Proposition XX

103. 1. Draw a straight line and a perpendicular to that line at its middle point; select any point in the perpendicular and from that point draw straight lines to the extremities of the given line. How do these lines compare in length? How do the angles made by these lines with the perpendicular compare in size? How do the angles made by these lines with the given line compare?

2. Select any point not in the perpendicular and from that point draw straight lines to the extremities of the given line. How do they compare in length?

3. Draw a straight line and find a point equidistant from its extremities; find another point equidistant from its extremities; connect these points by a line and if necessary extend it until it intersects the given line. At what point does it intersect the given line? What kind of angles does it make with the given line?

4. What line contains every point that is equidistant from the extremities of a straight line?

Theorem. If a perpendicular is drawn to a straight line at its middle point,

1. Any point in the perpendicular is equidistant from the extremities of the line.

2. Any point not in the perpendicular is unequally dis tant from the extremities of the line.

Data: Any straight line, as AB; a perpendicular to it at its middle point, as CD; any point in CD, as E; and any point not in CD, as F.

G

and, § 101,

That is,

AE BE.

E is equidistant from A and B.

2. From the point G, where AF cuts CD, draw GB.

That is,
F is unequally distant from A and B.
Therefore, etc.

Q.E.D.

104. Cor. I. Every point that is equidistant from the extremities of a straight line lies in the perpendicular at the middle point of that line.

105. Cor. II. If a perpendicular is erected at the middle point of a straight line, the lines joining the extremities of this line with any point in the perpendicular make equal angles with the line and also with the perpendicular.

§ 101

106. Cor. III. Two points each equidistant from the extremities of a straight line determine the perpendicular at the middle point of that line.

Ex. 45. How does the distance between two parallel lines at a given point compare with the distance between them at any other point?

Ex. 46. Can two angles which are not adjacent have a common vertex and a common side?

Ex. 47. If in an equilateral triangle a line is drawn from the vertex to the middle point of the base, how do the triangles thus formed compare in size?

Ex. 48. If two lines bisect each other, in what direction do the lines joining their opposite extremities extend with reference to each other?

Ex. 49. If two sides of a triangle are equal, and a line is drawn bisecting their included angle and intersecting the third side, how do the segments of the third side compare in length ?

Ex. 50. Perpendiculars are erected at the extremities of a line and terminate in any bisector of the line that is not perpendicular to the line. How do the perpendiculars compare in length ?

Ex. 51. If through the middle point of a straight line terminating in two parallel lines, a second straight line is drawn also terminating in the parallels, how do the parts of the second line compare in length?

Proposition XXI

107. 1. Make two triangles such that the sides of one shall be equal to the corresponding sides of the other. How do the triangles compare? How do the corresponding angles compare?

2. Under what conditions are two triangles equal?

Theorem. Two triangles are equal, if the three sides of one are equal to the three sides of the other, each to each.

Data: Any two triangles, as ABC and DEF, in which AB = DE, ACDF, and BC= EF.

To prove

triangles ABC and DEF equal.

Proof. Place ▲ DEF in the position ABF so that the equal sides, DE and AB, coincide, and the vertex F falls opposite C. Draw CF.

Data,

hence, § 106,

A and B are each equidistant from F and C;

AB 1 CF at its middle point,

Prove by placing the triangle so that

(1) DF will coincide with AC.
(2) EF will coincide with BC.

Q.E.D.

108. Sch. It is evident that in equal triangles the parts which are similarly situated are equal; that is, the angles included by the equal sides are equal, the angles opposite the equal sides are equal, the sides included between equal angles are equal, and the sides opposite the equal angles are equal.

109. In equal figures, the parts which are similarly situated are called Homologous parts.

Ex. 52. Draw two parallel lines intersecting two parallel lines, and draw a line joining two opposite points of intersection. How do the triangles thus formed compare?

Ex. 53. Perpendiculars are drawn from the line that bisects it and is not perpendicular to it. compare in length?

extremities of a line to any How do the perpendiculars

Ex. 54. The line BD is the bisector of the angle ABC whose sides are equal. Lines are drawn from any point of BD, as E, to A and C. How do AE and CE compare in length?

Ex. 55. In a triangle ABC angle A equals angle B; a line parallel to AB intersects AC in D and BC in E. How do the angles ADE and BED compare ?

Ex. 56. If D is the middle point of the side BC of the triangle ABC, and BE and CF are perpendiculars from B and C to AD, or AD produced, how do BE and CF compare in length ?

Proposition XXII

110. 1. Cut out a paper triangle ABC. Cut off the corners and place the vertices A, B, and C together. To how many right angles is the sum of the three angles equal?

2. If one angle of a triangle is a right angle, how does the sum of the other two angles compare with a right angle?

3. What is the greatest number of obtuse angles that a triangle may have? The greatest number of right angles?

4. If there are two triangles such that the sum of two angles of one is equal to the sum of two angles of the other, how do the third angles compare in size?

5. If there are two right triangles such that a side and an acute angle of one are equal to the corresponding parts of the other, how do the triangles compare?

6. Extend one side of a triangle through a vertex; through the same vertex draw a line parallel to the opposite side of the triangle. Since the figure thus formed contains two parallel lines and a transversal, what angles of the figure are equal? How does the exterior angle of the triangle compare with the sum of the two opposite interior angles?

Theorem. The sum of the angles of a triangle is equal to two right angles.

.. substituting Zs and Zt for Zs' and t' in the first equation, <r+ Ls + ≤t = 2 rt. .

Therefore, etc.

Q.E.D.

111. Cor. I. In a right triangle the sum of the two acute angles is equal to a right angle.

112. Cor. II. A triangle cannot have more than one right angle, nor more than one obtuse angle.

113. Cor. III. If two angles of one triangle are equal to two angles of another, the third angles are equal.

114. Cor. IV. Two right triangles are equal, if a side and an acute angle of one are equal to a side and an acute angle of the other, each to each.

115. Cor. V. Any exterior angle of a triangle is equal to the sum of the two opposite interior angles.

1. What interior angle is equal to t'?

Why?
Why?

3. To what, then, is the whole exterior angle equal?

Ex. 57. May a triangle be formed whose angles are 93°, 40°, and 61° respectively? 98°, 24°, and 58° ? 57°, 49°, and 74° ?

Ex. 58. Two angles of a triangle are together equal to 76°. What is the value of the third angle?