2. Chapter 9 - Geometry and Measurement5 4 3 2 1
Geo
metry
Lesson:
General Review
Prepared by:

3. BASIC CONCEPTS OF GEOMETRY POINT PLANE LINE
Chapter 9 - Geometry and Measurement
BASIC CONCEPTS OF GEOMETRY POINT PLANE LINE

4. Point
The most basic concept of geometry is the idea of a point in space.
A point has no length, no width, and no height, but it does have location. We will represent a point by a dot, and we will label points with letters. A
Point A

5. A plane is a flat surface that extends indefinitely.
Space extends in all directions indefinitely.

6. LINE
A line is straight arragement of points . A line has no width ,no thickness and extends without end in both direction.

7. Line AB or AB Line Segment AB or AB Ray AB or AB

8. Parallel lines Intersecting lines
Two lines in a plane can be either parallel or intersecting.
Parallel lines never meet
Intersecting lines meet at a point.
The symbol  is used to denote “is parallel to.” p q
Parallel lines
Intersecting lines
p  q

9. n  m Perpendicular lines
Two lines are perpendicular if they form right angles when they intersect.
The symbol  is used to denote “is perpendicular to.” n m
n  m
Perpendicular lines

10. Chapter 9 - Geometry and Measurement

11. We should not confuse difference between angle and angel
ANGLES
We should not confuse difference between angle and angel

12. The angle can be named ABC, CBA, B or x.
An angle is the union of two rays that have a commen endpoint.
An angle is made up of two rays that share the same endpoint called a vertex. A B C x
Vertex
The angle can be named ABC, CBA, B or x.

13. Angles are measured by an amount of rotation
We measure this rotation in units called
degrees .we show it as
360º

14. Classifying Angles Name Examples Angle Measure Acute Angle
Between 0° and 90°
Right Angle
Exactly 90°
Obtuse Angle
Between 90° and 180°
Straight Angle
Exactly 180º

15. AnGlES(song)
Everywhere you look There are angles. Acute, obtuse and right angles.
Everywhere you look There are angles. How many can you find?
Right angles have 90 degrees. Acute have less than these. Obtuse angles open wide, Wider than 90 degrees.
Everywhere you look There are angles. Acute, obtuse and right angles.
Everywhere you look There are angles. How many can you find?
Right angles have 90 degrees. Acute have less than these.
Obtuse angles open wide, Wider than 90 degrees.

16. Chapter 9 - Geometry and Measurement
When two lines intersect, four angles are formed. Two of these angles that are opposite each other are called vertical angles. Vertical angles have the same measure. a b c d
a = c
d = b

17. Two angles that share a common side are called adjacent angles
Two angles that share a common side are called adjacent angles. Adjacent angles formed by intersecting lines are supplementary. That is, they have a sum of 180 °. a b c d
a and b
b and c
c and d
d and a

18. Two angles that have a sum of 90° are called complementary angles.
Two angles that have a sum of 180° are called supplementary angles.

19. A line that intersects two or more lines at different points is called a transversal.

20. Parallel Lines Cut by a Transversal
If two parallel lines are cut by a transversal, then the measures of
corresponding angles are equal(1) and
alternate interior angles are equal(2).
Alternate exterior angles are equal(3).
Same-side interior angles
are suplementary a b c d e f g h

21. Corresponding angles are equal.b c d e f g h
c = g
a = e
d = h
b = f
Same-side interior angles
c e=180
d f=180

22. Alternate exterior angles
Alternate interior angles are angles on opposite sides of the transversal between the two parallel lines. a b c d
c = f
e = d e f g h
Alternate exterior angles b a = h g =

24. TRIANGLES

25. TRIANGLE: Triangle is a polygon with three sides
TRIANGLE: Triangle is a polygon with three sides. If we connect three noncollinear poinst we get a triangle
vertices
the plural of vertex is vertices
listening

26. A TRIANGLE Has three angles and three sides
The word triangle means “three angles”
symbol of a triangle
has three vertices

27. Regions of a triangle:
A triangle separrates a plane into three different regions
These regions are the triangle itself
And the interior and exterior region of the triangle
EXTERIOR REGION
ON THE TRIANGLE
INTERIOR

28. AUXILIARY ELEMENTS OF A TRIANGLE
MEDIAN ANGLE BISECTOR ALTITUDE

29. Chapter 9 - Geometry and Measurement
Perimeter of a Triangle
Perimeter =The perimeter of a triangle is the sum of the lenghts of its sides
P = side a + side b + side c

30. Area of a Triangle
The area of a triangle is half of the product of the lenght of a base and the height of the altitude drawn to that base.

31. Perimeter is always measured in units.
Helpful Hint
Perimeter is always measured in units.
The perimeter of every polygon may be found by adding all the sides.

32. Helpful Hints Area is always measured in square units.
When finding the area of figures, check to make sure that all measurements are the same units before calculations are made.

33. TYPES OF TRIANGLE
we can classify triangles accordingto the lengths of their sides or according to the measures of their angles
listening

35. A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees).
The hypotenuse of a right triangle is the side opposite the right angle. The legs of a right triangle are the other two sides.
hypotenuse
leg
leg

36. What is the sum of the measures of the interior angles of a triangle?
Chapter 9 - Geometry and Measurement
a+b+c=?
Angles on a Triangle
a+b+c=?
What is the sum of the measures of the interior angles of a triangle?

38. Chapter 9 - Geometry and MeasurementB A C B + C + A
The sum of the measures of the interior angles of a triangle is 180°.

39. Helpful Hint
The sum of the two acute angles in a right triangle is 90.
Equilateral triangle’s all angles are 60°
Isosceles triangle’s bases angles are congruent

40. Triangle Exterior Angle Theorem
Chapter 9 - Geometry and Measurement
Triangle Exterior Angle Theorem
The measure of an exterior angle (z) in a triangle is equal to the sum of the measures of its two nonadjacent (y),(x) interior angles.
listening

41. The sum of the measures of the exterior angles of a triangle is equal to 360°

42. Next week on monday I. exam

43. RELATIONS BETWEEN ANGLES AND SIDES
1.LONGER SIDE OPPOSITE LARGER ANGLE

44. 2.LARGER ANGLE OPPOSITE LONGER SIDE

45. If two sides of a triangle are congruent,
the angles opposite these sides are also
congruent.
If two angles of a triangle are congruent, the sides opposite these angles are also congruent

46. examples
Write the measures of the angles in each triangle in increasing order.
Write the lenghts of the sides from the smallest to the biggest of each triangle .

47. ,
Triangle inequality
1- The sum of the lengths of two sides of a triangle is
greater than the length of the third side.

48. The difference between the lengths of two sides of a triangle is less than the length of the third side.
If the first and the second properties are joined, we get

49. example cm find the possible integer Values of In the given figure,. .
In the given figure, cm
find the possible integer
Values of

50. Some properties (((gstrmmd1 2 3

51. examples

52. In a scalene triangle ABC, ,
and
and
In a scalene triangle ABC
In an equilateral triangle ABC

53. Unknown words
Ray :A ray is a straight line which extends infinitly in one direction from a fixed point.
Amount , a collection or mass especially of something which can not be counted
Flat:level and smooth,with no curved
Angle:The space between two lines or surface at the point
at which they touch each other .
Angel:A spiritual creature in religions.
Share:a part of something that has been divided
between several people.
Polygon:A flat shape with three or more straight sides.

54. Unknown words
Corresponding(shesabamisi)=The points, lines, and angles which match perfectly
when two congruent figures are placed one on top of the other

55. The difference of a winner and a loser

56. CONGRUENCE (n)(kongruenteloba)
CONGRUENT (adj) (kongruentuli)means
equal in all respects(things).
Objects which have same size and same shape
are called congruent objects.

57. So, the figures are not congruent
The figures have the same shape but they have different size.
So, the figures are not congruent

58. The figures have the same size and same shape.
So they are all congruent figures

59. Congruent Triangles .
if the corresponding angles and corresponding sides are congruent,
then these triangles are called congruent triangles.

60. corresponding angles are congruent to each other
all corresponding sides are congruent to each other.

61. Example: sides MN and ST sides NP and TX sides PM and XS
state the congruent parts without drawing the triangles.
sides MN and ST
sides NP and TX
sides PM and XS
which sides are congruent
Angles M and S
Angles N and T
which angles are congruent
Angles P and X

62. We can write the congruence in six different ways

63. B.WORKING WITH CONGRUENT TRIANGLES
1.The Side-Angle-Side (SAS) Congruence Postulate

65. example

66. homework
And page 152
Check yourself12..1and 2

67. 2.Angle- Side- Angle Congruence Postulate

69. 3.Side-Side-Side Congruence Postulate

70. example

71. Theorems step1 step2 step3
If a line parallel to one side of a triangle bisects another side of the triangle ,ıt also bisects the third side.
step1
step2
step3

72. step4

73. Triangle Midsegment Theorem
The line segment which joins the midpoint of two sides of a triangle is called a midsegment of the triangle
It is parallel to the third side and its lenght is equal to half the lenght of the third side

74. example1

75. example2

76. homework
Page 163 (check yourself15) .2and 3
Page and 18

77. ISOSCELES;EQUILATERAL AND RIGHT TRIANGLES
Properties of Isosceles and Equilateral Triangles
1* *(REMEMBER)
Properties
1.In any isosceles triangle

78. example

79. 2.ABC is an isosceles triangle with

80. example
solution

81. 3.ABC is an isosceles triangle with

82. 1.In any equilateral triangle
example

83. 2.In any equilateral triangle
example

84. 3.In any equilateral triangle

85. homework
Page 171 (check yourself17)

86. Now time to listen a friend of Pythagoras

87. Properties of Right Triangles

92. TIME TO LAUGH -1

93. General exam questions