  LESSON FIRST.LINES.Note to the Teacher.—In all the development-lessons, the pupils are to be occupied with the diagrams, and not with the printed matter. See Note A, Appendix. Refer to Diagram 1, and show that What are here drawn are intended to represent length only. They have a little width, that they may be seen. They are called lines. A line is that which has length only. POINTSShow that Position is denoted by a point. It occupies no space. It has some size, that it may be seen. The ends of a line are points. A line may be regarded as a succession of points. The intersection of two lines is a point. A point is named by placing a letter near it. Diagram 1. A point may be represented by a dot. The point is in the center of the dot. A point is that which denotes position only. A line is named by naming the points at its ends. Read all the lines in Diagram 1. CROOKED LINES.See Note B, Appendix. Does the line m n change direction at the point 1? At what other points does it change direction? It is called a crooked line. A crooked line is one that changes direction at some of its points. CURVED LINES.The line o p changes direction at every point. It is called a curved line. A curved line is one that changes direction at every point. STRAIGHT LINES.Does the line i j change direction at any point? It is called a straight line. A straight line is one that does not change direction at any point. OTHER LINES.The line q r winds about a line. It is called a spiral line. The line w x winds about a point. It also is called a spiral line. A spiral line is one that winds about a line or point. The line 7 8 1.To be read seven, eight, not seventy-eight. It is called a wave line. What kind of a line is a b? Why? What is a straight line? What kind of a line is 11 16? Why? What is a crooked line? What kind of a line is o p? Why? What is a curved line? Why? What kind of a line is 9 10? Why? What is a spiral line? What kind of a line is w x? Why? LESSON SECOND.REVIEW.Read all the straight lines. (Diagram 2.) Why is m n a straight line? Define a straight line. Read all the crooked lines. Why is 7 8 a crooked line? Define a crooked line. Read all the curved lines. Why is 5 6 a curved line? What is a curved line? Read all the wave lines. Read all the spiral lines. Why is 3 4 a spiral line? Why is u v a spiral line? What is a spiral line? Diagram 2. Diagram 3. LESSON THIRD.POSITIONS OF LINES.Let the pupils hold their books so that they will be straight up and down like the wall. VERTICAL LINES.The straight line a b points to the center of the earth. (Diagram 3.) It is called a vertical line. Name all the vertical lines. A vertical line is a straight line that points to the center of the earth. HORIZONTAL LINES.The straight line o p points to the horizon. Read all the horizontal lines. A horizontal line is a straight line that points to the horizon. OBLIQUE LINES.The line s t points neither to the center of the earth nor to the horizon. It is called an oblique line. Read all the oblique lines. An oblique line is a straight line that points neither to the horizon nor to the center of the earth. Note.—After going through with the lessons on angles, the pupils may be told that oblique lines are so called because they form oblique angles with the horizon. LESSON FOURTH.REVIEW.Read all the vertical lines. (Diagram 4.) Why is q r a vertical line? What is a vertical line? Read all the horizontal lines. Why is 5 6 a horizontal line? Define a horizontal line. Read all the oblique lines. Why is s t an oblique line. What is an oblique line? Note.—Lines that point in the same direction do not approach the same point. Diagram 4. Diagram 5. LESSON FIFTH.ANGLES.Do the lines a b and c d (Diagram 5.) point in the same direction? (See note, page 15.) Then they form an angle with each other. What other line forms an angle with a b? Which of the two lines c d, e f, has the greater difference of direction from the line a b? Then which one forms the greater angle with a b? What line forms a still greater angle with the line a b? If the lines a b, e f, were made longer, would their direction be changed? Then would there be any greater or less difference of direction? Then would the angles formed by them be any greater or less? Then does the size of an angle depend upon the length of the lines that form it? If the lines a b, e f, were shortened, would the angle formed by them be any smaller? If two lines form an angle with each other, and meet, the point of meeting is called the vertex. What is the vertex of the angle formed by the lines k j, i j?—i j, i l? An angle is named by three letters, that which denotes the vertex being in the middle. Thus, the angle formed by k j, i j, is read k j i, or i j k. Read the four angles formed by the lines m n and o p. The eight formed by r s, t u, and v w. LESSON SIXTH.REVIEW.Read all the lines that form angles with the line a b. (Diagram 6.) Which of them forms the greatest angle with it? Diagram 6. Which the least? Of the two lines c d, g h, which forms the greater angle with e f? Read all the angles whose vertices are at o on i j. Which angle is the greater, l o m, or m o j?—i o k, or i o l?—l o j, or m o j? Read all the angles formed by the lines v w and x y. Read all the angles above the line n p. Below the line n p. Above the line q r. At the right of the line 5 u. At the left. At the right of the line s t. At the left of the line s t. Which angle is the greater, n 1 3, or n 2 4? If the lines x y and v w were lengthened or produced, would the angles v z x, y z w be any greater? If they were shortened, would the angles be any less? Does the size of an angle depend upon the length of the lines which form it? Diagram 7. LESSON SEVENTH.RELATIONS OF ANGLES.ADJACENT ANGLES.Are the angles a e c, c e b (Diagram 7.), on the same side of any line? What line? By what other straight line are they both formed? Then, because they are both on the same side of the same straight line a b, and are both formed by the second straight line c d, they are called “adjacent angles.” The angles c e b, b e d are both on the same side of what straight line? Then what kind of angles are they? Why are they called adjacent angles? Read the adjacent angles below the line a b. Below the line c d. How many pairs of adjacent angles can be formed by two straight lines? Read all the adjacent angles formed by the lines l m and n p. VERTICAL ANGLES.Are the angles a e c, b e d formed by the same straight lines? Are they adjacent angles? They are called “vertical angles.” Vertical angles are angles formed by the same straight lines, but not adjacent to each other. Read the other pair of vertical angles formed by the lines a b, c d. Read all the vertical angles formed by the lines f g, i h. By l m, n p. Why are the angles l o n, n o m adjacent angles? Why are the angles l o n, p o m vertical angles? Diagram 8. LESSON EIGHTH.REVIEW.Read the pairs of adjacent angles above the line a b. (Diagram 8.) Why are they adjacent? What are adjacent angles? Read the adjacent angles below the line a b. On the right of the line c d. On the left. How many pairs of adjacent angles are formed by the intersection of two lines. Read the pairs of adjacent angles formed by the lines f g and i h. Read all the pairs of vertical angles formed by the lines a b, c d. Why are c e b and a e d called vertical angles? What are vertical angles? Read all the pairs of vertical angles formed by the lines h i, f g. How many pairs of vertical angles are formed by the intersection of two lines? Read all the pairs of vertical angles formed by the lines l m, n p. LESSON NINTH.KINDS OF ANGLES.RIGHT ANGLES.What do we call the angles a o c, c o b? (Diagram 9.) Are they equal to each other? Then they are called right angles. A right angle is one of two adjacent angles that are equal to each other. Are the adjacent angles c o b, b o d equal to each other? Then what are they called? Read the right angles below the line a b. On the left of c d. Read three right angles whose vertices are at p. Diagram 9. ACUTE ANGLES.Is the angle m p q greater or less than the right angle m p r? Then it is called an acute angle. An acute angle is one which is less than a right angle. Read four acute angles whose vertices are at p. Acute means sharp. Why is r p s an acute angle? What is an acute angle? OBTUSE ANGLES.Is the angle m p s greater or less than the right angle m p r? An obtuse angle is one which is greater than a right angle. What other obtuse angle has its vertex at p? Obtuse means blunt. Read three obtuse angles whose vertices are at x. Acute and obtuse angles are also called oblique angles. LESSON TENTH.REVIEW.Read all the right angles formed by the lines a b and c d. (Diagram 10.) Why are the adjacent angles c e b, b e d, right angles? What is a right angle? Read four right angles whose vertices are at n. Which is the greater, the right angle p q r, or the right angle t s u? Can one right angle be greater than another? Read six acute angles whose vertices are at n. Why is m n g an acute angle? What is an acute angle? Which is greater, the acute angle m n g, or the acute angle l n m? May one acute angle be greater than another? What three acute angles are equal to one right angle? Diagram 10. Which of the two acute angles v f w, y x z is the greater? Read four obtuse angles whose vertices are at n. Why is f n m an obtuse angle? What is an obtuse angle? What does obtuse mean? Acute? By what other name are both called? Which is greater, the large acute angle 1 4 2, or the small obtuse angle 1 4 3? How much greater than the right angle is the obtuse angle f n l? How much less than a right angle is f n i? Diagram 11. LESSON ELEVENTH.RELATIONS OF LINES.PERPENDICULAR LINES.What kind of angles do the lines a b and c d make with each other? (Diagram 11.) Then they are perpendicular to each other. What line is perpendicular to x y? Why is it perpendicular to it? What line is perpendicular to z 1? When is a line said to be perpendicular to another? Can a line standing alone be properly called a perpendicular line? Is the line g h perpendicular to the line i j? Why? What other line is perpendicular to the line i j? Read three lines that are perpendicular to the line a b. PARALLEL LINES.Do the lines k l, m n, differ in direction? Then do they form any angle with each other? They are said to be parallel to each other. Read four other lines that are parallel with k l. What line is parallel with 2 10? Why? Lines are parallel with each other when they do not differ in direction. OBLIQUE LINES.What kind of angles do the lines u t and 8 9 form with each other? Then they are said to be oblique to each other. Lines are oblique to each other when they form oblique angles. See Note C, Appendix. Diagram 12. LESSON TWELFTH.REVIEW.Read five lines that are perpendicular to the line a b. (Diagram 12.) Five that are perpendicular to c d. Two that are perpendicular to u v, and meet it. Three that do not meet it. Why are o p and m n perpendicular to each other? When are lines said to be perpendicular to each other? Read four lines that are parallel with e f. Why are the lines e f and g h said to be parallel to each other? Read four lines that are parallel to 5 6. Four that are parallel to o p. Is any line parallel to u v? Can a single line be properly called perpendicular? Parallel? If two lines are perpendicular to each other, what angle do they form? If parallel, what angle? If oblique? Diagram 13. LESSON THIRTEENTH.RELATIONS OF ANGLES.INTERIOR ANGLES.Is the angle a m n between the parallels, or outside of them? (Diagram 13.) It is called an interior angle. Why is b m n an interior angle? An interior angle is one that lies between parallel lines. Read the interior angles between the parallel lines g h and k l. Why is o p l an interior angle? What is an interior angle? EXTERIOR ANGLES.Is the angle a m e between the parallels, or outside of them? It is called an exterior angle. Read three other exterior angles formed by the lines a b, c d, and e f. Why is the angle c n f an exterior angle? An exterior angle is one that lies outside of the parallels. LESSON FOURTEENTH.REVIEW.Read all the interior angles formed by the lines a b, c d, and e f. Why is m n d an interior angle? What is an interior angle? Read all the exterior angles formed by the same lines. What is an exterior angle? Read all interior angles formed by the lines g h, k l, and i j. All the remaining interior angles in the diagram. All the exterior angles. Diagram 14. LESSON FIFTEENTH.RELATIONS OF ANGLES.OPPOSITE ANGLES.Are the angles e m b, b m n, on the same side of the intersecting line e f? Are they adjacent? Are e m b, m n d, on the same side of the intersecting line e f? Are they adjacent? Opposite angles lie on the same side of the intersecting line, but are not adjacent. Are the angles e m b, f n d, on the same side of the intersecting line? Are they adjacent? Then are they opposite? Are they interior or exterior angles? Then they are “opposite exterior angles.” Why are they exterior? Why are they opposite? Are the angles b m n, m n d, opposite angles? Are they interior or exterior angles? Then they are “opposite interior angles.” Why are they opposite? Why interior? Read the opposite exterior angles on the left of the line e f. Read the opposite interior angles on the same side. Are the opposite angles e m a, m n c, both exterior or interior? Then they are opposite exterior and interior angles. Read two pairs of opposite exterior and interior angles on the right of e f. On the left. ALTERNATE ANGLES.Do the angles b m n, m n c, lie on the same side of the intersecting line e f? Are they adjacent to each other? Are they vertical angles? Then they are alternate angles. Are the alternate angles b m n, m n c, exterior or interior? Then they are called “interior alternate angles.” Read another pair of interior alternate angles between a b and c d. Are the angles e m b, c n f, alternate angles? Why? Are they exterior or interior? Then what may they be called? Read another pair of exterior alternate angles. Why are e m a, d n f, alternate angles? Why exterior alternate? LESSON SIXTEENTH.REVIEW.Read the exterior opposite angles on the right of the line e f. (Diagram 14.) On the left. On the right of r s. On the left. Why are e m a, c n f, exterior angles? Why are they opposite angles? What are opposite angles? Read the interior opposite angles on the right of the intersecting line e f. On the left of it. On the right of r s. On the left. Read the interior alternate angles formed by the lines a b, c d, and e f. Which pair are acute angles? Which pair are obtuse angles? Why are b m n, m n c, interior angles? Why alternate? What are alternate angles? Read the exterior alternate angles of the same lines. Read the acute interior alternate angles of the parallels t u, v w. The obtuse. The acute exterior alternate angles. Obtuse. Read the pair of opposite exterior angles on the right of the line e f. On the left. On the right of r s. On the left. Diagram 15. LESSON SEVENTEENTH.REVIEW.Read thirteen or more angles whose vertices are at c. (Diagram 15.) Read four obtuse angles. Read two right angles. What three acute angles equal one right angle? Which is greater, the right angle 4, or the right angle 5? The obtuse angle 6, or the acute angle 7? Read twelve pairs of adjacent angles formed by the lines w x, &c. Read all the interior angles formed by the lines i j, k l, and m n. Read all the exterior angles formed by the same lines. Two pairs of opposite exterior angles. Two pairs of opposite interior angles. Four pairs of opposite exterior and interior angles. Two pairs of alternate interior angles. Two pairs of alternate exterior angles. Why are the angles i o m, m o j, called adjacent? What are adjacent angles? What kind of an angle is i o m? Why? What is an acute angle? What kind of an angle is m o j? Why? What is an obtuse angle? Why are a c f, f c b, right angles? What is a right angle? Why are m o i, j o p, vertical angles? What are vertical angles? Why is m o i an exterior angle? What is an exterior angle? Why is j o p an interior angle? What is an interior angle? Why are m o i, o p k, opposite angles? What are opposite angles? Why are j o p, o p k, alternate angles? What are alternate angles? Draw an obtuse angle which shall be only a little larger than a right angle. Draw one which shall be much greater than a right angle. Draw an acute angle which shall be only a little less than a right angle. Draw one which shall be much less than a right angle. Draw an obtuse angle with lines about one inch long. Draw an acute angle with sides three inches long. Which is greater, the obtuse angle, or the acute angle? Draw a right angle with lines an inch long. Draw one with lines five inches long. Which is the greater, first or the second? Diagram 16. LESSON NINETEENTH.POLYGONS.Name any thing besides your desk that has a flat surface. A flat surface is called a plane. How many sides has the plane Fig. A? (Diagram 16.) It is called a triangle. “Tri” means “three.” What other triangles do you see. Triangles are sometimes called trigons. A triangle is a plane figure having three sides. How many sides has the plane figure marked B? How many angles? It is called a quadrangle, or quadrilateral. “Quad” denotes “four.” What other quadrangles do you see? Why is Fig. B a quadrangle? A quadrangle is a plane figure having four sides. How many sides has the Fig. C? It is called a pentagon. What other pentagon do you see? Why is Fig. C a pentagon? A pentagon is a plane figure having five sides. In like manner,— A hexagon is a plane figure having six sides. A heptagon is a plane figure having seven sides. A nonagon has nine sides. A decagon has ten sides. All these figures are called polygons. “Poly” means “many.” What do you call a polygon of three sides? Of four sides? Of six sides? &c. If the length of each side of triangle A is one inch, how long are the three sides together? The sum of the sides of a polygon is its perimeter. Which of the triangles has unequal sides? Which has equal sides? The latter is called a regular polygon. Which pentagon has one side longer than any one of its other sides? Which has its sides all equal to each other? Are its angles also equal? It is therefore a regular polygon, or regular pentagon. Name a hexagon that is not regular. Name a regular hexagon. A regular octagon. A regular heptagon. A polygon is a plane figure bounded by straight lines. LESSON TWENTIETH.REVIEW.Name all the triangles. (Diagram 16.) Why is Fig. A a triangle? What is a triangle? What other name is sometimes given to triangles? Name all the quadrilaterals. Why is Fig. B a quadrilateral? What is a quadrilateral, or quadrangle? Name all the pentagons, hexagons, heptagons, octagons, and nonagons. Why is C a pentagon? What is a pentagon? A hexagon? A heptagon? &c. How many polygons in the diagram? What is a polygon? If each side of Fig. B is one inch, how many inches are there in its perimeter? When is a polygon regular? Name all the regular polygons in diagram 16. Name all the irregular polygons. Diagram 17. LESSON TWENTY-FIRST.TRIANGLES.ACUTE-ANGLED TRIANGLES.In the triangle 1, what kind of an angle is b a c? a c b? c b a? (Diagram 17.) Then it is called an acute-angled triangle. An acute-angled triangle is one whose angles are all acute. Read three other acute-angled triangles. OBTUSE-ANGLED TRIANGLES.In the triangle 4, what kind of an angle is l k m? Then it is called an obtuse-angled triangle. An obtuse-angled triangle is one that has one obtuse angle. Name two others. RIGHT-ANGLED TRIANGLES.In the triangle 3, what kind of an angle is g i j? Then it is called a right-angled triangle. A right-angled triangle is one that has one right angle. Name three other right-angled triangles. Upon which side does the triangle 3 seem to stand? Then i j is called the base of the triangle. What letter marks the vertex of the angle opposite the base? If, in the triangle 7, we consider t v the base, what point is the vertex? If v be considered the vertex, which side will be the base? In the triangle 3, what side is opposite the right angle? Then g j is called the hypothenuse of the triangle. The hypothenuse of a triangle is the side opposite the right angle. Read the hypothenuse of each of the triangles 5, 6, and 11. Either side about the right angle may be considered the base. Then the other side will be the perpendicular. In the triangle 3, if i j is the base, which side is the perpendicular? If g i be considered the base, which side is the perpendicular? In triangle 5, if n o is the base, which side is the perpendicular? LESSON TWENTY-SECOND.REVIEW.Name four acute-angled triangles. (Diagram 17.) Why is the triangle 8 acute-angled? What is an acute-angled triangle? Name three obtuse-angled triangles. Why is the triangle 9 an obtuse-angled triangle? What is an obtuse-angled triangle? Name four right-angled triangles. Why is the triangle 6 a right-angled triangle? What is a right-angled triangle? In the triangle 6, which side is the hypothenuse? Why? What is the hypothenuse? What two sides of the triangle 6 may be regarded as the base? If q r be considered the base, what do you call the side q s? Read the hypothenuse of each of the triangles 3, 5, 6, and 11. Diagram 18. LESSON TWENTY-THIRD.TRIANGLES. (Continued.)ISOSCELES TRIANGLES.Of the triangle 1, which two sides are equal to each other? Then it is called an isosceles triangle. An isosceles triangle is one that has two equal sides. Name eight isosceles triangles. Why is the triangle 2 an isosceles triangle? What kind of a triangle is it on account of its angles? Then it is an acute-angled isosceles triangle. Name four acute-angled isosceles triangles. What kind of a triangle is Fig. 4 on account of the angle k j l? What kind on account of its equal sides? Then it is called an obtuse-angled isosceles triangle. Name one other obtuse-angled isosceles triangle. What kind of a triangle is Fig. 6 on account of the angle q p r? What kind on account of its equal sides? Then it is called a right-angled isosceles triangle. Name one other right-angled isosceles triangle. Why is Fig. 12 a right-angled triangle? Why isosceles? EQUILATERAL TRIANGLES.Which of the isosceles triangles has all its three sides equal to each other? It is called an equilateral triangle. “Equi” means “equal.” “Latus” means a “side.” An equilateral triangle is one that has its three sides equal to each other. What kind of a triangle is Fig. 7 on account of its three equal sides? What kind on account of its two equal sides s t, s u, or t s, t u, or u s, u t? Then must not every equilateral triangle be also isosceles? What kind of a triangle is Fig. 2 on account of its equal sides d e, d f? If the side e f is longer than either of the other two sides, is it an equilateral triangle? Then is every isosceles triangle also equilateral? Name another isosceles triangle that is not equilateral. Name one that is equilateral. In any equilateral triangle the three angles are equal to each other. On account of its equal angles, it is also called an equiangular triangle. What is Fig. 8 called on account of its three equal sides? On account of its three equal angles? Every equilateral triangle is also equiangular. Every equiangular triangle is also equilateral. It is called a scalene triangle. What kind of a triangle is Fig. 5 on account of its right angle? What kind on account of its three unequal sides? Then it is a right-angled scalene triangle. What name can you give Fig. 11 on account of the angle g e f? On account of its three unequal sides? Then what may it be called? Diagram 19. LESSON TWENTY-FOURTH.REVIEW.Name eight isosceles triangles. (Diagram 19.) Why is Fig. 2 an isosceles triangle? What is an isosceles triangle? Name two right-angled isosceles triangles. Name five acute-angled isosceles triangles. Name one obtuse-angled isosceles triangle. Name two isosceles triangles that are also equilateral. Are all isosceles triangles equilateral? Name six isosceles triangles that are not equilateral. What does “equi” mean? “Latus”? What are equilateral triangles called on account of their equal angles? Are all equilateral triangles equiangular? Are all equiangular triangles equilateral? What are equilateral triangles? Name four scalene triangles. Name two right-angled scalene triangles. Why is Fig. 3 a right-angled triangle? Why scalene? What is a scalene triangle? Name one obtuse-angled scalene triangle. Name one acute-angled scalene triangle. PROBLEMS.From the same point draw two straight lines of any length, making an acute angle with each other. Make them equal to each other by measuring. Join their ends. What kind of a triangle is it on account of its angles? On account of its two equal sides? Write its two names inside of it. Draw an isosceles triangle whose equal sides shall each be less than the third side. Write its two names within it. Draw an oblique straight line twice as long as any short measure or unit. At one end draw a straight line perpendicular to it, and three times as long as the same measure. Connect the ends of the two lines by a straight line. What kind of an angle is that opposite the last line drawn? Are any two of its sides equal? Write its two names under it. Draw a horizontal straight line of any length. At one end draw a vertical line of equal length. Complete the triangle, and write two names inside. Draw a right-angled triangle whose base is of any length, and its perpendicular twice as long. Draw a right-angled triangle whose base is three times as long as any short measure, and its perpendicular five times as long as the same measure or unit. Diagram 20. QUADRILATERALS.How many sides has the figure a b d c? What is it called on account of the number of its sides? Name three other quadrilaterals whose vertices are marked. Name seven by numbers. Quadrilaterals are sometimes named by means of two opposite vertices. The quadrilateral a b d c, or c d b a, may be read a d, or b c, or c b, or d a. Name the quadrilateral, g h f e, four ways. How many angles has each figure? On account of the number of their angles they are called quadrangles. Has the quadrilateral a d any two sides parallel to each other? Then it is called a trapezium. A trapezium is a quadrilateral that has no two sides parallel. Name two other trapeziums. Why is Fig. 7 a trapezium? Has the quadrilateral e h any two sides parallel? Which two? Are the other two sides parallel? It is called a “trapezoid.” “Oid” means like. What does “trapezoid” mean? A trapezoid is a quadrilateral that has only one pair of sides parallel. Name another trapezoid. How many pairs of parallel sides has the quadrilateral i l? Name the horizontal parallels. Name the oblique parallels. It is called a “parallelogram.” A parallelogram is a quadrilateral whose opposite sides are parallel. Name five other parallelograms. Why is Fig. 4 a parallelogram? Why is not Fig. 6 a parallelogram? Why is not e h a parallelogram? What two names may you give to Fig. 5? Why is it a quadrilateral? Why a trapezium? What two names may we give to Fig. 6? Why is it a quadrilateral? Why a trapezoid? What two names may we give to Fig. 3? Why is it a parallelogram? Why a quadrilateral? LESSON TWENTY-FIFTH.REVIEW.How many quadrilaterals in the diagram. (Diagram 20.) Why is Fig. a d a quadrilateral? What is a quadrilateral? On account of the number of its angles, what may it be called? Name all the quadrilaterals. Name three trapeziums. Why is Fig. 5 a trapezium? What is a trapezium? Name two trapezoids. Why is Fig. 6 a trapezoid? Name its parallel sides. What is a trapezoid? Name six parallelograms. Why is Fig. 4 a parallelogram? Name its two pairs of parallel sides. What is a parallelogram? What two names can you give to Fig. 4? Why the first? Why the second? What two names may be given to Fig. 7? Why the first? Why the second? What two to Fig. 6? Why the first? Why the second? Diagram 21. LESSON TWENTY-SIXTH.KINDS OF PARALLELOGRAMS.RHOMBOIDS.How many quadrilaterals in the diagram? (Diagram 21.) How many parallelograms? Has the parallelogram a d any right angle? It is called a “rhomboid.” A rhomboid is a parallelogram which has no right angle. Name five other rhomboids. What three names may be given to Fig. 2? Why is it a quadrilateral? Why a parallelogram? Why a rhomboid? RHOMBS.Are the four sides of the rhomboid a d equal to each other? Are the four sides of the rhomboid e h equal to each other? If a triangle has its three sides equal to each other, what do you call it? Then when a rhomboid has its sides equal to each other, what may it be called? An equilateral rhomboid is called a rhombus. A rhombus is an equilateral rhomboid. See Note D, Appendix. What four names can you give to Fig. e h? Why a quadrilateral? Why a parallelogram? Why a rhomboid? Why a rhombus? RECTANGLES.Has the parallelogram i l any right angles? How many? It is called a “rectangle.” A rectangle is a right-angled parallelogram. Name four other rectangles. What three names may be given to Fig. i l? Why a quadrilateral? Why a parallelogram? Why a rectangle? SQUARES.Has the rectangle i l its four sides equal? Has the rectangle m p its four sides equal? It is called a “square.” A square is an equilateral rectangle. Name another “square.” What four names may be given to Fig. m p? Why a quadrilateral? Why a parallelogram? Why a rectangle? Why a square? LESSON TWENTY-SEVENTH.REVIEW.Name six rhomboids. (Diagram 21.) What three names may be given to Fig. 3? Why a quadrilateral? Why a parallelogram? Why a rhomboid? What is a quadrilateral? Parallelogram? Rhomboid? Name three rhombs. What four names may you give Fig. 5? Why a quadrilateral? Why a parallelogram? Why a rhomboid? Why a rhomb? What is a rhomboid? A rhomb? Name five rectangles. What three names may be given to Fig. 1? Why a quadrilateral? Why a parallelogram? Why a rectangle? What is a rectangle? Name two squares? By what four names may Fig. 7 be called? Why by the first? By the second? By the third? By the fourth? What is a square? What is a rectangle? What is a parallelogram? What is a quadrilateral? Diagram 22. LESSON TWENTY-EIGHTH.COMPARISON AND CONTRAST.TRAPEZIUM AND TRAPEZOID.In what respect are Figs. A and B alike? On this account, what name may be given to each? How does Fig. B differ from Fig. A? What particular name may you give to Fig. B? What one to Fig. A? RHOMBOID AND RECTANGLE.In what two respects are Figs. C and D alike? On account of the number of their sides, what may each be called? Because their opposite sides are parallel, what may each be called? In what respect do they differ? What particular name may be given to Fig. C? What one to Fig. D? What three names may you give to the figure with right angles? What three to the one without right angles? RHOMBOID AND RHOMBUS.In what three things are Figs. E and F alike? What three names may be given to each? How do they differ from each other? What particular name may you give to Fig. F? What four names has Fig. F? RECTANGLE AND SQUARE.In what three things are Figs. G and H alike? On account of the number of their sides, what may each be called? Because their opposite sides are parallel, what may each be called? Because they have right angles, what may they be called? In what respect is Fig. H different from Fig. G? What three names may be applied to Fig. G? What four to Fig. H? RHOMBUS AND SQUARE.In what three things are Figs. F and H alike? On account of the number of their sides, what name may be given to each? Because their opposite sides are parallel, what name may be given to each? Because both are parallelograms, and both have their sides equal, what name may be given to each? What particular name has Fig. F? What particular name has Fig. H? What four names may be given to Fig. F? What four to Fig. H? LESSON TWENTY-NINTH.REVIEW.What two names may be given to Fig. A. (Diagram 22.) To Fig. B? In what are they alike? In what do they differ? By what three names may Fig. C be called? By what three names may Fig. D be called? In what two things are they alike? In what one thing do they differ? What particular name has C? What one has D? What three names may be applied to Fig. E? What four to Fig. F? What property has F that E has not? What particular name has it on that account? What three names has Fig. G? What four has Fig. H? What property has Fig. H that G has not? What particular name has it in consequence? What four names may you give to Fig. F? What four to Fig. H? What three names may be applied to either? In what three things are they alike? In what respect do they differ? What particular name has Fig. F? What particular name has Fig. H? Diagram 23. LESSON THIRTIETH.MEASUREMENT OF SURFACES.In Fig. 1 (Diagram 23.) call the line a b a unit. Rectangle 1 is how many units long? How many high? Because its sides are equal, what is it called? Rectangle 2 is how many units long? How many squares does it contain? Rectangle 3 is how many units long? How many wide? How many squares does it contain? If it were four units long and one wide, how many squares would it contain? If it were five long and one wide? Six long? &c. Rectangle 4 is how many units long? How many wide? How many squares does it contain? How many squares in that part which is two units long, m n, and one unit wide, m l? On account of the second unit in width, l k, how many times two squares are there? If the width were one unit more, how many times two squares would there be? Rectangle 5 is how many units long? How many units wide or high? How many squares does it contain? How many squares in that part which is three units long, o p, and one unit wide, o t? The second unit in width, t q, gives how many more squares? How many times three squares? If another unit were added to the width, how many more squares would be made? How many times three squares? If it were four units wide, how many times three squares would there be? Rectangle 6 is how many units long? How many squares in that part which is four units long and one high? How many times four squares in that part which is four long and two high? How many times four squares when it is four long and three high? If another unit were added to the height, how many more squares would be added? How many times four squares would there be? If a rectangle were five units long and one unit wide, how many square units would it contain? If it were two units wide, how many times five square units would it contain? If it were three units wide? Four? &c. If your ruler is ten inches long and only one inch wide, how many square inches are there in it? If it were two inches wide, how many times ten square inches would it contain? If your arithmetic-cover is seven inches long and five inches wide, how many square inches are there in it? If a wall of this room is twenty feet long, how many square feet are there in that part which is one foot high? Two high? Three high? Four high? If the same wall is sixteen feet high, how many square feet in it? Fig. 5 has how many times three squares? Which has the greater number of squares? What difference is there between two times three squares and three times two squares? Draw a rectangle of any width whose length is three times the width. How many squares has it if the width be taken as the unit? Make it twice as wide as before. How many squares has it now? What two numbers multiplied together will give the number of squares? Make it three times as wide. How many squares has it now? What two numbers multiplied together will give the number of squares? The cover of a geography is one foot long and one foot wide, how many square feet in it? How many inches long is the same cover? How many wide? How many square inches does it contain? How many square inches are equal to one square foot? How many feet long is the same table? How many feet wide? How many square feet does it contain? One square yard equals how many square feet? Draw a square whose side is a unit of any length. Draw another whose side is two units of the same length. The second square is how many times as large as the first one? How many squares in half the second square? Which is greater, two square inches, or two inches square? Two inches square is how many times two square inches? Draw a square whose side is three inches. How many square inches does it contain? How many times as many squares as the square of one inch? How many square inches in the bottom row? How many in all? Which is greater, three inches square, or three square inches? Three inches square is how many times three square inches? PROBLEMS.An equilateral triangle has each of its sides one inch long, what is its perimeter? If each side were two inches long, what would be its perimeter? An isosceles triangle has its two equal sides each three inches long, and its third side five inches long, what is its perimeter? A right-angled isosceles triangle has its base five inches, and its hypothenuse seven inches long, what is its perimeter? A square geography-cover is nine inches long on one side, how long all round? How many square inches in it? A slate is sixteen inches long and twelve wide, how many inches all round it? A rectangle is five inches long and three wide, how long all round? How many square inches in it? A slate is one foot long and eight inches wide, what is its perimeter? A room is twenty-four feet long and twenty-one feet wide, how many feet all round it? How many square feet in the floor? How many pieces of paper each a foot square would exactly cover it? A yard of carpet is two feet wide, how many square feet in it? If they start from opposite ends of a straight walk twenty-five feet long, and walk towards each other, how many feet will Charles have to walk to meet Henry who has walked fifteen feet? A lot is forty rods long and thirty wide, how long must the fence be? What length of fence will divide it into four equal parts? Diagram 24. LESSON THIRTY-SECOND.THE CIRCLE AND ITS LINES.If the straight line c a were a string made fast at c, with a sharp pencil-point at the other end a, and the pencil-point were moved towards d, what line would be drawn? What kind of a line would it be? If the pencil-point continued to move in the same direction until it returned to the starting-point a, what curved line would be drawn, naming it by all the points in it which are marked? What point is at the centre of this figure? A circle is a plane figure bounded by a curved line, all points of which are equally distant from the centre. The curved line is called a “circumference.” The circumference of a circle is the curve which bounds it. Name a straight line that joins two points in the circumference. It is called a “chord.” A chord is a straight line that joins two points of a circumference. Read six chords in the diagram. Which two of these chords pass through the centre? They are called “diameters.” A diameter is a chord that passes through the centre. Name a line that joins the centre with a point of the circumference. It is called a “radius.”—(Plural, radii.) A radius is a straight line that joins the centre to a point of the circumference. Read five radii. Which is farther from the centre, the point a or the point d? Can the radius c d be greater than the radius c a? Or greater than c v, or c o? What do we call the lines o d, c d, c o? What part of the diameter o d is the radius o c? Name a chord that is produced without the circle. It is called a “secant.” A secant is a chord produced. Name two secants. If the chord d i were made a secant, would it become longer or shorter? In how many points does the straight line l m touch the circumference? It is called a “tangent.” A tangent is a straight line that touches a circumference in only one point. Name three tangents. LESSON THIRTY-THIRD.REVIEW.Read six chords. (Diagram 24.) Why is i d a chord? What is a chord? Name two diameters. Why is a j a diameter? What is a diameter? Is every chord a diameter? Name five radii. Why is c a a radius? What is a radius? A diameter is equal to how many radii? Are all radii equal to each other? Are all chords equal to each other? Are all diameters equal to each other? Name two secants. Why is either one a secant? What is a secant? Name three tangents. Why is a b a tangent? What is a tangent? Is a tangent inside of a circle or outside of it? Is a chord inside or outside of a circle? Is a secant within or without a circle? If the radius is three inches, how long is the diameter? Diagram 25. LESSON THIRTY-FOURTH.ARCS AND DEGREES.What small part of the circumference of circle 1 (Diagram 25.) is marked? It is called an “arc.” An arc is any part of a circumference. Read five arcs that are marked. Which is longer, the arc e d, or the arc e f? b d, or b d e? a b d, or a b d e? Name an arc which is half of the circumference. It is called a “semi-circumference.” “Semi” means “half.” A semi-circumference is half of a circumference. Read three arcs, each of which is one-fourth of the circumference. If the whole circumference were divided into three hundred and sixty equal arcs, would each arc be large or small? Each of these arcs would be called a “degree.” [Degrees are marked (°).] A degree of a circumference is a three hundred and sixtieth part of it. How many degrees in a semi-circumference? How many degrees in one-fourth of a circumference? If a fourth of a circumference were divided into three equal parts, how many degrees would there be in each part? Is an arc of ninety-one degrees greater or less than one-fourth of a circumference? Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference? Can there be more than three hundred and sixty degrees in a circumference? If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot. If a degree were divided into sixty equal parts, each part would be called a minute. If a minute were divided into sixty equal parts, each part would be called a second. How many degrees in the large circle of Fig. 2? How many in the smaller one? Has a large circle any more degrees than a small circle? In the large circle how many degrees from a to b? In the small circle how many from a to b? Which is greater, an arc of ninety degrees of the large circle, or one of ninety degrees of the small one? Which is greater, an arc of a degree of the large circle, or one of a degree of the small one? The angle a o b has its vertex at what part of the larger circle? At what part of the smaller circle? On how many degrees of the larger circle does the angle stand? Then it is said to be an angle of 90°. If the angle a o f is an angle of 30°, how many degrees must there be in the arc a f? If the arc f e is an arc of 60°, what is the size of the angle f o e? An angle of 10° stands upon an arc of how many degrees? Of 8°? Of 1°? The angle a o b is what kind of an angle? Upon how many degrees does it stand? Then a right angle is an angle of how many degrees? If an angle stand upon less than 90°, what kind of an angle is it? If an angle stand upon more than 90°, what kind of an angle is it? Can an angle have as many degrees as a hundred and eighty? LESSON THIRTY-FIFTH.REVIEW.Read nine arcs whose ends are marked. (Diagram 26.) Read three arcs each of which is one-fourth of a circumference. Read two arcs each of which is one-half of a circumference. Why is e g an arc? What is an arc? How many degrees in the arc f h? In e h? If the arc f h were divided into three equal parts, how many degrees would there be in each? How many degrees in a circumference? In a semi-circumference? How many more degrees in a large circumference than in a small one? If the arc i f is 40°, what is the size of the angle f o i? If the angle f o g is an angle of 130°, what is the size of the arc f i h g? How many degrees in each of the adjacent angles f o h, h o e? When two adjacent angles are equal to each other, what is each called? How many degrees in a right angle? Diagram 26. LESSON THIRTY-SIXTH.PARTS OF THE CIRCLE.The part of the circle bounded by the chord a b and the arc a b is called a segment. Read three segments, each less than half a circle, thus,—the segment bounded by the chord a d and the arc a b d. A segment is a part of a circle bounded by an arc and a chord. Read two segments that are each half a circle. What is the chord called? What is the arc called? A semicircle is half a circle. Read four segments each larger than a semicircle. The part of the circle between the two radii o f, o i, and the arc f i, is called a “sector.” Read four sectors each less than one-fourth of a circle. 2.Thus, a sector bounded by the two radii o g, o h, and the arc g h. A sector is a part of a circle bounded by two radii and an arc. What part of the whole circle is the sector f o h? It is called a “quadrant.” A quadrant is a sector which is one-fourth of a circle. Read a sector which is greater than a quadrant. If the chord e f be regarded a diameter, what do you call the semicircle below it? If it be regarded as two radii, what is the semicircle called? Then a semicircle is both a segment and a sector. LESSON THIRTY-SEVENTH.REVIEW.Name ten segments. (Diagram 26.) What is a segment? Of the segments named, which are less than a semicircle? Which are greater? Which two are semicircles? Which two are on the chord a f? Name nine sectors. Why is g o i a sector? What is a sector? Which four of the sectors named are each less than a quadrant? Which three are quadrants? Which two are greater than a quadrant? What part of the circle is both a segment and a sector? How many quadrants in a circle? How many semicircles? |
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