Prove that the sum of the three medians of a triangle is greater than half the sum of the sides.

QUESTIONS ON SECTION II.

I. Give the meaning and derivation of the words triangle, perimeter, isosceles, equilateral, hypotenuse, median.

2.

If a triangle is isosceles, the angles at its base will be equal. Enunciate the obverse, converse and contrapositive theorems.

3. Apply Theorem 7 to find the height of a tower.

4. Prove Theorem 6 in the manner of Theorem 8.

5. Why cannot Theorem 15 be proved in the same manner as Theorem 5?

6. Prove that only one perpendicular can be drawn from a given point to a given straight line.

7. Prove fully the corollary to Theorem 19.

8. Enumerate the five cases in which the equality of three parts in a pair of triangles involves the equality in all respects.

9. Mention cases, and draw the figures, in which two triangles are equal in three respects but not in all.

IO. Prove fully the corollaries to Theorem 20.

II.

Prove Theorem 19 by conceiving the figure to be folded down over the line AB, O falling on a point Oʻ, and RO, QO, PO, on RO', QO', PO', and using Theorems 12 and 13.

I2.

In Theorem 9, prove fully that ACD is greater than ABC.

13. Why is it necessary, in the enunciation of Theorem 9, to say interior and opposite angles?

14. What is the magnitude indirectly measured in Theorem 9?

15. Enunciate Theorem 10 formally. Is it merely the obverse of Theorem 6, or does it contain an additional geometrical fact?

16. Prove Theorem 10 by reversal and superposition, using Theorem 9.

17. Shew how Theorem 12 depends ultimately on the Axioms.

18. Which Theorem in this Section proves that as you increase the angle between the legs of a pair of compasses you also increase the distance between their points?

19. Shew the relation of Theorems 14, 15 and 16 to Theorem 5.

20. Enunciate the contra-positives of Theorems 9 and 18.

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