85. Classification of triangles with reference to character of their angles. A right triangle is a triangle one of whose angles is a right angle. An obtuse, triangle is a triangle one of whose angies is an obtuse angle. An acute triangle is a triangle all of whose angles are acute angles. An equiangular triangle is one in which all the angles are equal.

Right

Obtuse

Acute

Equiangular

The angle opposite

86. The base of a triangle is the side upon which the triangle is supposed to stand, as AB. the base is called the vertex angle, as angle ACB; the vertex of a triangle is the vertex of the vertex angle of the triangle.

The altitude of a triangle is the perpendicular from the vertex to the base or base extended, as CD.

87. In an isosceles triangle, the legs are the equal sides, and the base is the remaining side.

88. In a right triangle, the hypotenuse is the side opposite the right angle, and the legs are the sides adjacent to the right angle.

89. Altitudes, bisectors, medians. In any triangle, any side may be taken as the base; hence the altitudes of a triangle are the three perpendiculars drawn one from each vertex to the side opposite.

C

A bisector of an angle of a triangle is a line which divides this angle into two equal parts. This bisector is usually produced to meet the side opposite the given angle.

A median of a triangle is a line drawn from a vertex of the triangle to the middle point of the opposite side. How many medians has a triangle?

90. Two mutually equiangular triangles are triangles having their corresponding angles equal.

91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.

Homologous sides of two mutually equiangular triangles are sides opposite homologous angles in those triangles.

We shall now proceed to determine first, the properties. of a single triangle, as far as possible, then those of two triangles.

92. Property of a triangle immediately inferred. The sum of any two sides of a triangle is greater than the third For a straight line is the shortest line between two points (Art. 15.)

side.

Ex. 1. Point out the hypothesis and conclusion in the general enunciation of Prop. III; also point them out in the particular enunciation. (As each of the next fifteen Props. is studied, let the pupil do the same for it.)

Ex. 2. Find the angle whose complement is 18°; whose supplement is 76°

Ex. 3. If the complement of an angle is known, what is the shortest way of finding the supplement of the angle? If the supplement is known, what is the shortest way of finding the complement?

Ex. 4. In 25 minutes, how many degrees does the minute-hand of a clock travel? How many does the hour-hand?

Ex. 5. Draw three straight lines so that they shall intersect in three points; in two points; in one point.

PROPOSITION IV. THEOREM

93. Any side of a triangle is greater than the difference between the other two sides.

B

A

Given AB any side of the ▲ ABC, and AC>BC.

To prove

Proof.

AB>AC- BC.

AB+BC>AC,

Art. 92.

(the sum of any two sides of a triangle is greater than the third side). Subtracting BC from each member of the inequality,

if equals be subtracted from unequals, the remainders are unequal in the same order).

Q. E. D.

94. COR. The perpendicular is the shortest line that can be drawn from a given point to a given line.

Hence, DEF. The distance from a point to a line is the perpendicular drawn from the point to the line.

Ex. 1. If one side of an equilateral triangle is 4 inches, what is its perimeter ?

Ex. 2. Is it possible to form a triangle whose sides are 6, 9 and 17 inches? Try to do this with the compasses and ruler.

Ex. 3. Is it possible to form a triangle in which one side is 10 inches and the difference of the other two sides is 12 inches?

Ex. 4. On a given line as base, by exact use of ruler and compasses, construct an equilateral triangle.

PROPOSITION V. THEOREM

95. If, from a point within a triangle, to lins be azurn to the extremities of one side of the triangle, the sum of the other two sides of the triangle is greater than the sum of the two lines so drawn.

B

Given P any point within the triangle ABC, and PA and PC lines drawn from P to the extremities of the side AC.

To prove

Proof.

Then

AB+ BC > AP+ PC.

Produce the line AP to meet BC at Q.

AB + BQ > AP + PQ,

Art. 15.

(a straight line is the shortest line connecting two points).

Adding these inequalities,

AB+BQ+PQ+ QC > AP+ PQ+ PC. Ax. 9.

Substituting BC for its equal BQ + QC,

Ex. On a given line as base, by exact use of ruler and compasses, construct an isosceles triangle each of whose legs is double the base.

PROPOSITION VI. THEOREM

96. Two triangles are equal if two sides and the included angle of one are equal, respectively, to two sides and the included angle of the other.

Given the triangles ABC and DEF in which AB=DE, AC=DF, and ≤ A= ≤ D.

Proof. Place the ▲ ABC upon the ▲ DEF so that the line AC concides with its equal DF.

Geom. Ax. 2.

Then the line AB will take the direction of DE, (for LA = LD by hyp.).

Also

the point B will fall on E,

(for line AB = line DE by hyp.).

Hence the line BC will coincide with the line EF, Art. 66. (only one straight line can be drawn connecting two points).

:. A ABC and DEF coincide.

:. Δ ΑΒΟ A DEF,

(geometric figures which coincide are equal).

Art. 47.

Q. E. D.

Ex. 1. What kind of proof is used in Prop. VI? (See Art. 62).

Ex. 2. If 4 A, B and C=60°, 70°, 50°, AB=16, AC≈19, BC=18; also D=60°, DE=16, DF=19: find 4 E and F and side EF without measuring them.