1. ABCD is a quadrilateral having AB = CD and < BAD> Z ADC. Show that BCD > < ABC.

2. In ▲ ABC, AB > AC and D is the middle point of BC. If any point P in the median AD be joined to B and C, BP > CP.

If AD be produced to any point Q show that BQ <QC.

3. D is a point in the side AB of the ABC. AC is produced to E making CE = BD. BE and CD are joined. Show that BE > CD.

4. If two chords of a circle be unequal the greater subtends the greater angle at the centre.

5. Two circles have a common centre at O. A, B are two points on the inner circumference and C, D two on the outer. < AOC > < BOD. Show that ACBD.

A point E is taken not in

6. CD bisects AB at rt. s.
Prove that EA, EB are unequal.

CD.

7. In ▲ ABC, AB > AC. Equal distances BD, CE are cut off from BA, CA respectively.

8. In ▲ ABC, AB > AC. E making BD = CE. Prove

Prove BE> CD..

AB, AC are produced to D, CD > BE.

PARALLELOGRAMS

THEOREM 19

Straight lines which join the ends of two equal and parallel straight lines towards the same parts are themselves equal and parallel.

.. BD =

AC,

(I-8, p. 40.)

and BDA AD} (1-2, p. 16.)

= CAD

transversal AD

makes BDA

=

Z CAD,

.. BD | AC.

(I-6, p. 36.)

THEOREM 20

In any parallelogram:

(1) The opposite sides are equal;
(2) The opposite angles are equal;
(3) The diagonal bisects the area;
(4) The diagonals bisect each other.

E

B

Hypothesis.-ABCD is a gm, AC, BD its diagonals.

63. Definitions.-A parallelogram of which the angles are right angles is called a rectangle.

A rectangle of which all the sides are equal to each other is called a square.

A figure bounded by more than four straight lines is called a polygon.

The name polygon is sometimes used for a figure having any number of sides.

A polygon in which all the sides are equal to each other and all the angles are equal to each other is called a regular polygon.

64.-Exercises

1. The diagonals of a rectangle are equal to each other. 2. If the diagonals of a gm are equal to each other, the gm is a rectangle.

3. A rectangle has two axes of symmetry.

4. A square has four axes of symmetry.

F

H

B

C

5. The st. line joining the middle points of the sides of a A is the base, and equal to half of it.

NOTE.-D, E are the middle points of AB, AC. Produce DE to F making EF = DE. Join FC.

- 6. Of two medians of a ▲ each cuts the other at the point of trisection remote from the vertex.

NOTE.-Medians BE, CF cut at G. Bisect BG, CG at H, K. Join FH, HK, KE, EF.

7. The medians of a ▲ pass through one point.

Definition. The point where the medians of a ▲ intersect is called the centroid of the A.

x 8. A st. line drawn through the middle point of one side of a A, to a second side, bisects the third side.

9. In any gm the diagonal which joins the vertices of the obtuses is shorter than the other diagonal.

× 10. If two sides of a quadrilateral be ||, and the other two be equal to each other but not ||, the diagonals of the quadrilateral are equal.

11. Through a given point draw a st. line, such that the part of it intercepted between two given || st. lines is equal to a given st. line.

Show that, in general, two such lines can be drawn.

12. Through a given point draw a st. line that shall be equidistant from two other given points.

Show that, in general, two such lines can be drawn.

13. Draw a st. line || to a given st. line, and such that the part of it intercepted between two given intersecting lines is equal to a given st line.

14. BAC is a given ▲, and P is a given point. Draw a st. line terminated in the st. lines AB, AC and bisected at P.

X 15. Construct a A having given the middle points of the three sides.

16. If the diagonals of a gm cut each other at rt. Ls, the gm is a rhombus.

x.17. Every st. line drawn through the intersection of the diagonals of a ||gm, and terminated by a pair of opposite sides, is bisected, and bisects the ||gm.

18. Bisect a given ||gm by a st. line drawn through a given point.

19. Divide a given A into four congruent As.