PROPOSITION XXVII

100. If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior non-adjacent angles.

E

B

Let ABC be any triangle having the side BC produced to D.

It is required to prove that the exterior angle ACD is equal to the sum of the two interior non-adjacent angles, viz. the angles CAB and CBA.

Proof. From C draw the straight line CE parallel to BA. (Prop. XXVI.) Then ECD = ≤ CBA and ≤ ACE = ≤ CAB. Why? Therefore the sum of Zs ACE and ECD equals the sum of Zs CAB and CBA

That is,

ACD equals the sum of Zs CAB and CBA.

101. COROLLARY I. The sum of the three angles of a triangle is equal to two right angles.

102. COROLLARY II. The two acute angles of any right triangle are complementary; i.e. their sum is a right angle.

DEFINITION. One angle is said to be the complement of another when the sum of the two is a right angle.

103. COROLLARY III.

Each angle of a triangle is the supple

ment of the sum of the other two.

104. COROLLARY IV. If two triangles have two angles of the one equal respectively to two angles of the other, their third angles are also equal; or, if the sum of two angles in the one is equal to the sum of two angles in the other, their third angles are equal.

105. COROLLARY V. If the boundaries of one angle are respectively parallel or perpendicular to the boundaries of another, these two angles are either equal or supplementary.

Make diagrams representing all the cases.

106. COROLLARY VI. If two triangles have two angles of the one equal respectively to two angles of the other, and the sides opposite one pair of equal angles also equal, the triangles are identically equal.

For then the third angles are also equal, and the theorem falls under Proposition V. This theorem may be easily proved by superposition, making use of Proposition XII.

EXERCISES

1. If an isosceles triangle is right-angled, each of the angles, at the base is half a right angle.

2. If two isosceles triangles have their vertical angles equal, they are mutually equiangular; i.e. each angle of the one is equal to an angle of the other.

3. If one angle of a triangle is equal to the sum of the other two, it must be a right angle.

4. Divide a right triangle into two isosceles triangles, and hence show that the midpoint of the hypotenuse is equidistant from A the three vertices.

D

B

SUGGESTION. Construct ACD = ZA, and show that BCD = 2 B. Hence DA= DC = DB.

5. Each angle of an equilateral triangle is two-thirds of a right angle.

PROPOSITION XXVIII

107. Every point on the bisector of an angle is equidistant from the boundaries of the angle; and every point within the angle which is equidistant from the boundaries is on the bisector; that is, the bisector of an angle is the locus of points within the angle which are equidistant from its boundaries.

First, let AD be the bisector of the angle BAC and let P be any point on AD.

It is required to prove that P is equidistant from AB and AC.

Proof. From P draw the lines PM and PN perpendicular, respectively, to AB and AC.

Then As PAM and PAN are identically equal. Why? (Apply Prop. V.)

Therefore PM equals PN; that is, P is equidistant from AB and AC.

Next, let

be any point equidistant from AB and AC within the angle BAC.

It is required to prove that Q lies on the bisector of the angle BAC.

Proof. From Q draw QS and QT perpendicular to AB and AC, respectively, and join QA.

By hypothesis, QS and QT are equal.

Then As QSA and QTA are identically equal. Why?

(Apply Prop. XIX.) Therefore

QAS equals QAT; that is, Qlies on the bisector of the angle BAC.

Since the bisector of the angle BAC contains all points within the angle which are equidistant from the boundaries, and no point which is not equidistant from the boundaries, it is the locus of such points.

108. COROLLARY. The locus of points equidistant from two intersecting straight lines consists of the two lines which bisect the angles formed by the given lines.

EXERCISES

1. Prove that the locus of points equidistant from two intersecting straight lines consists of two straight lines at right angles.

2. If the straight line drawn from one vertex of a triangle to the midpoint of the opposite side is equal to half of this side, prove that the triangle has one right angle.

3. If, in a right triangle, a perpendicular is drawn from the vertex of the right angle to the opposite side, the two triangles so formed are equiangular with each other and with the whole triangle.

4. A straight line drawn perpendicular to the base BC of an isosceles triangle ABC meets the side AB at E and the side CA produced at F. Prove that the triangle EAF is isosceles.

5. If a point is equidistant from two parallel straight lines, any linesegment drawn through it and terminated by the parallel lines is bisected at the point.

6. If a point is equidistant from two parallel straight lines, any two straight lines drawn through it intercept equal segments of the parallel lines.

7. Construct a triangle having given two angles and the length of the perpendicular from the third vertex to the opposite side.

8. If two exterior angles of a triangle be bisected, and from the point of intersection of the bisecting lines a straight line is drawn to the third vertex, it bisects the third angle. (Apply Proposition XXVIII.)

SECTION III

CLOSED RECTILINEAR FIGURES OF MORE THAN
THREE SIDES

109. On page 13 a closed figure was defined as one which can be traversed by starting at any point of it, and moving continuously along the lines of the figure in order, returning to the same point without passing twice over any portion of the figure.

A closed rectilinear figure is one which is made up wholly of line-segments and the points in which they intersect, two and two, in order.

The line-segments are called the sides of the figure, and the points in which the sides intersect are called the vertices of the figure.

Two sides which intersect in a vertex are called adjacent sides.

The angles formed by pairs of adjacent sides are called the angles of the figure.

A straight line joining any two vertices not on the same side is a diagonal.

In the diagram, the points A, B, C, D, E are vertices, the line-segments AB, BC, CD, etc., are sides, and AC, AD, CE, etc., are diagonals.

B

If the number of sides of any closed rectilinear figure is n, the number of diagonals which can pass through any one vertex is n-3, and since each diagonal passes through two vertices, the total number of diagonals is n(n − 3).