Chapter 9 - Geometry and Measurement 5 4 3 2 1 Geo metry Lesson: General Review Prepared by:

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

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

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

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

Line AB or AB Line Segment AB or AB Ray AB or AB

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

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

Chapter 9 - Geometry and Measurement

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

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.

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

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

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.

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

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

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

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

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

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

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 =

TRIANGLES

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

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

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

AUXILIARY ELEMENTS OF A TRIANGLE MEDIAN ANGLE BISECTOR ALTITUDE

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

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.

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.

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.

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

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

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?

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

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

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

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

Next week on monday I. exam

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

2.LARGER ANGLE OPPOSITE LONGER SIDE

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

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 .

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

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

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

Some properties (((gstrmmd 1 2 3

examples

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

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.

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

The difference of a winner and a loser

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

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

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

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

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

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

We can write the congruence in six different ways

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

example

homework And page 152 Check yourself12..1and 2

2.Angle- Side- Angle Congruence Postulate

3.Side-Side-Side Congruence Postulate

example

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

step4

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

example1

example2

homework Page 163 (check yourself15) .2and 3 Page 191 . 17and 18

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

example

2.ABC is an isosceles triangle with

example solution

3.ABC is an isosceles triangle with

1.In any equilateral triangle example

2.In any equilateral triangle example

3.In any equilateral triangle

homework Page 171 (check yourself17) .1 .2.3.4.6

Now time to listen a friend of Pythagoras

Properties of Right Triangles

TIME TO LAUGH -1

General exam questions