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.
Diagram 16.
Diagram 16.
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? (Diagram16.)
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 apentagon.
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.
An octagon has eight sides.
A nonagon has nine sides.
A decagon has ten sides.
All these figures are calledpolygons.
“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 aregular 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 aregular polygon, orregular 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.
Name all the triangles. (Diagram16.)
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.
Diagram 17.
Diagram 17.
In the triangle 1, what kind of an angle isb a c?a c b?c b a? (Diagram17.)
Then it is called anacute-angled triangle.
An acute-angled triangle is one whose angles are all acute.
Read three other acute-angled triangles.
In the triangle 4, what kind of an angle isl k m?
Then it is called anobtuse-angled triangle.
An obtuse-angled triangle is one that has one obtuse angle.
Name two others.
In the triangle 3, what kind of an angle isg i j?
Then it is called aright-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?
Theni jis called thebaseof the triangle.
What letter marks the vertex of the angle opposite the base?
Then the pointgis said to be the vertex of the triangle.
If, in the triangle 7, we considert vthe base, what point is the vertex?
Ifvbe considered the vertex, which side will be the base?
In the triangle 3, what side is opposite the right angle?
Theng jis called thehypothenuseof 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, ifi jis the base, which side is the perpendicular?
Ifg ibe considered the base, which side is the perpendicular?
In triangle 5, ifn ois the base, which side is the perpendicular?
Name four acute-angled triangles. (Diagram17.)
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?
Ifq rbe considered the base, what do you call the sideq s?
Read the hypothenuse of each of the triangles 3, 5, 6, and 11.
Diagram 18.
Diagram 18.
Diagram 18.
Of the triangle 1, which two sides are equal to each other?
Then it is called anisosceles 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 anacute-angled isosceles triangle.
Name four acute-angled isosceles triangles.
What kind of a triangle is Fig. 4 on account of the anglek j l?
What kind on account of its equal sides?
Then it is called anobtuse-angled isosceles triangle.
Name one other obtuse-angled isosceles triangle.
What kind of a triangle is Fig. 6 on account of the angleq p r?
What kind on account of its equal sides?
Then it is called aright-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 anequilateral 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 sidess t,s u, ort s,t u, oru 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 sidesd e,d f?
If the sidee fis 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 isnotequilateral.
Name one thatisequilateral.
In any equilateral triangle the three angles are equal to each other.
On account of its equal angles, it is also called anequiangular 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.
Name a triangle that has no two sides equal to each other.
It is called ascalene 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 aright-angled scalene triangle.
What name can you give Fig. 11 on account of the angleg e f?
On account of its three unequal sides?
Then what may it be called?
Diagram 19.
Diagram 19.
Diagram 19.
Name eight isosceles triangles. (Diagram19.)
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 arenotequilateral.
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.
Diagram 20.
Diagram 20.
QUADRILATERALS.
How many sides has the figurea 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 quadrilaterala b d c, orc d b a, may be reada d, orb c, orc b, ord 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 calledquadrangles.
Has the quadrilaterala dany two sides parallel to each other?
Then it is called atrapezium.
A trapezium is a quadrilateral that has no two sides parallel.
Name two other trapeziums.
Why is Fig. 7 a trapezium?
Has the quadrilaterale hany 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.
Why is Fig. 6 a trapezoid?
How many pairs of parallel sides has the quadrilaterali 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 note ha 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?
How many quadrilaterals in the diagram. (Diagram20.)
Why is Fig.a da 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.
Diagram 21.
Diagram 21.
How many quadrilaterals in the diagram? (Diagram21.)
How many parallelograms?
Has the parallelograma dany 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?
Are the four sides of the rhomboida dequal to each other?
Are the four sides of the rhomboide hequal 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 NoteD, Appendix.
Name two other rhombuses, or rhombs.
What four names can you give to Fig.e h?
Why a quadrilateral? Why a parallelogram? Why a rhomboid? Why a rhombus?
Has the parallelogrami lany 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?
Has the rectanglei lits four sides equal?
Has the rectanglem pits 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?
Name six rhomboids. (Diagram21.)
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.
Diagram 22.
Diagram 22.
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 onewithoutright angles?
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?
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?
On this account, what particular name may be applied to Fig. H?
What three names may be applied to Fig. G?
Whatfourto Fig. H?
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?
What two names may be given to Fig. A. (Diagram22.)
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.
Diagram 23.
Diagram 23.
In Fig. 1 (Diagram23.) call the linea b aunit.
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 high or wide?
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 units high or wide?
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?
Fig. 7 has how many times two squares?
Which has the greater number of squares?
What difference is there between two times three squares and three times two squares?