14. A method of finding the distance between two points, A and B, separated by a stream:

Construct any convenient line AC. At

=

Measure the angle BAC and construct angle CAE BAC. Construct ZACD = LACB. AD = AB. Why?

15. A method of finding the distance between two points, A and B, both on the further side of a stream: Find OA and OB as in ex. 14; then find FE as in ex. 13.

FL

E

केर

B

REVIEW QUESTIONS

1. What are the three elements of a geometrical figure?

2. What three qualities of a geometrical figure are discussed and compared?

3. Two figures that have the same size are.

4. Two figures that have the same form are

what?

what?

5. Two figures that have the same form and size are.

what?

6. How many straight lines can be drawn through two points?

7. What may an angle be said to represent.

8. Upon what does the size of an angle depend?

9. Define Adjacent angles; Straight angles; Right angles; Acute angles; Obtuse angles; Supplementary angles; Complementary angles; Vertical angles.

10. State six general axioms of equality.

11. Define Perpendicular.

12. If one straight line meets another straight line, the sum of the two adjacent angles is . . . what?

13. If one straight line crosses another straight line, the vertical angles are . . . what?

14. If the sum of two adjacent angles equals two right angles,.. what?

15. What can be said of the supplements of the same angle? Complements of the same angle?

16. Define Triangle; Altitude; Median; Angle-bisector.

17. Define Superposition.

18. What are its three main steps?

19. What is its principal use?

20. What is a Right triangle? Hypotenuse? Arm?

21. How many times was superposition used in proving triangles congruent? In what theorems?

22. How were the other cases of congruent triangles proved?

23. What two theorems of congruent triangles were proved by placing triangles adjacent?

24. Summarize congruent triangles into four cases.

25. If one line is to be proved equal to a second line, what is the usual method?

26. In proving lines equal, which one of the four cases of congruent triangles cannot be used?

27. In proving a line is bisected, what equality is to be proved? 28. In proving an angle is bisected, what equality is to be proved? 29. State seven methods of proving angles equal.

CHAPTER VI

PARALLEL LINES

The two basic methods are:

*35. If two straight lines are cut by a transversal making the corresponding angles equal, the lines are parallel. *38. If two straight lines are cut by a transversal making the alternate-interior angles equal, the lines are parallel.

87. Define Parallel Lines.

88. What are Alternate-interior angles of parallel lines? Corresponding angles?

89. What is the usual method of proof called?

What is the Indirect Method of proof?

State the four steps of the Indirect Method.

*90. If two straight lines are cut by a transversal making the two interior angles on the same side of the transversal supplementary, the lines are parallel.

*91. Two straight lines perpendicular to a third straight line are parallel.

*92.

PROBLEM: Construct through a point a line parallel to a given line.

93. State the axiom of parallel lines.

94. Two straight lines parallel to a third straight line are parallel to each other.

95. Define Quadrilateral.

96.

Define Trapezoid.

98. Define Rhombus.

100.

Define Square.

97. Define Parallelogram. 99. Define Rectangle. *101. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

*102. If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

*103. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

104. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

SUMMARY

105. State four methods of proving quadrilaterals parallelograms.

State six methods of proving lines parallel.

1. If two sides of a triangle are produced their own lengths, through their common vertex, the line joining their ends is parallel to the third side.

2. If two parallel lines are cut by a transversal, the bisectors of two alternate-interior angles are parallel.

3. If two parallel lines are cut by a transversal, the bisectors of two corresponding angles are parallel.

4. If two lines are cut by a transversal making the alternateexterior angles equal, the lines are parallel.

5. If two lines are cut by a transversal making the two exterior angles on the same side of the transversal supplementary, the lines are parallel.

6. If side AC of triangle ABC is bisected at E, and BE is drawn and then extended its own length to F, AF is parallel to BC.

7. If a quadrilateral has all angles right angles, the figure is a parallelogram.

8. In ▲ABC, E and F are the mid-points of AB and AC respectively; EF is extended its own length to H. Prove AE= and || HC. 9. In the same figure prove EBC H a parallelogram.

10. In trapezoiḍ ABCD, AD || BC; E and F are the mid-points of AB and CD respectively; AF is drawn and extended to meet BC extended at H. Prove AF=FH.

11. In the same figure, what can be said of EF and BH Why? 12. If two straight lines are cut by a transversal making alternateinterior angles equal, any other transversal makes alternate-interior angles equal.

13. The line bisecting the exterior angle at the vertex of an isosceles triangle is parallel to the base.

14. If on the equal sides of an isosceles triangle equal distances are measured from the vertex, the line joining these points is parallel to the base.

15. If a line is drawn connecting the ends of the bisectors of the base angles of an isosceles triangle, this line is parallel to the base.

16. If a line is drawn connecting the ends of the perpendiculars from the ends of the base to the sides of an isosceles triangle, this line is parallel to the base.

17. If a line is drawn connecting the ends of the medians to the sides of an isosceles triangle, this line is parallel to the base.

18. If an exterior angle of a triangle equals twice an opposite interior angle, the triangle is isosceles.

19. A method of finding the distance between a moving boat and a distant object such as a lighthouse:

Let A be the lighthouse and C the A boat, moving along CD. Find the angle ACD. As the boat moves, this angle increases. When the angle is twice as large, such as ABD, the distance CB, which the boat has gone, equals AB. Why?

D

B