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92. To construct a parallelogram. It is required to construct a parallelogram, given two adjacent sides, x and y, and their included

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On the sides of A mark off AB and AD equal to x and y respectively.

With B as a center and radius y construct an arc. With D as center and radius x construct an arc cutting the arc last constructed at C.

Draw lines BC and DC. Then ABCD is the required parallelogram.

Exercise 85

1. With ruler and compasses construct a rectangle with base 2 in. and altitude 1 in.

2. Construct a square with side 1 in.

3. How many degrees in the sum of the angles of a rectangle? Of a square?

4. Construct a parallelogram MNOP, given two adjacent sides MN and NP, and their included angle M.

5. With a protractor measure the four angles of the parallelogram MNOP. Find the number of degrees in the sum of the angles M and N; in the sum of angles N and 0; in the sum of angles O and P; in the sum of angles P and M.

6. Construct a parallelogram given two adjacent sides and their included angle, which is obtuse. Measure the angles with a protractor. Compare the opposite angles. Find the sum of each pair of adjacent angles.

7. State any fact you have discovered about the opposite angles of a parallelogram; about the adjacent angles.

8. How many degrees are there in the sum of the angles of parallelogram MNOP?

9. If, in a parallelogram MNOP, ZM=60°, how many degrees in each of the other angles?

10. Draw the diagonals of the parallelogram constructed in exercise 4. Call the point of intersection A. Measure and compare the lengths of MA and AO, and of NA and AP. What conclusion can you draw from these measurements? Test this conclusion in other parallelograms.

11. Construct a rectangle and draw its diagonals. Measure them and the parts into which they divide each other. What do these measurements show?

12. A lot is in the form of a trapezoid ABCD. AB=100 ft., BC=40 ft., CD=60 ft., and angles B and C are right angles. Draw the trapezoid on the scale of 20 ft. to 1 in. 13. Measure the opposite sides of a parallelogram. How do they compare in length?

14. Measure and find the sum of the angles in each of two trapeziums. What do you think the sum of the angles of a trapezium is?

15. Find at least one exercise in which each of the facts stated below is illustrated.

93. Principles concerning quadrilaterals. The following principles concerning quadrilaterals have been illustrated in the above exercises. The pupil should memorize these principles.

I. The sum of the angles of a quadrilateral is 360°.

II. In a parallelogram the opposite angles are equal and the adjacent angles are supplementary.

III. The opposite sides of a parallelogram are equal.

IV. The diagonals of a parallelogram bisect each other. V. The diagonals of a rectangle are equal.

Exercise 86

1. In a parallelogram ABCD, ▲ A=35° 20′ 15′′. Find the size of each of the other angles.

2. The four angles of a quadrilateral are equal. How many degrees in each? What kind of quadrilateral is it?

3. Two boys lay out a tennis court. To test their work they measure the diagonals. One diagonal is 86 ft. and the other is 85 ft. Is the tennis court laid out properly? Give reasons for your answer.

4. A figure is known to be a parallelogram. It is found by measuring that one angle is 90°. What kind of parallelogram is it? Why?

5. One side of a square is s. What is the perimeter? State your answer as a formula, using p to represent the perimeter.

6. The perimeter of a square is p. What is the length of one side? State your answer as a formula.

7. Make a formula for finding the perimeter, p, of a parallelogram whose adjacent sides are a and b.

8. One angle of a parallelogram contains x degrees. How many degrees in each of the other angles?

9. Three angles of a parallelogram contain a degrees, b degrees, and a degrees, respectively. How many degrees in the fourth angle?

10. A trapezoid has two right angles. The third angle contains 47° 30'. Find the size of the fourth angle.

11. A quadrilateral has three equal sides. Is it necessarily a parallelogram? May it be a trapezoid? May it be a trapezium?

12. Do the diagonals of a parallelogram bisect the angles?

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15. Let each pupil copy a geometric design such as may be found in wall paper or linoleum patterns.

94. Polygons. A portion of a plane bounded by straight lines is called a polygon. The bounding lines are called the sides of the polygon.

Thus, AB, BC, CD, DE, and EA are the sides of the polygon ABCDE.

The points of intersection of the sides are called the vertices of the polygon. A, B, C, D, and E are the vertices of the polygon ABCDE.

A straight line joining two non

E

D

A

B

FIG. 85.

adjacent vertices of a polygon is called a diagonal of the polygon.

AC is a diagonal of the polygon ABCDE.

95. Kinds of polygons. Polygons are named according to the number of sides.

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Exercise 87

1. With a ruler construct an example of each of the polygons named in § 95.

2. Construct a pentagon and draw all of its diagonals.

3. Can you construct a polygon with only two sides? What is the least number of sides that a polygon may have? Can you name the greatest number of sides that a polygon may have?

4. What kinds of polygons are used in the woodwork in the schoolroom? What kind is most common?

5. What forms of tiles have you seen in floors, hearths, and mantels?

6. What kinds of polygons are the faces of a cube? The faces of the pyramid on page 115?

7. How many sides has a sixpointed star?

8. How many diagonals has a quadrilateral? A triangle? A hexagon?

9. Name as many kinds of polygons as you can that are shown in Figure 86.

10. Make a copy of Figure 86. First draw a circle and divide it into 8 equal parts to locate the vertices of the octagon. Draw lines AF, BE, HC, and GD to get the sides of the square JQRK. I is the midpoint of AH. M is the midpoint of JK.

11. Make a copy of the pattern given in Figure 87.

H

M

PR

E

A

B

FIG. 86.

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FIG. 87.

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