Ex. 598. If two of the bisectors of the angles of a triangle meet at a point I the perpendiculars from I to the sides are all equal.

Ex. 599. The perpendicular bisectors of two sides of a triangle meet at a point which is equidistant from the vertices of the triangle.

Ex. 600. In the equal sides PQ, PR of an isosceles triangle PQR points X, Y are taken equidistant from P; QY, RX intersect at Z. Prove that A" ZQR, ZXY are isosceles.

Ex. 601. ABC is a triangle right-angled at A; AD is drawn perpendicular to BC. Prove that the angles of the triangles ABC, DBA are respec

tively equal.

Ex. 602. From a point O in a straight line XOX' two equal straight lines OP, OQ are drawn so that▲ POQ is a right angle. PM and QN are drawn perpendicular to XX'. Prove that PM=ON.

Ex. 603. If points P, Q, R are taken in the sides AB, BC, CA of an equilateral triangle such that AP BQ= CR, prove that PQR is equilateral.

=

Ex. 604. ABC is an equilateral triangle; DBC is an isosceles triangle on the same base BC and on the same side of it, and ▲ BDC=14 BAC. Prove that AD=BC.

Ex. 605. How many sides has the polygon, the sum of whose interior angles is three times the sum of its exterior angles?

[What is the sum of all the exterior and interior angles? What is the sum of an exterior angle and the corresponding interior angle ?]

Ex. 606. If two isosceles triangles have equal vertical angles and if the perpendiculars from the vertices to the bases are equal, the triangles are congruent.

Ex. 607. If, in two quadrilaterals ABCD, PQRS,

AB=PQ, BC=QR, CD=RS, ▲ B= ▲Q, and ▲C= ▲R,

the quadrilaterals are congruent.

Prove this (i) by superposition (see 1. 10 and 11);

(ii) by joining BD and QS and proving triangles congruent.

Ex. 608. If two quadrilaterals have the sides of the one equal respectively to the sides of the other taken in order, and have also one angle of the one equal to the corresponding angle of the other, the quadrilaterals are congruent.

[Draw a diagonal of each quadrilateral, and prove triangles congruent.]

Ex. 609. If points X, Y, Z are taken in the sides BC, CA, AB of an equilateral triangle, such that ▲ BAX= 2 CBY= ¿ACZ, prove that, unless AX, BY, CZ pass through cne point, they form another equilateral triangle.

Ex. 610. If points X, Y, Z are taken in the sides BC, CA, AB of any triangle, such that ▲ BAX= ▲ CBY=ACZ, prove that, unless AX, BY, CZ pass through one point, they form a triangle whose angles are equal to the angles of the triangle ABC.

Ex. 611. If AA', BB', CC' are diameters of a circle, prove

AABCAA'B'C'.

Ex. 612. On the sides AB, BC of a triangle ABC, squares ABFG, BCED are described (on the opposite sides to the triangle); prove that

AABDA FBC.

Ex. 613. On the sides of any triangle ABC, equilateral triangles BCD, CAE, ABF are described (all pointing outwards); prove that AD, BE, CF are all equal.

Ex. 614. The side BC of a triangle ABC is produced to D; ACB is bisected by the straight line CE which cuts AB at E. A straight line is drawn through E parallel to BC, cutting AC at F and the bisector of LACD at G. Prove that EF=FG.

Ex. 615. ABC, DBC are two congruent triangles on opposite sides of the same base BC; prove that either AD is bisected at right angles by BC, or AD and BC bisect one another.

Ex. 616. In a triangle ABC, the bisector of the angle A and the perpendicular bisector of BC intersect at a point D; from D, DX, DY are drawn perpendicular to the sides AB, AC produced if necessary.

Prove that

[Join BD, CD.]

AX AY and BX=CY.

INEQUALITIES. *

Ex. 617. Draw a scalene triangle, measure its sides and arrange them in order of magnitude. Under each side in your table write the opposite angle and its measure, thus:-

Are the angles now in order of magnitude?

Ex. 618. In fig. 136, AD=AC; if ▲A=88°, find LADC and LACD. What is the sum of LB and DCB?

*This section, pp. 119--132, may be omitted at a first reading.

These signs are easily distinguished if it is borne in mind that the greater quantity is placed at the greater end of the sign.

THEOREM 16.

If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.

Construction From AB, the greater side, cut off AD = AC.

Proof

Join CD.

In AACD, AD = AC,

.. LACDL ADC.

But since the side BD of the ▲ DBC is produced to A,
.. ext. ▲ ADC > int. opp. ▲ B,

.. LACD > LB.

But ACB > its part ▲ ACD,

.. L ACB > L B.

1. 12.

I. 8, Cor. 2.

Q. E. D.

Ex. 619. In a ▲ ABC, BC=7 cm., CA 6.7 cm., AB=7.5 cm. ; which is the greatest angle of the triangle? Which is the least angle? Verify by drawing.

Ex. 620. If one side of a triangle is known to be the greatest side, the angle opposite that side must be the greatest angle. (Notice that I. 16 only compares two angles; here we are comparing three.)

Ex. 621. The angles at the ends of the greatest side of a triangle are acute.

Ex. 622. In a parallelogram ABCD, AB > AD; prove that

LADB LBDC.

[What angle is equal to BDC?]

Ex. 623. In a quadrilateral ABCD, AB is the shortest side and CD is the longest side; prove that

A

LB > LD, and LA > LC.

[Draw a diagonal.]

D

fig. 137.

Ex. 624. Assuming that the diagonals of a parallelogram ABCD bisect one another, prove that, if BD > AC, then 4 DAB is obtuse.

[Let the diagonals intersect at O, then OB > OA and OD > OA; what follows?]

Ex. 625. Prove Theorem 16 by means of the following construction :-from AB cut off AD=AC, bisect BAC by AE, join DE.

A

B

E

C

fig. 138.

THEOREM 17.

[CONVERSE OF THEOREM 16.]

If two angles of a triangle are unequal, the greater angle has the greater side opposite to it.

NOTE. The method of proof adopted in the above theorem

is called reductio ad absurdum.

Ex. 626. In a AABC, LA=68° and

B=28°. Which is the greatest

side of the triangle? Which is the shortest side?