TRIANGLES.

85. A triangle is a plane figure bounded by three straight lines.

The three straight lines which bound a triangle are called its sides. Thus, AB, BC, CA, are the

sides of the triangle ABC.

The angles of the triangle are the angles formed by the sides with each other; as BAC, ABC, ACB. The ver

tices of these angles are also called the D A vertices of the triangle.

86. An exterior angle of a triangle is the angle formed between any side and the continuation of another side; as CAD.

The angles BAC, ABC, BCA are called interior angles of the triangle. When we speak of the

angles of a triangle we mean the three interior angles.

87. An equilateral triangle is one whose three sides are equal.

88. An isosceles triangle is one which has two equal sides.

89. A scalene triangle is one which has three unequal sides.

90. A right-angled triangle is one which has a right angle.

The side opposite the right angle is called the hypotenuse, and the other two sides the legs.

91. An acute-angled triangle is one which has three acute angles.

It will be shown hereafter that every triangle must have at least two acute angles.

92. An obtuse-angled triangle is one which has an obtuse angle.

When all the angles are equal it is called an equiangular triangle.

93. The base of a triangle is the side upon which it is supposed to stand.

Either side may be taken as the base; but in an isosceles triangle the side which is not one of the equal sides is called the base.

94. When one side of a triangle has been taken as the base, the angle opposite is called the

vertical angle, and its vertex is called the vertex of the triangle.

95. The altitude of a triangle is the perpendicular let fall from the vertex A to the base, or the base produced.

D

B

Thus, in the triangle ABC, AB is the base, ACB is the vertical angle, C is the vertex, and CD is the altitude.

When two triangles have the angles of the one equal respectively to the angles of the other, the angles which are equal are called homologous angles, and the sides which are opposite the equal angles are called homologous sides; hence in equal triangles the homologous sides are equal.

Proposition 19. Theorem.

96. Either side of a triangle is less than the sum, and greater than the difference, of the other two sides.

Proposition 20. Theorem.

97. The sum of the three angles of any triangle is equal

...ZABC+2CBE+/EBD=2ABC+2C+A. (Ax. 2)

But

Z ABC + /CBE+2 EBD = 2 rt. ≤ s.

The sum of all the s on the same side of a st. line at a point = 2rt. ≤8 (53). ... ZABC +≤ C + ≤ A = 2 rt. s.

(Ax. 1)

Q.E.D.

98. COR. 1. Since EBD = ≤ A, and ≤ CBE = /C,

../CBD ZA+ZC.

Hence, the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

99. COR. 2. If two angles of a triangle are given, or merely their sum, the third angle can be found by subtracting this sum from two right angles.

100. COR. 3. If two triangles have two angles of the one equal to two angles of the other, the third angles are equal. 101. COR. 4. A triangle can have but one right angle, or but one obtuse angle.

102. COR. 5. In any right-angled triangle the two acute angles are complementary.

103. COR. 6. Each angle of an equiangular triangle is two-thirds of a right angle.

Proposition 21. Theorem.

104. Two triangles are equal when two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.

Hyp. Let ABC, DEF be two

As, having AB = DE, AC = DF.

A

B

▲ DEF (29) so that the point A
shall fall on D, and the side AB on DE.

Now, since B falls on E, and C on F,

... BC must coincide with EF. (Ax. 11)

Hence, since BC coincides with EF,

<B

... BC= EF, B = ZE, C= /F.

Therefore the two As coincide in all their parts, and hence are equal (29).

Q.E.D.

NOTE.-When all the parts of one triangle are respectively equal to all the parts of another triangle, the triangles are said to be " equal in all respects." Such triangles are said to be identically equal, or congruent.

This proposition is proved by "the method of superposition," i.e., it is shown that one of the triangles could be placed on the other so as to cover it exactly without overlapping; and then, since all their parts would coincide, the two triangles are equal by definition (29).

Proposition 22. Theorem.

105. Two triangles are equal when a side and the two adjacent angles of the one are equal respectively to a side and the two adjacent angles of the other. Hyp. Let ABC, DEF be two As having A= 2 D, 2 B = < E, AB = DE.

To prove ABC = A DEF. Proof. Apply the ▲ ABC to the A DEF so that the point A shall fall on D, and the side AB

on DE.

AA

Because AC and BC fall upon DF and EF, respectively, ... the point C, falling upon both lines DF and EF, must fall at their point of intersection F.

... the two As coincide in all their parts, and are equal.

Q.E.D.

106. COR. 1. Two right-angled triangles are equal when the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other.

107. COR. 2. Two right-angled triangles are equal when a side and an acute angle of the one are equal respectively to a side and homologous acute angle of the other.