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PRACTICAL MENSURATION

LINES AND ANGLES

LESSON 105

A Line is that which has length only.

A Straight Line is one that does not change its direction.

A Curved Line is one that continually changes its direction.

Parallel Lines are those that have the same direction.

The difference in direction between two straight lines that meet in a common point is called an Angle.

When one straight line meets another straight line so as to form two equal angles, the lines are said to be Perpendicular to each other, and the angles thus formed are called Right Angles.

When an angle is less than a right angle it is called an Acute Angle.

When an angle is greater than a right angle it is called an Obtuse Angle.

1. Draw two straight lines so as to form one angle; two angles; four angles.

2. Draw two straight lines so as to form an acute angle; a right angle; an obtuse angle.

3. Draw two straight lines so as to form one right angle; two right angles; four right angles; one obtuse and one acute angle; two obtuse and two acute angles.

4. Will the size of an angle be increased by lengthening its sides?

5. If two straight lines are drawn from a point, one due east and the other due north, what kind of an angle do they form?

6. If two straight lines are drawn from a point, one due east and the other northwest, what kind of an angle do they form?

SURFACES AND PLANE FIGURES

LESSON 106

A Surface is that which has the dimensions length and breadth.

A Plane Surface is a level surface. When no part of a surface is plane, it is called a Curved Surface.

1. What kind of surface is the floor of your schoolroom? The ceiling? The walls? The blackboard ?

2. What kind of surface is that of a ball ? Of a slate pencil ?

3. Name three objects having a surface like that of a ball. Three having a surface like a slate pencil. Three having a surface like the top of a table.

A Plane Figure is a plane surface bounded by straight or curved lines.

A Polygon is a plane surface bounded by three or more straight lines.

The Perimeter of a polygon is the distance around it. A polygon of three sides is called a Triangle.

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The Base of a triangle is the side upon which it seems to stand, as AB (Fig. 1).

The Vertex is the point opposite the base, as F (Fig. 2).

The Altitude is the perpendicular distance from the vertex to the base, as FM (Fig. 2).

LESSON 107

Considered with reference to the relative size of their angles, triangles are distinguished as right-angled, acuteangled, or obtuse-angled.

A Right-angled Triangle, or right triangle, has one right angle (Fig. 1).

An Acute-angled Triangle has three acute angles (Fig. 2). An Obtuse-angled Triangle has one obtuse angle (Fig. 4). Considered with reference to the relative length of their sides, triangles are distinguished as equilateral, isosceles, or scalene.

An Equilateral Triangle has three equal sides (Fig. 2).
An Isosceles Triangle has two equal sides (Fig. 3).
A Scalene Triangle has three unequal sides (Fig. 4).

A polygon of four sides is called a Quadrilateral.

QUADRILATERALS

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Rhombus

Trapezoid

Trapezium The first four of the figures above are called Parallelograms, because their opposite sides are parallel.

The Base of a parallelogram is the side upon which it seems to stand, as ab, Fig. A.

The Altitude of a parallelogram is the perpendicular distance between the base and the side opposite, as de, Fig. C.

The Diagonal of a quadrilateral is a straight line joining its opposite angles, as ac, Fig. A, and fy, Fig. D.

LESSON 108

A Rectangle is a plane surface having four right angles, as Fig. A, above.

A Square is a rectangle whose four sides are of equal length, as Fig. B, above.

A Rhomboid is an oblique-angled parallelogram, as Fig. C.

A Rhombus is a rhomboid whose four sides are of equal length, as Fig. D.

A Trapezoid is a quadrilateral having only two of its sides parallel, as Fig. E, p. 121.

A Trapezium is a quadrilateral having none of its sides parallel, as Fig. F, p. 121.

1. Why are the pages of this book quadrilaterals ? Why parallelograms?

2. Point out any parallelograms in your schoolroom, and tell why they are parallelograms.

3. In what particular respect are a rectangle and a square alike? Wherein are they unlike?

4. If you cut a rectangle of paper through the diagonal into two parts, what plane figure is each part, and what part of the rectangle is each part ?

5. Compare the base of each triangle with the length of the rectangle.

6. Compare the rectangle with the rhomboid, and state wherein they are unlike.

7. If you cut a square into two equal triangles, what kind of triangle is each ?

8. In what respect are a trapezoid and a rhomboid unlike?

9. Wherein does a rhombus resemble a square ?

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