COROLLARY 1. If one angle of a parallelogram is a right angle, all its angles are right angles.

In other words:

All the angles of a rectangle are right angles.

For the sum of two consecutive ▲3=2 rt. 2o; (Theor. 14.) if one of these is a rt. angle, the other must be a rt. angle. And the opposite angles of the parTM are equal; ..all the angles are rt. angles.

COROLLARY 2. All the sides of a square are equal; and all its angles are right angles.

COROLLARY 3. The diagonals of a parallelogram bisect one another.

Let the diagonals AC, BD of the parm

ABCD intersect at O.

To prove AO=OC, and BO=OD.

1. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

2. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

3. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

4. The diagonals of a rhombus bisect one another at right angles.

5. If the diagonals of a parallelogram are equal, all its angles are right angles.

6. In a parallelogram which is not rectangular the diagonals are unequal.

EXERCISES ON PARALLELS AND PARALLELOGRAMS.

(Symmetry and Superposition.)

1. Shew that by folding a rhombus about one of its diagonals the triangles on opposite sides of the crease may be made to coincide.

That is to say, prove that a rhombus is symmetrical about either diagonal.

2. Prove that the diagonals of a square are axes of symmetry. Name two other lines about which a square is symmetrical.

About

3. The diagonals of a rectangle divide the figure into two congruent triangles is the diagonal, therefore, an axis of symmetry? : what two lines is a rectangle symmetrical?

4. Is there any axis about which an oblique parallelogram is symmetrical? Give reasons for your answer.

5. In a quadrilateral ABCD, AB = AD and CB-CD; but the sides are not all equal. Which of the diagonals (if either) is an axis of

symmetry?

6. Prove by the method of superposition that

(i) Two parallelograms are identically equal if two adjacent sides of one are equal to two adjacent sides of the other, each to each, and one angle of one equal to one angle of the other.

(ii) Two rectangles are equal if two adjacent sides of one are equal to two adjacent sides of the other, each to each.

7. Two quadrilaterals ABCD, EFGH have the sides AB, BC, CD, DA equal respectively to the sides EF, FG, GH, HE, and have also the angle BAD equal to the angle FEH. Shew that the figures may be made to coincide with one another.

(Miscellaneous Theoretical Examples.)

8. Any straight line drawn through the middle point of a diagonal of a parallelogram and terminated by a pair of opposite sides, is bisected at that point.

9. In a parallelogram the perpendiculars drawn from one pair of opposite angles to the diagonal which joins the other pair are equal.

10. If ABCD is a parallelogram, and X, Y respectively the middle points of the sides AD, BC; shew that the figure AYCX is a paral· lelogram.

11. ABC and DEF are two triangles such that AB, BC are respectively equal to and parallel to DE, EF; shew that AC is equal and parallel to DF.

12. ABCD is a quadrilateral in which AB is parallel to DC, and AD equal but not parallel to BC; shew that

(i) the LA+the LC=180° the LB+ the LD;

(ii) the diagonal AC = the diagonal BD;

(iii) the quadrilateral is symmetrical about the straight line joining the middle points of AB and DC.

13. AP, BQ are straight rods of equal length, turning at equal rates (both clockwise) about two fixed pivots A and B respectively. If the rods start parallel but pointing in opposite senses, shew that

(i) they will always be parallel;

(ii) the line joining PQ will always pass through a certain fixed point.

14.

(Miscellaneous Numerical and Graphical Examples.)

Calculate the angles of the triangle ABC, having given.
int. LA of ext. LA; 3B=4C.

15. A yacht sailing due East changes her course successively by 63°, by 78°, by 119°, and by 64°, with a view to sailing round an island. What further change must be made to set her once more on an Easterly

course?

16. If the sum of the interior angles of a rectilineal figure is equal to the sum of the exterior angles, how many sides has it, and why? 17. Draw, using your protractor, any five-sided figure ABCDE, in which

LB=110°, LC=115°, 4D=93°, LE=152°.

Verify by a construction with ruler and compasses that AE is parallel to BC, and account theoretically for this fact.

18. A and B are two fixed points, and two straight lines AP, BQ, unlimited towards P and Q, are pivoted at A and B. AP, starting from the direction AB, turns about A clockwise at the uniform rate of 7° a second; and BQ, starting simultaneously from the direction BA, turns about B counter-clockwise at the rate of 33° a second.

(i) How many seconds will elapse before AP and BQ are parallel? (ii) Find graphically and by calculation the angle between AP and BQ twelve seconds from the start.

(iii) At what rate does this angle decrease?

THEOREM 22.

If there are three or more parallel straight lines, and the intercepts made by them on any transversal are equal, then the corresponding intercepts on any other transversal are also equal.

Let the parallels AB, CD, EF cut off equal intercepts PQ, QR from the transversal PQR; and let XY, YZ be the corresponding intercepts cut off from any other transversal XYZ.

It is required to prove that XY = YZ.

Through X and Y let XM and YN be drawn parallel to PR.

Proof.

Since CD and EƑ are parallel, and XZ meets them, .. the XYM the corresponding YZN.

=

And since XM, YN are parallel, each being parallel to PR, .. the MXY = the corresponding ▲ NYZ.

Now the figures PM, QN are parallelograms, .. XM = the opp. side PQ, and YN = the opp. side QR; and since by hypothesis PQ = QR,

.. the triangles are identically equal; Theor. 17.

... XY = YZ.

Q.E.D.

COROLLARY. In a triangle ABC, if a set of lines Pp, Qq, Rr, ..., drawn parallel to the base, divide one side AB into equal parts, they also divide the other side AC into equal parts.

Through p, q, and r let pl, q2, r3 be drawn par1 to AB.

Then, by Theorem 22, these parls divide BC into four equal parts, of which Pp evidently contains one, Qq two, and Rr three. In other words,

Similarly if the given parls divide AB into n equal parts,

If from the extremities of a straight line AB perpendiculars AX, BY are drawn to a straight line PQ of indefinite length, then XY is said to be the orthogonal projection of AB on PQ.