REVISION LESSON ON TRIANGLES.

1. State the properties of a triangle relating to

(i) the sum of its interior angles;

(ii) the sum of its exterior angles.

What property corresponds to (i) in a polygon of n sides? With what other figures does a triangle share the property (ii)?

2. Classify triangles with regard to their angles. Enunciate any Theorem or Corollary assumed in the classification.

3. Enunciate two Theorems in which from data relating to the sides a conclusion is drawn relating to the angles.

In the triangle ABC, if a=36 cm., b=2.8 cm., c=36 cm., arrange the angles in order of their sizes (before measurement); and prove that the triangle is acute-angled.

4.

Enunciate two Theorems in which from data relating to the angles a conclusion is drawn relating to the sides.

In the triangle ABC, if

(i) A=48° and B=51°, find the third angle, and name the greatest side.

(ii) A=B=621°, find the third angle, and arrange the sides in order of their lengths.

5. From which of the conditions given below may we conclude that the triangles ABC, A'B'C' are identically equal? Point out where ambiguity arises; and draw the triangle ABC in each case.

6. Summarise the results of the last question by stating generally under what conditions two triangles

(i) are necessarily congruent;

(ii) may or may not be congruent.

7. If two triangles have their angles equal, each to each, the triangles are not necessarily equal in all respects, because the three data are not independent. Carefully explain this statement.

(Miscellaneous Examples.)

8. (i) The perpendicular is the shortest line that can be drawn to a given straight line from a given point.

(ii) Obliques which make equal angles with the perpendicular are equal.

(iii) Of two obliques the less is that which makes the smaller angle with the perpendicular.

9. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles opposite to one pair of equal sides equal, then the angles opposite to the other pair of equal sides are either equal or supplementary, and in the former case the triangles are equal in all respects.

10. PQ is a perpendicular (4 cm. in length) to a straight line XY. Draw through Pa series of obliques making with PQ the angles 15°, 30°, 45°, 60°, 75°. Measure the lengths of these obliques, and tabulate the results.

11. PAB is a triangle in which AB and AP have constant lengths 4 cm. and 3 cm. If AB is fixed, and AP rotates about A, trace the changes in PB, as the angle A increases from 0° to 180°.

Answer this question by drawing a series of figures, increasing A by increments of 30°. Measure PB in each case, and tabulate the results.

12. From B the foot of a flagstaff AB a horizontal line is drawn passing two points C and D which are 27 feet apart. The angles BCA and BDA are 65° and 40° respectively. Represent this on a diagram (scale 1 cm. to 10 ft.), and find by measurement the approximate height of the flagstaff.

13. From P, the top of a lighthouse PQ, two boats A and B are seen at anchor in a line due south of the lighthouse. It is known that PQ=126 ft., LPAQ=57°, LPBQ=33°; hence draw a plan in which 1" represents 100 ft., and find by measurement the distance between A and B to the nearest foot.

14. From a lighthouse L two ships A and B, which are 600 yards apart, are observed in directions S. W. and 15° East of South respectively. At the same time B is observed from A in a S. E. direction. Draw a plan (scale 1" to 200 yds.), and find by measurement the distance of the lighthouse from each ship.

PARALLELOGRAMS.

DEFINITIONS.

1. A quadrilateral is a plane figure bounded by four straight lines.

The straight line which joins opposite angular points in a quadrilateral is called a diagonal.

fow sided figure A parallelogram is a quadrilateral

whose opposite sides are parallel.

[It will be proved hereafter that the opposite sides of a parallelogram are equal, and that its opposite angles are equal.]

3. A rectangle is a parallelogram which has one of its angles a right angle.

L

[It will be proved hereafter that all the angles of a rectangle are right angles. See page 59.]

4. A square is a rectangle which has two adjacent sides equal.

[It will be proved that all the sides of a square are equal and all its angles right angles. See page 59.]

sidia

5. A rhombus is a quadrilaterál which has all its sides equal, but its angles are not right angles,

6. A trapezium is a quadrilateral which has one pair of parallel sides.

THEOREM 20. [Euclid I. 33.]

The straight lines which join the extremities of two equal and parallel straight lines (towards the same parts are themselves equal and parallel.

A

B

Let AB and CD be equal and parallel straight lines; and let them be joined towards the same parts by the straight lines AC and BD.

It is required to prove that AC and BD are equal and parallel.

Join BC.

Proof. Then because AB and CD are parallel, and BC meets them,

... the ABC= the alternate DCB.

.. the triangles are equal in all respects;

so that AC = DB,........

and the ACB = L DBC.

But these are alternate angles;

... AC and BD are parallel.

That is, AC and BD are both equal and parallel.

Proved.

.(i)

.(ii)

Q.E.D.

2.

The opposite sides and angles of a parallelogram are equal to one another, and each diagonal bisects the parallelogram.

Let ABCD be a parallelogram, of which BD is a diagonal. It is required to prove that

Proof. Because AB and DC are parallel, and BD meets them,

Again, because AD and BC are parallel, and BD meets them, .. the ADB = the alternate CBD.

and BD is common to both;

.. the triangles are equal in all respects;

so that ABCD, and AD = CB;

Proved.

Theor. 17.

(i)

(ii)

.(iv)

Proved.

and the

CDB = the

.. the whole

ADC = the whole ▲ CBA..........(iii)

Q.E.D.