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15. Construct an equilateral arch.

HINT. Draw an equilateral triangle and bisect its sides.

FIG. 67.

This picture from Durham Cathedral, England, shows the use of geometric forms in architectural decoration.

[graphic]

CHAPTER X

PARALLEL LINES, QUADRILATERALS, AND
REGULAR POLYGONS

87. Parallel lines. On page 125 the pupil learned what a plane surface is. Give some examples of plane surfaces. Give examples in the schoolroom of two lines in the same plane. Give examples of lines not in the same plane. Can you give examples of two lines in the same plane that will not meet however far they are produced? (Produced means extended.)

Two lines in the same plane that will not meet however far they are produced are called parallel lines.

The opposite edges of a ruler, the top and bottom of the blackboard, may be taken as examples of parallel lines.

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A line that intersects two or more lines is called a trans

versal of those lines.

In Figure 69 XY is a transversal of MN and RS.

The distance between two parallel lines is the length of the perpendicular between them. In Figure 69 AB is the distance between MN and RS.

Exercise 83

1. Point out examples of parallel lines about the school

room.

2. Give examples of lines in the same plane that are not parallel.

3. Give an example of two lines not in the same plane.

4. Use the opposite edges of a ruler for making lines MN and RS and make a figure similar to Figure 69, omitting the line AB. With the protractor measure the angles made by the transversal. Copy and complete the following table.

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6. Construct a parallel to a line AB through a point P. HINT. First draw through P a line XY cutting AB at a point C.

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Through P draw a line DE making EPX equal to LBCX.

The line DE is the required line.

Exercise 84

1. Using the protractor, make an angle of 60°. Construct an angle equal to it, using ruler and compasses. Test the accuracy of your construction by using the protractor.

2. Through a given point Q draw a line parallel to a given line MN.

3. Construct two parallel lines AB and CD. Construct MN and PQ perpendicular to AB. Are MN and PQ also perpendicular to CD? Are MN and PQ equal?

N

D

-B

M

FIG. 72.

4. Are two parallel lines the same distance apart at all points?

5. A contractor wishes to lay a concrete walk 6 ft. from the wall of a building and parallel to it. State two ways of locating the edge of the walk so that it will be parallel to the wall.

6. Construct parallel lines 1 in. Apart.

7. Construct the design given in Figure 73. Notice the parallel lines. Upon what kind of triangles is this design based?

8. Construct the design given in Figure 74. Notice the size of the angles, and the parallel lines.

9. Construct the linoleum pattern of Figure 75. It is based upon how many sets of parallel lines?

89. Quadrilaterals. A portion of a plane bounded by four lines is called a quadrilateral.

[graphic]

FIG. 73.

FIG. 74.

[graphic]

FIG. 75.

A quadrilateral is classified according to the number of its parallel sides.

A quadrilateral with no parallel sides is called a trapezium.

A quadrilateral with only one pair of parallel sides is called a trapezoid.

A quadrilateral with two pairs of parallel sides is called a parallelogram.

90. Kinds of parallelograms. A parallelogram whose sides are all equal is called a rhombus.

A parallelogram whose angles are all right angles is called a rectangle.

A rectangle whose sides are all equal is called a square.

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91. Base. Altitude. Diagonal. The side upon which a figure is supposed to rest is called the base.

The parallelogram and the trapezoid are said to have an upper base and a lower base. MN is the upper base and OP the lower base of the parallelogram MNOP.

The perpendicular distance between the bases of a parallelogram or of a trapezoid is called the altitude. RS is the altitude of the parallelogram MNOP.

The diagonal of a quadrilateral is a line joining two opposite vertices.

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