22. A triangle is called equilateral if it has its three sides equal, isosceles if it has at least two sides equal,

scalene if it has no two sides equal, equiangular if it has its three angles equal.

Select each kind from the figures on this page.

23. A triangle is called a right triangle if it has one right angle, an obtuse triangle if it has one

obtuse angle, an acute triangle if all its angles are acute.

Select each kind from the figures on this page.

The side of a right triangle opposite the right angle is called the hypotenuse in dis

Hypotenuse

tinction from the other two sides, which are sometimes called its legs..

24. The side of a triangle on which it is supposed to stand is called its base. called the vertex angle, and its vertex is the vertex of the triangle.

The angle opposite the base is

The altitude of a triangle is the perpendic

ular from the vertex to

Vertex

Altitude

Base,

Vertex

Altitude

Base

the base or the base produced. Evidently any side may be taken as the base, and hence a triangle has three different altitudes.

1. Is every equilateral triangle also isosceles? Is every isosceles triangle also equilateral?

2. Is a right triangle ever isosceles? Is an obtuse triangle ever isosceles? Draw figures to illustrate your answers.

3. In the figure on page 4 determine by measuring sides which of the triangles HNP, LKW, IHN, MIJ, KVU, OKJ, LVW, are isosceles, which are equilateral, and which are scalene.

4. Determine whether J, K, V of the same figure may be the vertices of a triangle; also whether J, O, G may be.

5. Pick out ten obtuse triangles in this figure; also ten acute triangles.

CONGRUENCE OF GEOMETRIC FIGURES.

26. In comparing geometric figures it is assumed that they may be moved about at will, either in the same plane or out of it, without changing their shape or size.

27. Two figures are said to be similar if they have the same shape. This is denoted by the symbol~, read is similar to.

For a more precise definition see §§ 255, 256.

Two figures are said to be equivalent or simply equal if they have the same size or magnitude. This is denoted by the symbol, read is equivalent to or is equal to.

Two figures are said to be congruent if, without changing the shape or size of either, they may be so placed as to coincide throughout. This is denoted by the symbol , read is congruent to.

=

~

to denote

In the case of line-segments and angles, congruence is determined by size alone. Hence in these cases we use the symbol congruence, and read it equals or is equal to.

28. It is clear that if each of two figures is congruent to the same figure they are congruent to each other.

Hence if we make a pattern of a figure, say on tracing paper, and then make a second figure from this pattern, the two figures are congruent to each other.

29. If ABC≈AA'B'C', the notation of the triangles may be so arranged that AB = A'B',

BC= B'C', CA =C'A', LA=L A', Z B=ZB' and C=c'. In this case AB is said to correspond to A'B', BC to B'C', CA to C'A', ZA to ZA', etc.

B

Hence, we say that corresponding parts of A congruent triangles are equal.

B

1. Using tracing paper, draw triangles congruent to the triangles MIN, NHP, OAB, OFE, OKL, UKV, OGL on page 4, and by applying the pattern of each triangle to each of the others determine whether any two are congruent.

2. Find as in § 28 whether any two of three accompanying triangles are congruent, and if so arrange the notation so as to show the corresponding parts.

3. Give examples of figures which are similar, equal, or congruent, different from those in § 27.

4. If two figures are congruent, does it follow that they are equal? Similar?

5. If two figures are similar, does it follow that they are equal? Congruent?

6. If two figures are equal, are they similar? Congruent?

TESTS FOR CONGRUENCE OF TRIANGLES.

31. The method of determining whether two triangles are congruent by making a pattern of one and applying it to the other is often inconvenient or impossible. There are other methods in which it is necessary only to determine whether certain sides and angles are equal.

These methods are based upon three important tests for congruence of triangles.

32.

First Test for Congruence of Triangles.

If two triangles have two sides and the included angle of one equal respectively to two sides and the included angle of the other, the triangles are congruent. This may be shown by the following argument:

Let ABC and A'B'C' be two triangles in which AB= A'B', AC A'C', and ZA ZA'.

We are to show that ▲ ABC≈▲ A'B'C'.

Place ABC upon AA'B'C' so that Z4 coincides with ZA', which can be done since it is given that ZA=ZA'. Then point B will coincide with B' and C with c', since it is given that AB = A'B' and AC = A'C'.

Hence, side BC will coincide with B'C' (§ 8).

Thus, the two triangles coincide throughout and hence are congruent (§ 27).

The process just used is called superposition. It B' may sometimes be necessary to move a figure out of its plane in order to superpose it upon another, as in the case of the accompanying triangles.

C'

B

33. The equality of short line-segments is conveniently tested by means of the dividers or compasses.

Place the divider points on the end-points of one segment AB and then see whether they will also coincide with the end-points of the other segment A'B'. If so, the two segments are equal.

The equality of two angles may be tested by means of

the protractor.

Place the protractor on one angle BOC as shown in the figure and read the scale where OC crosses it. Then place the protractor on the other angle B'O'C' and see whether O'C" crosses the scale at the same point. If so, the two angles are equal.

B

1. Using the protractor determine which pairs of the following angles on page 4 are equal :

HPG, LGW, GWL, AOB, VLW, LVW.

2. By the test of § 32 determine whether, on page 4,

AJKUAGWL, also whether ▲ MIH≈▲ KVW.

First find whether two sides of one are equal respectively to two sides of the other, and if so compare the included angles.

3. Could two sides of one triangle be equal respectively to two sides of another and still the triangles not be congruent? Illustrate by constructing two such triangles.

4. Show by the test of § 32 that two right triangles are congruent if the legs of one are equal respectively to the legs of the other. Can this be shown directly by superposition?

5. Find the distance AB when, on account of some obstruction, it cannot be measured directly.

SOLUTION. To some convenient point C' measure the distances AC and BC. Continuing in the direction AC lay off C'A' = AC, and in the direction BC lay off CB' BC. Then 21 = 22 (see § 74). Test this with the protractor. Show that the length AB is found by measuring A'B'.

B'

Α'

B