Third Year Geometry

Third Year GeometryI. Points, Lines, and Planes

Ideas cannot be seen, they exist in our mind. Numbers cannot be seen, but we use numerals to represent them. Points cannot be seen, dots are physical representation of points.

Examples :

B

1. Fig. !. A

C

Points can be named by a capital letters. We have point A,

point B, and point C.

2. Fig. 2. l In Geometry, a line connotes a straight line. The arrowheads

A B indicate that the line has no end point. A line is named by the

names of any two points of the line or by small letter near one

of the arrowheads. In the figure a line can be named as

line l , or line AB, or in symbol l or AB .

B

3. Fig.3. Collinear points is a set of points which are in the line.

A C D

In the figure, point A, point C, and point D are collinear,

but point B is not collinear with points A,C, and D.

4. The subsets of a line are line segment or segment, ray, and half-line.

a. Line Segment = Set of points which is formed by two distinct points of a line and all the points

of the line between them.

Example. read as segment AB or in symbol AB . A B

b. Ray

= Set of points which is formed by a line segment and all the points of the line on

only one side of the line segment.

Example. read as ray AB or in symbol AB . A B

c. Half-line = Set of points which is formed by all the points of a line on one side of the fixed

point but excluding the fixed point.

Example.

o read as half-line AB. The tiny circle indicates that the vertex A B of a ray is not included in the set of points.

5.Opposite rays = are two distinct rays that have the same vertex and which lie on the same line.

Example. ray BA and ray BC are opposite rays, they have the A B C same vertex which is B.6.Plane= A plane is a perfectly flat surface, extending infinitely far in all directions. A portion of a

plane has length and width but no thickness. A plane may be named by the name of a point

in the plane such as plane M below. If three or more points are in a plane then the points are

said to be coplanar points.

M B

A C

Exercises A. Use the figure below to answer the following. A. In the figure name :

D B 1.Three distinct lines

A C

________________________________________

2.nine distinct line segments

F

________________________________________

E G

H 3.ten different rays

________________________________________

4.six half-lines

________________________________________

5.three set of 3 points that are collinear

________________________________________

6.Three sets of opposite rays

________________________________________

B. Identify the term described. Write your answer on the space provided for.

_____________________ 1. Subset of a line having only one endpoint.

_____________________ 2. Subset of a line having two endpoints.

_____________________ 3. Has length only_____________________ 4. Has length and width but no thickness

_____________________ 5. Three or more points lying on the same line.

_____________________ 6. Endpoint of a ray.

_____________________ 7. Three or more points in the plane

_____________________ 8. Subset of a line which is formed by all the points of a line on one side

of a fixed point but excluding the fixed point.

_____________________ 9. Shows location only

II. Angles is a set of points formed by two non-collinear rays which have a common vertex. Angles can be named

by three capital letters. Protractor is the instrument used to measure an angle. A

1.

= Named as angle AOB or angle BOA or just simply angle O. The vertex of an

O B

angle can also be used to name angle.

A 2. O 1

= another way of naming an angle is by the use of numerals or small letters.

in this example, angle AOB, angle BOA, angle O, and angle 1 all refer B

to the same angle.3. A

= in this figure four angles are formed namely : angle AOB, angle BOD,

O

C B angle DOC and angle COA. In this case we cannot use the vertex as the name

D of the angle, because the vertex is the point of intersection of two lines AB

and CB, which form four distinct angles.

A. Kinds of angles Acute angle is an angle that measures less than 900

ex. Right angle is an angle that measures 900

ex. Obtuse angle is an angle that measures more than 900 but less than 1800.

ex. Straight angle Is an angle that measures exactly 1800.ex. Reflex angle is an angle that measures more than 1800 but less than 3600.

ex. Complete or whole angle is an angle that measures exactly 3600.

.

Exercises. A. Use your protractor to sketch the following angle and write if it is acute, right, obtuse. straight, reflex or whole angle.

1. 4502.18003.9004.12005.2240B. Use the figure to answer the following: A B 1.m HID + m DIE = ______________ C D E

2.m DEI + m IEJ = ______________ F K

3.m CDI - m CDH = ______________ G H J

I

4.m LHD - m LHI = ______________

5.m IJE + m JEB = ______________ L MIII. Transversal A transversal line is a line that intersects two or more lines in distinct points.

l

Examples.

O = line l intersects line m and line n at points O and P respectively.1. m

line l is called the transversal line, because it intersects two lines at distinct points. P n

p T = line p intersects line q , line r , and line s at points T, U and V 2. q respectively. Line p is called the transversal line. U r s VAngles Formed by Lines Cut by a TransversalExamples.

1. Fig. 1.a 1.Corresponding angles are an exterior angle and a non-adjacent 1 2

interior angle on the same side of the transversal. 3 4

ex. 1. angle 1 and angle 5 are corresponding angles 5 6

2. angle 3 and angle 7 are corresponding angles 7 8

3. angle 2 and angle 6 are corresponding angles

4. angle 4 and angle 8 are corresponding angles

2.Alternate exterior angles are two exterior angles which are

non-adjacent and which lie on opposite side of the transversal.

ex. 1. angle 1 and angle 8 are alternate exterior angles

2. angle 2 and angle 7 are alternate exterior angles

3.Alternate interior angles are two interior angles which are

non- adjacent and lie on opposite sides of the transversal.

ex. 1. angle 3 and angle 6 are alternate interior angles

2. angle 4 and angle 5 are alternate interior angles

4. Same-side interior angles are non-adjacent interior angles lie on the same side of the transversal.

ex.

1. angle 3 and angle 5 are same side interior angles

2. angle 4 and angle 6 are same side interior angles

5. Same-side exterior angles are non-adjacent exterior angles lie on the same side of the transversal.

ex. 1. angle 1 and angle 7 are same-side exterior angles

2. angle 2 and angle 8 are same-side exterior angles

6. Vertical angles are pair of non-adjacent angles formed by the intersection of two straight lines. ex.

1. angle 1 and angle 4 are vertical angles

2. angle 2 and angle 3 are vertical angles

3. angle 5 and angle 8 are vertical angles

4. angle 6 and angle 7 are vertical angles

7. Adjacent-supplementary angles also known as Linear Pair. Two angles that have a common side and whose other sides lie on the same line.

ex.

1. angle 1 and angle 2 are adjacent-supplementary angles

2. angle 2 and angle 4 are adjacent-supplementary angles

3. angle 4 and angle 3 are adjacent-supplementary angles

4. angle 1 and angle 3 are adjacent-supplementary angles

5. angle 5 and angle 6 are adjacent-supplementary angles

6. angle 6 and angle 8 are adjacent-supplementary angles

7. angle 7 and angle 8 are adjacent-supplementary angles

8. angle 5 and angle 7 are adjacent-supplementary angles

Exercise : Use the figure to answer the following.

Name the following : A

1.4 pairs of corresponding angles B D

________________________________________________ C

2.2 pairs of alternate exterior angles

_____________________________________________ E F G

3.2 pairs of alternate interior angles

H

_________________________________________________

4.2 pairs of same side interior angles

_________________________________________________

5.4 pairs of vertical angles

_________________________________________________

IV. Polygons

IV.A. Polygon is formed by the union of three or more line segments lying on the same plane. Classification of polygons.

Triangle

= is a polygon with three sides.

Quadrilateral= is a polygon with four sides.

Pentagon

= is a polygon with five sides.

Hexagon

= is a polygon with six sides.

Heptagon

= is a polygon with seven sides.

Octagon

= is a polygon with eight sides.

Nonagon

= is a polygon with nine sides.

Decagon

= is a polygon with ten sides.

Undecagon

= is a polygon with eleven sides.

Dodecagon

= is a polygon with twelve sides

n-gon

= is a polygon with n sides.

* Triangle is a three sided polygon, formed by connecting three points not in a straight line.

The sum of the measures of the three angles of a triangle is equal to 1800.

The exterior angle is equal to the sum of the measures of the remote interior angles.

1. Kinds of triangle according to angles Equiangular triangle is a triangle with three equal sides. Acute triangle is a triangle that has all angles less than 900. Right triangle is a triangle with a 900 angle.

Obtuse triangle is a triangle with an obtuse angle.2. Kinds of triangle according to sides Equilateral triangle is a triangle with three equal sides.

Scalene triangle is a triangle with no sides equal.

Isosceles triangle is a triangle with at least two sides equal.

IV.B. Measuring the angles of a Triangle.

Examples:

1. Find x 870 Solution : Since the sum of the three angles of the triangle is 1800.

then, x + 87 + 43 = 180

x 430

x = 180 ( 87 + 43 )

x = 180 130

x = 50

2.

Solution : 2x + 2 + 4x + 46 = 180

4x

6x + 48 = 180 combine similar terms 6x = 180 48 transpose constant term

2x + 2 460 at the right side of the

equation.

6x = 132

divide both side by

6 6

the coefficient of x.

x = 22Exercises : A. Use the figure to identify each of the following polygons. Write your answer on the

space provided for.

__________________ 1. __________________ 6.

__________________ 2. __________________ 7.

___________________ 3. __________________ 8.

___________________ 4. __________________ 9.

___________________ 5.

__________________ 10. B. Solve for the unknown value of each angle of the triangle. B

a =

5a angle B =1. C 4a + 3 A angle A =

B

x =

x0

angle A =2.

angle B =

C x0 (x-30)0 A

IV.C Measuring the interior and exterior angles of a regular polygon. Regular polygon is a polygon having all angles and all sides are equal.

The interior angles of a polygon can be measured using the formula ( n - 2 ) x 180, where n is thetotal number of sides/angles.

Examples :

1. Find the measure of each angle of a regular polygon with 6 sides. Using the formula ( n 2 ) x 180 , we have

( 6 2 ) x 180

= ( 4 ) x 180

=720, since the polygon has 6 angles then

=720 6 = 120 is the measure of each angle.

2. Find the measure of each angle of a regular polygon with 8 sides

Using the formula ( n 2 ) x 180, we have

=( 8 2 ) x 180

=( 6 ) x 180

=1080, since the polygon has 8 angles then,

=1080 8 = 135 is the measure of each angle.

Exercises. A. Find the measure of each interior and exterior angle of a regular polygon with:

Interior Angle

Exterior Angle1.24 sides.

___________

____________2.18 sides.

___________

____________3.72 sides.

___________

____________4.26 sides.

___________

____________5.29 sides.

___________

____________B. Use the figure to solve for the value of x , and the measure of each angle below.

Write your answer on the space provided for.

B C A 820 680 D E 1080 H 14x - 4 F G 15x + 18 J I1.x

= __________________9.m BAE=__________________

2.m AGI=__________________10.m DEH=__________________

3.m GAF=__________________11.m HEF= __________________4.m BAC=__________________12.m DEA=__________________5.m EFJ=__________________13.m CAF=__________________6.m AFE=__________________14.m GAE=__________________7.m JFI=__________________15.m GAB=__________________8.m HEF=__________________V. Circles

- The word circle was derived from the Latin word circus which means ring or racecourse.

- It is a set of all points in a plane that are equidistant from a fixed point called the center.

ex.

F center ; named as circle F

radius

- Circles are named by their centers.

- The distance from the center to a point on the circle is called the radius.

- The line through any two points of the circle that passes through the center is called the

diameter.

- The segment that joins any two points of the circle without passing the center is called chord.

- The chord, when extended to both direction is called a secant.

ex.

diameter

M N M N chord Secant

- A line that lies on the exterior of the circle and intersects the circle at exactly one point

is called the tangent line.

- The point of intersection of the circle and the tangent is called the point of tangency.

AB is the tangent line , while point O is the point of A

tangency. F O B

- An angle whose vertex is the center of the circle is called central angle.

- A central angle separates the circle into arcs. Arcs are the broken parts of the circle.

All points of the circle interior to central angle form a minor arc. All points of the circle exterior to the central angle is called a major arc, at least 3letters are needed to name a major arc.

- The endpoints of a segment containing the diameter of a circle separate the circle into 2 arcs

called the semi-circle.Exercises. A. Given the circle below, Identify what each of the following illustrates. Write your answer on the space provided for.

B A C O D E F M G H I J K L

__________________ 1.BOH

__________________ 6. GD __________________ 2. CH

__________________ 7. BH

__________________ 3. JL

__________________ 8. CHB

__________________ 4. K

__________________ 9. OH __________________ 5. OB

__________________ 10. GDB. Write T if the statement is correct and F if it is false. Write your answer on the space provided for.

____________ 1. The longest chord in a circle is its diameter.

____________ 2. The radius of a circle is twice the measure of its chord.

____________ 3. A secant was formed when the chord is extended.

____________ 4. A circle is a closed curve figure with set of all points equidistant from the

center.

____________ 5. Tangent is a line that lies on the exterior of the circle and intersect at exactly

one point of that circle.

____________ 6. The point of intersection between the circle and the line which lie in the

exterior of the circle is called the chord.

____________ 7. In the figure above the points G, K, H, D, B and C forms a minor arc.

____________ 8. In the figure, the distance between O and K is also known as the radius.

____________ 9. COB in the figure is also called the central angle.____________ 10. The intersection of line AI and line FE is point M.

VI. Solid GeometrySolid or space figures are geometric figures having three dimensions: the length, the width, and the height. If a two dimensional figures are called a polygon, a three dimensional figures are called a polyhedron.A polyhedron has faces, edges and vertices. The flat surfaces formed by a polyhedron and their interiors are called faces. Pairs of faces intersect at line segments are called the edges. Three or more edges intersect at a point is called the vertex.Illustrative example.

edge vertex

faces

A polyhedron is regular if all its faces are shaped like congruent regular polygons. Since all of the faces of a regular polyhedron are regular and congruent, all of the edges of a regular polyhedron are also congruent.

A. Prisms

A polyhedron is a prism, if and only if, two of its faces are joined by a congruent polygon in distinct parallel planes, while its other faces are joined by a parallelogram.

Illustrative examples :

cube rectangular prism square prism triangular prism

B. Pyramids

If one of the base of a polyhedron is a single point, then the polyhedron is a pyramid. A pyramid has several lateral faces and lateral edges but only one base.Illustrative examples :

Square pyramid

Rectangular pyramid

Triangular pyramid

VI.A. Area of polygonsa. Area of a square - the area of a square is given by A = s2, where s is the side.

example. 5cm Solution : 5cm A = 52

= 25cm2b.Area of a Rectangle the area of a rectangle is given by A = l x wI, where l is the length

and w is the width.

example.

4cm

Solution :

A = 4 x 2

2 cm

= 8cm2C.Area of a triangle the area of a triangle is given by A = bh, where b is the base and

h is the height.

example.

Solution :

A= bh

A = ( 15) ( 5 ) h= 5m A = ( 75 )

A = 75/2 b = 15m

A = 37.5m2D. Area of a circle the area of a circle is given by A = r2 , where r is the radius and = 3.14. example.1.

Solution :

r = 8cm

A = r2

A = 3.14 ( 8 )2

A = 3.14 ( 64 )

A = 200.96cm2

example.2.

Solution

Since d=2r, then d/2 = r. So we have,

r = 6/2 = 3.

A = r2

d = 6m

A = 3.14 ( 3 )2

A = 3.14 ( 9 )

A = 28.26m2VI.B. Surface area of a polyhedron = The surface area is the sum of the areas of the bases and faces of a solid figure. It is

expressed in square units.

A. Surface area of rectangular prism is given by SA = 2 [ ( l x w ) + ( l x h ) + (w x h )]

ex. Solution : SA = 2 [ ( l x w ) + ( l x h ) + (w x h )]

= 2 [ ( 12 x 10 ) + ( 12 x 2 ) + ( 10 x 2 ) ]

l = 12cm h =2cm

= 2 ( 120 + 24 + 20 ) w =10cm

= 2 ( 164 )

= 328cm2B. Surface area of a cube is given by SA = 6s2

ex. SA = 6s2

SA = 6 ( 8 )2 8cm

SA = 6 ( 64 )

SA = 384cm2C. Surface area of a square prism is given by SA = 2 x are of square + 4 x area of lateral faces.

ex.

Solution : SA = 2 ( 4 )2 + 4 ( 8 x 4 )

SA = 2 ( 16 ) + 4 ( 32 )

4m

SA = 32 + 128

SA = 160m2

8mD. Surface area of triangular prism is given by SA = 2 x area of a triangle + LA, wherein LA is

the sum of the area of the lateral sides.

ex.

Solution : SA = 2 ( ) ( 4 ) ( 3 ) + ( 7 x 5 ) + ( 7 x 3 ) + ( 7 x 4 ) 5in SA = 2 ( 12/2 ) + 35 + 21 + 28 3in

SA = 24/2 + 84 7in

SA = 12 + 84 4in

SA = 96in2E. Surface area of square pyramid is given by SA = area of base + 4 ( area of lateral sides )

ex.

Solution : SA = 32 + 4 [ ( 3 x 5 ) ]

SA = 9 + 4 [ ( 15 ) ] slanted height h = 5m

SA = 9 + 4 ( 15/2 )

SA = 9 + 60/2

SA = 9 + 30 3m

SA = 39m2F. Surface area of Triangular pyramid is given by SA = area of base + Area of lateral sides

ex.

Solution : SA = ( 3 x 5 ) + 5x9/2 + 7x9/2 + 3x9/2 h = 9 cm

SA = 1/ 2 ( 15 ) + 45/2 + 63/2 + 27/2

SA = 15/2 + 22.5 + 31.5 + 13.5 7cm

SA = 7.5 + 67.5

SA = 75cm2 3cm 5cmG. Surface area of rectangular pyramid is given by SA = area of base + area of lateral faces

ex.

Solution : SA = ( 4 x 2 ) + 2 ( 5x2/2 ) + 2 ( 5x4/2 )

SA = 8 + 2 ( 10/2 ) + 2 ( 20/2 )

SA = 8 + 2 ( 5 ) + 2 ( 10 ) h = 5cm

SA = 8 + 10 + 20

SA = 38cm2

4cm 2cmH. Surface area of Sphere - is given by SA = 4r2

ex. radius = 5mm Solution :

SA = 4 ( 3.14 ) ( 5 )2

SA = 12.56 ( 25 )

SA = 314mm2I. Surface area of cylinder - is given by SA = 2r2 + 2rh or SA = 2r ( r x h ).base ( b )

example . Find the surface area of a cylinder whose

base has the radius of 4cm and whose

height is 8 cm. altitude h

Solution : Given : r = 4cm

h = 8cm radius r

SA = 2r2 + 2rh base ( a )

SA = 2 ( 3.14 ) ( 4 )2 + 2 ( 3.14 ) ( 4 ) ( 8 )

SA = 6.28 ( 16 ) + 6.28 ( 32 )

SA = 100.48 + 200.96

SA = 301.44cm2J. Surface area of a Cone - is given by SA = r2 + r h, where r is the radius of the base and

h is the slant height.

example. Find the surface area of a cone whose base has a

radius of 3 dm and whose slant height is 9 dm.

slant height Solution: Given : r = 3 dm

h = 9 dm SA = r2 + r h radius

SA = 3.14 ( 3 )2 + 3.14 ( 3 ) ( 9 )

SA = 3.14 ( 9 ) + 3.14 ( 27 )

SA = 28.26 + 84.78

SA = 113.04dm2Exercises. A. Write T if the statement is true and F if it is false. Write your answer on the space

provided for.

_________ 1. The three dimensional figure is also known as polyhedron.

_________ 2. A polyhedron is a representation of a plane._________ 3. A plane is a perfectly flat surface, just like the sheet of a paper.

_________ 4. A regular polyhedron is a figure which has no-congruent parts.

_________ 5. A polyhedron is a prism, if and only if, two of its faces are joined by a congruent polygon in distinct parallel planes, while its other faces are joined by a parallelogram.

_________ 6. A pyramid is a solid figure with two equal bases._________ 7. The perimeter is the sum of all the sides of a figure.

_________ 8. The diameter of a circle is twice the measure of its circumference.

_________ 9. The circumference refers to the perimeter of a circle.

_________ 10. The surface area can also be applied in a plane figure.

B. Find the area of the following figures.

5in

Area1.

______________ 5 in 7cm

2.

_______________

4cm

3. 5m

________________

4m

4.

r = 6in

________________C. Analyze and solve for the following problems.

1.What is the area of a square swimming pool if the measure of each side is 7m?

Given :

Solution:

2.Mr. Domingo is replacing the side of the triangular portion of his house having a base of 18m

and height of 7m.That portion was damaged by termites. What is the area of the section that

Mr. Domingo needs to replace?

Given :

Solution :

3.The diameter of a circular field is 1000m. One fourth of the field will be planted with corn,

and the rest with rice. How many square meters will be planted with rice?

Given :

Solution :

D. Find the surface area of the following.

Solutions here :1.

4cm 16cm

12cm

5mm2. 3mm

7mm 4mm

3.

10cm 6cm

4. h = 8m 6m

2m 4m

5.

h = 12ft r = 4ft