To describe a triangle when two sides and an angle opposite one of them are given.

a

b

85. Make an angle A, equal to the given angle; produce the sides, and mark off on one of them a line A C, equal to one of the given sides, and with C as centre and the other side as radius, describe an arc cutting the second side of the angle. When C is

A

B'

Fig. 74.

E

joined with the point of intersection, the required triangle is formed (fig. 74).

C

It is evident that: 1st. If the side C B, be less than the perpendicular from C on AE, the solution is impossible.

3rd. If a be greater than b, the arc cuts A E in one point only, and one triangle can be drawn fulfilling the conditions (fig. 75).

4th. If a be greater than the perpendicular, but less than b, the arc cuts A E in the points B and B', and either of the triangles ABC, A B'C, fulfils the given conditions.

To construct a right-angled triangle, having given the hypothenuse and one angle adjacent to it.

86. Draw a line equal to the hypothenuse, and make at one extremity an angle equal to the given angle; from the other extremity, draw a perpendicular

to the opposite side (fig. 76). It is evident that there

a

Fig. 76.

is only one solution.

87. From these constructions it follows that two triangles are equal,

Ist. When three sides of the one are equal to three sides of the other, each to each.

2nd. When two sides and the are respectively equal to two sides

included angle of one
and included angle of the other.

a

b

b

a

Fig. 77.

3rd. When two angles and the included side of one are respectively equal to two angles and included side of the other.

4th. When the triangles are rightangled, and the hypothenuse and a side of one are respectively equal to the hypothenuse and side of the other. 5th. When the triangles are right-angled, and have the hypothenuse and another angle of the one equal respectively to the hypothenuse and one angle of the other.

88. These propositions may also be proved independently of the constructions.

(a) If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another, the triangles will be equal in ail respects.

Let the two sides BA, A C of the triangle BAC (Fig. 78a), be respectively equal to the two sides E D, D F of the triangle EDF, and let the angle B A C=ED F, then will the triangles be equal in all respects.

Imagine the triangle A B C placed on D E F so that the point A falls on D, and the line AB on DE, then A C will fall on DF, because the angles A and D are equal. Also the point B will fall on E, because A B=DE; and C will fall on F, because AC=DF. Consequently BC will coincide with EF, the angle B will coincide with E, and the angle C with the angle F; hence the triangles are equal in all respects.

(b) If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another, these triangles will be equal in all respects.

(1) Let the angles B and C be respectively equal to E and F, and let the corresponding sides B C, E F, common to the equal angles, be equal: then shall the triangles be equal in all respects. For if the triangle A B C be placed on the triangle DEF, so that the base B C coincides with the equal base E F, BA will fall on E D, since the angle B=the angle E: and CA will fall on F D, since the angle C=the angle F.

A A

E

F

Therefore A will fall on D, and the triangles will coincide, consequently they are equal in all respects.

(2) Let the angles B, C, be respectively equal to E, F, and let the corresponding sides A B, DE, opposite to the equal angles C and F, be equal; the triangles are equal in all respects.

For if the triangle ABC be placed on DEF, so that A B coincides with the equal side D E, B C will fall on EF, since the angle B=the angle E, and AC cannot fall otherwise than on DF: for if it had any other direction we should have two straight lines from the same point, making the exterior angle equal to the interior angle, which is impossible. Consequently C falls on F, and the triangles coincide.

(c) When two triangles have two sides of the one respectively equal to two sides of the other, but the included angle of the one greater than the included angle of the other, the base of that which has the greater angle is greater than the base of the other.

B

да

Fig. 78 6.

Let A BC, DE F be the triangles, let A B=DE, AC=DF and the angle BAC be greater than ED F. Place D E F on

E

ABC so that the side DE coincides with A B, then evidently DF falls within the angle BAC on a line A F'. Bisect the

angle F'A C by the line AG, and and join F'G. Then since AFA C, the angle F'A G=C AG, and AG is common to the triangles F'A G and CAG; therefore, F'G=GC and BC =BG+GF', and is therefore greater than B F'.

(d) When three sides of one triangle are respectively equal to the three sides of another, the triangles are equal in all respects.

Let A B D E, A C=D F, and also B C E F, then the angle A will be equal to the angle D.

For if the angle A were greater than the angle D, B C would be greater than EF: and if the angle A were less than the angle D, BC would be less than E F; but B C is neither greater nor less than E F; therefore the angle A is neither greater nor less than D, but is equal to it. The equality of the triangles follows by proposition (a).

a

89. Except in the case of a right-angled triangle, we cannot

conclude that two triangles are equal which have two sides and an angle opposite one of them of the one, respectively equal to two sides and an angle of the other. If, however, it is known that of the two given sides in each triangle, the C side opposite the given angle is greater than the other, we may conclude the

Fig. 79. equality of the triangles.

90. In a triangle there are six parts, namely, three sides and three angles, and we have considered every possible arrangement of three that can be obtained from these six parts (§ 87), except that of the three angles. It is evident that the three angles of a triangle may be respectively equal to the three angles of another, while the triangles are unequal (fig. 78 b).

Questions for Examination.

1. Define a triangle.

2. Prove that if in a triangle two sides are equal, the two opposite angles are equal.

3. Prove that if in a triangle two angles are equal, the two opposite sides are equal.

4. When two angles of a triangle are unequal, the greater side is opposite the greater angle.

5. When two sides of a triangle are unequal, the angle opposite the greater is greater than that opposite the less.

6. An equilateral triangle is equiangular.

7. An equiangular triangle is equilateral. 8. Show how to trisect a given angle.

Describe a trisector.

9. Construct a triangle, having given :-Ist. The three sides. 2nd. Two sides and the included angle. 3rd. Two angles and included side. 4th. Two sides and an angle opposite one. Show when two triangles can be constructed fulfilling the conditions in this case, and when only one.

10. Construct a right-angled triangle, having given :-Ist. The hypothenuse and an angle adjacent to it. 2nd. The hypothenuse and another side.

II. Prove that two triangles are equal :-Ist. When two sides and the included angle of one are respectively equal to the corresponding parts of the other. 2nd. When two angles and the included side of the one are respectively equal to the corresponding parts of the other. 3rd. When the three sides of the one are respectively equal to the three sides of the other.

Theorems and Problems.

1. The line that joins the vertex to the middle point of the base of a triangle is less than half the sum of the two sides.

2. The perimeter of a triangle is greater than the sum of the lines joining the middle points of the sides with the opposite angles.

3. The perimeter of a triangle is greater than the sum of the straight lines which join any point in the interior of the triangle with the three angles, and is less than twice this sum.

4. If two angles of a triangle be bisected, the perpendiculars from the point of intersection of the bisectors to the sides are equal. 5. If two exterior angles of a triangle be bisected, the perpendiculars from the point of intersection of the bisectors to the sides are equal.

6. From two summer-houses I wish to make two straight walks of the same length to the same point in a distant road : how shall I find the point?

7. Describe an isosceles triangle on a given base, each of whose sides shall be double of the base.

8. Construct a triangle, having given the base, sum of the sides, and one of the angles at the base.

CHAPTER V.

PARALLELS AND QUADRILATERALS.
Parallels.

91. Two straight lines in the same plane, which do not meet however far they may be produced, are called parallel lines (fig. 80). Such would be two lines drawn on writing paper along the edges of a ruler. It is of