82. Combined statement. For brevity, denote the angles of a triangle by the single capital letters A, B, C, and the opposite sides by the corresponding small letters a, b, c; then, by Theorems 8 and 14 the following statements are true: (1) If b is equal to c, then B is equal to C;

(2) Ifb is greater than c, then B is greater than C; (3) If b is less than c, then B is less than C.

These three statements may be conveniently combined into one general statement as follows:

According as one side of a triangle is greater than, equal to, or less than, another side, so is the angle opposite the first side greater than, equal to, or less than, the angle opposite the second side.

Here the word If is replaced by the distributive phrase According as, and the word then by the words so is.

Converse of 81.

83. THEOREM 15. If one angle of a triangle is greater than another, then the side opposite the greater angle is greater than the side opposite the less. In the triangle ABC, let the angle ABC be greater than ACB.

A

B

To prove that the side AC is greater than AB.

The side AC is either equal to, less than, or greater than, the side AB.

Now AC is not equal to AB; for then the angle B would be equal to C (59), contrary to the hypothesis.

Again, 4C is not less than AB; for then the angle B would be less than C (81), contrary to the hypothesis.

It only remains that the side AC is greater than AB.

84. Combined statement. By 62 and 83 the following statements are true:

If B is equal to C, then b is equal to c;

If B is greater than C, then b is greater than c;

If B is less than C, then b is less than c.

These three statements are respectively converse to those of Art. 82; and, like them, may be combined into one complete statement as follows:

According as one angle of a triangle is greater than, equal to, or less than, another, so is the side opposite the first angle greater than, equal to, or less than, the side opposite the second angle.

Perpendicular and oblique lines.

85. THEOREM 16. Of all the straight lines that can be drawn from a given point to a given line:

(1) the perpendicular is the least;

(2) any two that make equal angles with the perpendicular are equal;

(3) one that makes a greater angle with the perpendicular is greater than one that makes a less angle. Let OP be the perpendicular

from the given point to the
given line LL'. Let ON, OQ be
any lines making equal angles
NOP, QOP, with OP. Let OR
make with OP the angle POR
greater than the angle POQ or L

NOP.

(1) To prove that OP is less than OQ.

N P ♦ Ꭱ

The angle OQP is less than the exterior angle OPL (79). Now the angles OPL and OPQ are equal, being right angles. Hence OQP is less than OPQ; therefore the opposite side OP is less than OQ (83).

(2) To prove that the lines ON, OQ are equal. [Apply 65.] (3) To prove that OR is greater than OQ or ON.

The angle OQR is greater than the right angle OPR (79); which equals the right angle OPN; which is greater than the interior angle ORQ (79).

Hence the angle OQR is greater than ORQ (30); and therefore the side OR is greater than oQ (83).

86. Cor. In an isosceles triangle, a line joining the vertex to any point in the base is less than either side; and a line joining the vertex to any point in the base extended is greater than either side.

Sum of two sides.

87. THEOREM 17. Any side of a triangle is less than the sum of the other two.

Let ABC be a triangle.

To prove that any side AB is less than

the sum of the other two sides AC and BC.

Prolong the side AC until the prolongation CD equals the side CB; and draw BD.

B

In the isosceles triangle BCD, the angle CBD equals CDB. Hence the whole angle ABD is greater than the angle CDB. Therefore, in the triangle ADB, the side AD, opposite the greater angle, is greater than the side AB (83).

Now AD equals the sum of AC and CD, which equals the sum of AC and CB.

Therefore the sum of AC and CB is greater than AB.

88. Cor. I. Any side of a triangle is greater than the difference of the other two.

89. Cor. 2. Any straight line is less than the sum of the parts of a broken line having the same extremities.

MCM. ELEM. GEOM. 4

90. Cor. 3. If from the ends of a side of a tria straight lines are drawn to a point within the triang sum is less than the sum of the other two sides of the tr

Definition. The sum of the three sides is called the eter of the triangle.

Ex. The sum of the lines joining any point within a triang three vertices is less than the perimeter of the triangle, and than half the perimeter.

A case of unequal triangles.

91. THEOREM 18. If two triangles have two of one respectively equal to two sides of the othe the included angle in the first greater than t cluded angle in the second, then the third side first is greater than the third side of the second

Let the two triangles ABC and A'B'C' have the sides respectively equal to A'B', B'C', and the included angl greater than A'B'C'.

To prove that the third side AC is greater than A'C'. Draw the line BC" making the angle ABC" equal t less angle B'. Take BC" equal to B'C'; and draw AC",

First let the point c" fall within the triangle ABC. M and N be points on the prolongations of BC and BC". The triangles ABC" and A'B'C' are equal; and the AC", A'C' are equal (64).

In the isosceles triangle BCC", the angles cc''N and C below the base are equal (60).

Hence the whole angle AC"C, being greater than one of the equal angles, is greater than ACC", which is a part of the other. Therefore, in the triangle AC"C, the side AC, being opposite the greater angle, is greater than 4C" (83).

Therefore AC is greater than A'C'.

Next let c" fall without the triangle ABC.

The proof for the second figure is left to the student; also the consideration of the intermediate case in which C" falls on AC.

Combined statement.

92. Cor. I. If two triangles have two sides of one equal to two sides of the other, then according as the vertical angle of the first is greater than, equal to, or less than, the vertical angle of the second, so is the base of the first greater than, equal to, or less than, the base of the second. [Combine 64 and 91.]

93. Cor. 2 (Converse of 91). If two triangles have two sides of the first equal respectively to two sides of the second, and the base of the first greater than the base of the second, then the vertical angle of the first is greater than that of the second. [Prove by exclusion, using 92; see 83.]

Combined statement.

94. Cor. 3. If two triangles have two sides of the first equal to two sides of the second, then, according as the base of the first is greater than, equal to, or less than, the base of the second, so is the vertical angle of the first greater than, equal to, or less than, the vertical angle of the second. [66, 93.]