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CHAPTER IX

INEQUALITY

The two basic methods are:

130. The whole is greater than any one of its parts. (Art. 7a.)

131. A straight line is the shortest line between two points. (Axiom.)

132. State six general axioms of inequality.

133. Supplements or complements of unequal angles are unequal in reverse order.

*134. An exterior angle of a triangle is greater than either opposite interior angle.

*135. If two sides of a triangle are unequal, the opposite angles are unequal and the greater angle is opposite the greater side.

*136. If two angles of a triangle are unequal, the sides opposite are unequal and the greater side is opposite the greater angle.

137. The sum of two lines drawn from a point to the ends of a line is greater than the sum of two other lines similarly drawn but enveloped by them.

138. If two lines are drawn from a point in a perpendicular, cutting off unequal distances from the foot of the perpendicular, the more remote is the greater.

139. (Conversely.) If two unequal lines are drawn from a point in a perpendicular, cutting off distances from the foot of the perpendicular, the greater cuts off the greater distance.

*140. a. The perpendicular is the shortest line that can be drawn from a point to a line.

b. (Conversely.)

*141. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, the third side of the first is greater than the third side of the second.

*142. If two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, the included angle of the first is greater than the included angle of the second.

SUMMARY

143. State four methods of proving angles unequal. 144. State seven methods of proving lines unequal.

145. State an additional method of proving lines perpendicular.

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1. If point E be taken on side AB of triangle ABC, prove AB + AC greater than EB + EC.

2. In equilateral triangle ABC, extend AC to D. Prove angle ABD greater than angle ADB; _ ABD > LA; LA>D.

3. In equilateral triangle ABC, extend AC to D. Prove AD greater than BD, and BD greater than AB.

4. If in triangle ABC the median AE be drawn, making the acute angle AEB, prove AB less than AC.

5. H is a point on the base BC of isosceles triangle ABC. D is a point on the side AB extended through A. Fis a point on BC extended through C. K is a point outside of the triangle, to the right of A. Prove AB greater than A H.

6. In the same figure, prove AF greater than AB. 7. In the same figure, prove BD greater than DC. 8. In the same figure, prove B K greater than KC.

9. One side of a triangle is less than the sum of the other two and greater than their difference.

10. The sum of the lines drawn from a point within a triangle to its vertices is less than the perimeter of the triangle; is greater than one-half the perimeter.

CHAPTER X

LOCI

The basic principle is the definition:

The locus of points fulfilling certain conditions is the line or lines that contain all points under these conditions and no others.

147. Define Locus.

148. a. Place a point one inch from a given point 0. Place a second point one inch from 0. Place many points in the same way.

Draw a line through these points. What kind of line is it? This line is called the locus of points one inch from 0.

b. Place a point one inch from a given straight line AB Place a second point one inch from AB. Place many points in the same way.

Draw lines through these points. What kind of lines are they?

These lines form the locus of points one inch from AB.

c. Place a point midway between two parallel lines AB and CD. Place a second point in the same way. Place many points.

Draw a line through these points. What kind of line is it?

This line is the locus of points equally distant from AB and CD.

149. The general statement of 148a is:

The locus of points at a given distance from a given point is a circle with the given point as center and the given distance as radius.

150. The general statement of 1486 is:

The locus of points at a given distance from a given line consists of two lines, parallel to the given line, on opposite sides of it, and at the given distance from it.

151. The general statement of 148c is:

The locus of points equidistant from two parallel lines is a parallel line, midway between the two given lines.

*152. The locus of points equidistant from the ends of a line is the perpendicular bisector of that line.

*153. The locus of points equidistant from two intersecting lines is the pair of lines bisecting the angles.

SUMMARY

154. a. State a method of finding the locus of points under given conditions.

b. State a method of finding a point under two given conditions.

c. State a method of proving a line is the locus of points under given conditions.

d. State the five fundamental locus figures.

e. State two methods of proving a line passes through a given point.

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1. Find a point at a given distance from a given line, and equidistant from two lines that meet.

2. Find a point at a given distance from a given line, and equidistant from two parallel lines. When will there be no case?

3. Find a point equidistant from two lines that meet, and equidistant from two parallel lines.

4. Find a point equidistant from two given points, and equidistant from two given intersecting lines

5. Find a point equidistant from two given points, and equidistant from two parallel lines.

6. In a given line, find a point equidistant from two given points. When is there no case?

7. In a given line, find a point at a given distance from a given line.

8. In a given line, find a point equidistant from two intersecting lines.

9. In a given line, find a point equidistant from two parallel lines. 10. Find a point equidistant from three given points, not in a straight line.

11. Find a point equidistant from three given non-parallel lines.

12. Any two angle-bisectors of a triangle meet in a point equidistant from the three sides.

13. The three angle-bisectors of a triangle meet in a common point.

14. Any two perpendicular bisectors of the sides of a triangle meet in a point equidistant from the three vertices.

15. The three perpendicular bisectors of the sides of a triangle meet in a common point.

16. The three altitudes of a triangle meet in a common point.

17. Any two medians of a triangle meet in a point, which divides each median into two parts, one double the other.

18. The three medians of a triangle meet at a common point.

19. Find the locus of the vertices of triangles, having the same base and a given altitude.

20. Find the locus of the middle points of lines drawn from a given point to a given line. Prove the result.

156. State the four steps in Analysis of Theorems. State the four cases of concurrent lines in triangles.

REVIEW QUESTIONS 1. Summarize congruent triangles into four cases. 2. What is the principal use of congruent triangles? 3. What is a converse theorem? An opposite theorem? 4. Define Isosceles triangle; Scalene; Equilateral. 5. Give the definition of parallel lines, and the axiom.

6. Make three other statements beginning with "only one straight line."

7. What is the Indirect Method of proof?

8. Define Quadrilateral; Trapezoid; Parallelogram; Rhombus; Rectangle; Square.

9. State four methods of proving a quadrilateral is a parallelogram. 10. State seven methods of proving lines parallel.

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