Ex. 550. Draw an acute angle and bisect it; from any point on the bisector drop perpendiculars on the arms of the angle; measure the perpendiculars.

Ex. 551.

Ex. 552. of the circle.

Repeat Ex. 550 for an obtuse angle.

From the centre of a circle drop a perpendicular on a chord

Ex. 553. Cut out of paper an acute-angled triangle; by folding construct the perpendiculars from each vertex to the opposite side.

Ex. 554. Cut out a paper triangle ABC (4o B and C being acute); by folding construct AD perpendicular to BC. Again fold so that A, B and C all fall on D.

Ex. 555. Cut out of paper an equilateral triangle ABC (see Ex. 540). Construct two of the perpendiculars from the vertices to the opposite sides; let them intersect at O. Fold so that A falls on O, and then so that B and

C fall on O. What is the resulting figure?

CONSTRUCTION OF TRIANGLES FROM GIVEN DATA.

We have seen how to construct triangles having given
(i) the three sides (Ex. 99-102, and p. 104);
(ii) two sides and the included angle (Ex. 87, 88, 513);
(iii) one side and two angles (Ex. 89, 90, 514, 515).

1. 14, 10, 11 prove that if a set of triangles were constructed from the same data, such as those given above, they would all be congruent.

In Ex. 146-150, we saw that, given the angles, it is possible to construct an unlimited number of different triangles.

If two angles of a triangle are given, the third angle is known; hence the three angles do not constitute more than two data.

We have still to consider the case in which two sides are given and an angle not included by these sides.

G. S.

8

Ex. 556.

Construct a triangle ABC having given BC = 2·4 in.,

CA 1.8 in., and B = 32°.

First make BC= 2.4 in. and

CBD = 32°.

A must lie somewhere on BD, and must be 1.8 in. from C.
Where do all the points lie which are 1.8 in. from C?

How many points are there which are on BD and also 1.8 in. from C?

We see that it is possible to construct two unequal triangles which satisfy the given conditions. This case is therefore called

Ex. 558.

Prove (theoretically) that the two triangles obtained in

Ex. 557 (iv) are congruent.

We may summarise the cases of congruence of triangles as

MISCELLANEOUS EXERCISES.

CONSTRUCTIONS.

Ex. 559. Construct angles of (i) 135°; (ii) 105°; (iii) 221° (without protractor or set square).

Ex. 560. On a given straight line describe an isosceles triangle, having each of its equal sides double the base.

Are the base angles double the vertical angle?

Ex. 561.

Describe a circle and on it take three points A, B, C; join Bisect angle BAC and draw the perpendicular bisector of BC. Produce the two bisectors to meet.

BC, CA, AB.

Ex. 562. Having given two angles of a triangle, construct the third angle (without protractor).

Ex. 563. Draw an isosceles triangle ABC; on the side AB describe an isosceles triangle having its angles equal to the angles of the triangle ABC (without protractor).

Ex. 564. Describe a right-angled triangle having given its hypotenuse and one acute angle.

Ex. 565. Construct a triangle ABC having AB=3 in., BC=5 in., and the median to BC=2.5 in. Measure CA.

Ex. 566. Construct a triangle ABC having given AB = 10 cm., AC=8 cm., and the perpendicular from A to BC=7.5 cm. Measure BC. Is there any ambiguity?

[First draw the line of the base, and the perpendicular.]

Ex. 567. Construct a triangle ABC having given AB=11.5 cm., BC=4.5 cm., and the perpendicular from A to BC=8·5 cm. Measure AC. Is there any ambiguity?

Ex. 568.

angles.

Construct a quadrilateral having given its sides and one of its

Ex. 569. Four of the sides, taken in order, of an equiangular hexagon are 1, 3, 3, 2 inches respectively: construct the hexagon and measure the remaining sides.

[What are the angles of an equiangular hexagon ?]

Ex. 570. Construct an isosceles triangle having given the base and the perpendicular from the vertex to the base. Give a proof.

[See Ex. 455.]

Ex. 571. A, B are two points on opposite sides of a straight line CD; in CD find a point P such that LAPC=LBPD. Give a proof.

Ex. 572. A, B are two points on the same side of a straight line CD; in CD find a point P such that APC=LBPD. Give a proof.

[From A draw AN perpendicular to CD and produce it to A' so that NA' NA; if P is any point in CD, ▲ APN and A'PN can be proved equal; in fact, A and A' are symmetrical points with regard to CD.]

Ex. 573. On a given base construct an isosceles triangle having given the sum of the vertical angle and one of the base angles.

Ex. 574. Construct a triangle, having one angle four times each of the other two. Find the ratio of the longest side to the shortest.

[First calculate the angles.]

Ex. 575. On a given base construct an isosceles triangle, having its vertical angle equal to a given angle. Give a proof.

Ex. 576.

Give a proof.

Construct an equilateral triangle with a given line as median.

Ex. 577. Through one vertex of a given triangle draw a straight line cutting the opposite side, so that the perpendiculars upon the line from the other two vertices may be equal. Give a proof.

[See Ex. 670.]

Ex. 578. From a given point, outside a given straight line, draw a line making with the given line an angle equal to a given angle. (Without protractor.) Give a proof.

[Use parallels.]

Ex. 579. Through a given point P draw a straight line to cut off equal parts from the arms of a given angle XOY. Give a proof.

[Use parallels.]

Ex. 580. Draw a triangle ABC in which B is less than LC. In AB find a point P such that PB = PC.

Ex. 581. In the equal sides AB, AC of an isosceles triangle ABC find points X, Y such that BX = XY=YC. Give a proof.

Ex. 582.

THEOREMS.

How many diagonals can be drawn through one vertex of (i) a quadrilateral, (ii) a hexagon, (iii) a n-gon?

Ex. 583. How many different diagonals can be drawn in (i) a quadrilateral, (ii) a hexagon, (iii) a n-gon?

Ex. 584.

The bisectors of the four angles formed by two intersecting straight lines are two straight lines at right angles to one another.

Ex. 585. If the bisector of an exterior angle of a triangle is parallel to one side, the triangle is isosceles.

Ex. 586. The internal bisectors of two angles of a triangle can never be at right angles to one another.

Ex. 587. AB, CD are two parallel straight lines drawn in the same sense, and P is any point between them. Prove that BPD=LABP+LCDP.

Ex. 588. ABC is an isosceles triangle (AB=AC). A straight line is drawn at right angles to the base and cuts the sides or sides produced in D and E. Prove that A ADE is isosceles.

Ex. 589. From the extremities of the base of an isosceles triangle straight lines are drawn perpendicular to the opposite sides; show that the angles which they make with the base are each equal to half the vertical angle.

Ex. 590. The medians of an equilateral triangle are equal.

Ex. 591. The bisector of the angle A of a triangle ABC meets BC in D, and BC is produced to E. Prove that LABC+LACE=24ADC.

Ex. 592. From a point O in a straight line XY, equal straight lines OP, OQ are drawn on opposite sides of XY so that YOP=YOQ. Prove that A PXY= A QXY.

Ex. 593. The sides AB, AC of a triangle are bisected in D, E; and BE, CD are produced to F, G, so that EF = BE and DG=CD. Prove that FAG is a straight line.

Ex. 594. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet, show that they contain an angle equal to an exterior angle at the base of the triangle.

Ex. 595. The bisectors of the angles B, C of a triangle ABC intersect at I; prove that BIC 90°+LA.

Ex. 596. XYZ is an isosceles right-angled triangle (XY = XZ); YR bisects Y and meets XZ at R; RN is drawn perpendicular to YZ. Prove that RN XR.

Ex. 597. The perpendiculars from the vertices to the opposite sides of an equilateral triangle are equal to one another.