Bernhard Marks.

Marks' first lessons in geometry, objectively presented online

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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?

[Illustration: 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_.

Read three other interior angles between the same parallels.

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.

Why is _d n f_ an exterior angle?

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.

[Illustration: 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?

Then they are called opposite angles.

_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.

_Alternate angles lie on different sides of the intersecting line,
and are neither adjacent nor vertical._

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.

[Illustration: 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 six pairs of vertical angles formed by the same lines.

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?




LESSON EIGHTEENTH.


PROBLEMS.

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?

[Illustration: 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._

An octagon has eight 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.

[Illustration: 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?

Then the point _g_ is said to be the vertex of the triangle.

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.

[Illustration: 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.

Name a triangle that has no two sides equal to each other.

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?

[Illustration: 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.

[Illustration: 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.

Why is Fig. 6 a 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?

[Illustration: 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.

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?


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?

[Illustration: 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?


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Online LibraryBernhard MarksMarks' first lessons in geometry, objectively presented → online text (page 2 of 6)