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### Center: The point which allpoints of the circle are equidistant to.

### GEOMETRY

### GEOMETRY

### Geometry

### Types of Triangles

### Acute Triangles

### Right Triangles

### Basic ShapesTwo Dimensions

Terms

Center

Radius

Chord

Diameter

Circumference

Formulas

Circumference formula

Area formula

Basic Terms and FormulasAngles in Geometry

Fernando Gonzalez - North Shore High School

Intersecting Lines

- Two lines that share

one common point.

Intersecting lines can

form different types of

angles.

Supplementary Angles

- Two angles that equal 180º

Corresponding Angles

- Angles that are

vertically identical

they share a common vertex and have a line running through them

Exterior Angle Sum Theorem

What is the Exterior Angle Sum Theorem?

- The exterior angle is equal to the sum of the interior angles on the opposite of the triangle.

40

70

70

110

110 = 70 +40

Exterior Angle Sum Theorem

- There are 3 exterior angles in a triangle. The exterior angle sum theorem applies to all exterior angles.

128

52

64

64

116

116

128 = 64 + 64 and 116 = 52 + 64

Linking to other angle concepts

- As you can see in the diagram, the sum of the angles in a triangle is still 180 and the sum of the exterior angles is 360.

160

20

100

80

80

100

80 + 80 + 20 = 180 and 100 + 100 + 160 = 360

Interior Angle Sum Theorem

What is the Interior Angle Sum Theorem?

- The interior angle is equal to the sum of the interior angles of the triangle.

40

70

70

110

110 = 70 +40

Interior Angle Sum Theorem

- There are 3 interior angles in a triangle. The interior angle sum theorem applies to all interior angles.

128

52

64

64

116

116

128 = 64 + 64 and 116 = 52 + 64

Linking to other angle concepts

- As you can see in the diagram, the sum of the angles in a triangle is still 180.

160

20

100

80

80

100

80 + 80 + 20 = 180

Parallel Lines with a Transversal

Interior and exterior Angles

Vertical Angles

By

Sonya Ortiz

NSHS

Transversal

- Definition:
- A transversal is a line that intersects a set of parallel lines.
- Line A is the transversal

A

Interior and Exterior Angles

- Interior angels are angles 3,4,5&6.
- Interior angles are in the inside of the parallel lines
- Exterior angles are angles 1,2,7&8
- Exterior angles are on the outside of the parallel lines

1

2

3

4

5

6

7

8

Vertical Angles

- Vertical angles are angles that are opposite of each other along the transversal line.
- Angles 1&4
- Angles 2&3
- Angles 5&8
- Angles 6&7
- These are vertical angles

1

2

3

4

5

6

7

8

Summary

- Transversal line intersect parallel lines.
- Different types of angles are formed from the transversal line such as: interior and exterior angles and vertical angles.

Parallelograms

- A parallelogram is a a special quadrilateral whose opposite sides are congruent and parallel.

A

B

D

C

- Quadrilateral ABCD is a parallelogram if and only if
- AB and DC are both congruent and parallel
- AD and BC are both congruent and parallel

Kinds of Parallelograms

- Rectangle
- Square
- Rhombus

Rectangles

- Properties of Rectangles
- 1. All angles measure 90 degrees.
- 2. Opposite sides are parallel and congruent.
- 3. Diagonals are congruent and they bisect each other.
- 4. A pair of consecutive angles are supplementary.
- 5. Opposite angles are congruent.

Squares

- Properties of Square
- 1. All sides are congruent.
- 2. All angles are right angles.
- 3. Opposite sides are parallel.
- 4. Diagonals bisect each other and they are congruent.
- 5. The intersection of the diagonals form 4 right angles.
- 6. Diagonals form similar right triangles.

Rhombus

- Properties of Rhombus
- 1. All sides are congruent.
- 2. Opposite sides parallel and opposite angles are congruent.
- 3. Diagonals bisect each other.
- 4. The intersection of the diagonals form 4 right angles.
- 5. A pair of consecutive angles are supplementary.

Pythagorean Theorem

- The Pythagorean theorem
- This theorem reflects the sum of the

squares of the sides of a right triangle

that will equal the square of the hypotenuse.

C2 =A2 +B2

To further solve for the length of C

Take the square root of C

41 = 6.4

This finds the length of the Hypotenuse

of the right triangle.

The theorem will help calculate distance when traveling

between two destinations.

Triangles

- Find the sum of the angles of a three sided figure.

Quadrilaterals

- Find the sum of the angles of a four sided figure.

Pentagons

- Find the sum of the angles of a five sided figure.

Hexagon

- Find the sum of the angles of a six sided figure.

Heptagon

- Find the sum of the angles of a seven sided figure.

Octagon

- Find the sum of the angles of an eight sided figure.

What is the angle sum formula?

- Angle Sum=(n-2)180
- Or
- Angle Sum=180n-360

A presentation by

Mary McHaney

SQUARE Characteristics:

- Four equal sides
- Four Right Angles

RECTANGLE Characteristics

- Opposite sides are equal
- Four Right Angles

Square and Rectangle share

- Four right angles
- Opposite sides are equal

SQUARE AND RECTANGLE DO NOT SHARE:

- All sides are equal

SO

- A SQUARE IS RECTANGLE
- A RECTANGLE IS NOT A SQUARE

Triangles Are Classified Into 2 Main Categories.

Triangles Classified by Their SidesScalene Triangles

- These triangles have all 3 sides of different lengths.

Isosceles Triangles

- These triangles have at least 2 sides of the same length. The third side is not necessarily the same length as the other 2 sides.

Equilateral Triangles

- These triangles have all 3 sides of the same length.

These Triangles Have All Three Angles That Each Measure Less Than 90 Degrees.

These triangles have exactly one angle that measures 90 degrees. The other 2 angles will each be acute.

ObtuseTriangles

These triangles have exactly one obtuse angle, meaning an angle greater than 90 degrees, but less than 180 degrees. The other 2 angles will each be acute.

Quadrilateral Objectives

- Upon completion of this lesson, students will:
- have been introduced to quadrilaterals and their properties.
- have learned the terminology used with quadrilaterals.
- have practiced creating particular quadrilaterals based on specific characteristics of the quadrilaterals.

Parallelogram

- A quadrilateral that contains two pairs of parallel sides

Rectangle

- A parallelogram with four right angles

Square

- A parallelogram with four congruent sides and four right angles

Group Activity

Each group design a different quadrilateral and prove that its creation fits the desired characteristics of the specified quadrilateral. The groups could then show the class what they created and how they showed that the desired characteristics were present.

If you look around you, you’ll see angles are everywhere. Angles are measured in degrees. A degreeis a fraction of a circle—there are 360 degrees in a circle, represented like this: 360°.

- You can think of a right angle as one-fourth of a circle, which is 360° divided by 4, or 90°.
- An obtuse angle measures greater than 90° but less than 180°.

Complementary Angles

Complementary angles are two adjacent angles whose sum is 90°

60 °

30 °

60 ° + 30 ° = 90°

Supplementary angles are two adjacent angles whose sum is 180°Supplementary Angles

120°

60°

120° + 60° = 180°

First look at the picture. The angles are complementary angles.

Set up the equation:

12 + x = 180

Solve for x:

x = 168°

Application12°

x

RIGHT ANGLES

- RIGHT ANGLES ARE 90 DEGREE
- ANGLES.

Recall:

- Equation of a straight line: Y=mX+C
- Slope of Line = m
- Y-Intercept = C

Parallel Lines Symbol: “||”

- Two lines are parallel if they never meet or touch.

Look at the lines below, do they meet?

Line AB is parallel to Line PQ or AB || PQ

Slopes of Parallel Lines

- If two lines are parallel then they have the same slope.

Example:

Line 1: y = 2x + 1

Line 2: y = 2x + 6

THINK: What is the slope of line 1?

What is the slope of line 2?

Are these two lines parallel?

Perpendicular Lines

- Two lines are perpendicular if they intersect each other at 90°.

Look at the two lines below:

A

D

C

B

Is AB perpendicular to CD? If the answer is yes, why?

Slopes of Perpendicular Lines

- The slopes of perpendicular lines are negative reciprocals of each other.

Example:

Line 3: y = 2x + 5

Line 4: y = -1/2 x + 8

THINK: What is the slope of line 3?

What is the slope of line 4?

Are these two lines perpendicular. If so, why?

Show your working.

Parallel Lines

Do not intersect.

If two lines are parallel then their slopes are thesame.

Perpendicular Lines

Intersect at 90°(right angles).

If two lines are perpendicular then their slopes are negative reciprocals of each other.

What do you need to knowQuestions

- Write an equation of a straight line that is parallel to the line y = -1/3 x + 7

State the reason why your line is parallel to that of the line given above.

- Write an equation of a straight line that is perpendicular to the line y = 4/5 x + 3.

State the reason why the line you chose is perpendicular to the line given above.

Triangle

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