1 / 7

# Proofs of Theorems and Glossary of Terms - PowerPoint PPT Presentation

Proofs of Theorems and Glossary of Terms. Just Click on the Proof Required. Menu. Theorem 4 Three angles in any triangle add up to 180°. Theorem 6 Each exterior angle of a triangle is equal to the sum of the two interior opposite angles.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Proofs of Theorems and Glossary of Terms' - portia

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Theorem 4 Three angles in any triangle add up to 180°.

Theorem 6 Each exterior angle of a triangle is equal to the sum of the two interior opposite angles

Theorem 9 In a parallelogram opposite sides are equal and opposite angle are equal

Theorem 14 Theorem of Pythagoras : In a right angle triangle, the square of the hypotenuse is the sum

of the squares of the other two sides

Theorem 19 The angle at the centre of the circle standing on a given arc is twice the angle at any point of the circle standing on the same arc.

Go to JC Constructions

5

3

1

2

Theorem 4:Three angles in any triangle add up to 180°C.

Use mouse clicks to see proof

Given: Triangle

Proof:Ð3 + Ð4 + Ð5 = 1800Straight line

Ð1 = Ð4 and Ð2 = Ð5 Alternate angles

ÞÐ3 + Ð1 + Ð2 = 1800

Ð1 + Ð2 + Ð3 = 1800

Q.E.D.

To Prove:Ð1 + Ð2 + Ð3 = 1800

Construction:Draw line through Ð3 parallel to the base

Constructions

Quit

45

135

3

0

180

1

2

4

Theorem 6:Each exterior angle of a triangle is equal to the sum of the two interior opposite angles

Use mouse clicks to see proof

To Prove:Ð1 = Ð3 + Ð4

Proof:Ð1 + Ð2 = 1800 …………..Straight line

Ð2 + Ð3 + Ð4 = 1800 ………….. Theorem 2.

Þ Ð1 + Ð2 = Ð2 + Ð3 + Ð4

Þ Ð1 = Ð3 + Ð4

Q.E.D.

Constructions

Quit

c

a

d

Theorem 9:In a parallelogram opposite sides are equal and opposite angle are equal

Use mouse clicks to see proof

Given: Parallelogram abcd

To Prove:|ab| = |cd| and |ad| = |bc|

3

4

Construction:Draw the diagonal |ac|

1

Proof: In the triangle abc and the triangle adc

2

Ð1 = Ð4 …….. Alternate angles

Ð2 = Ð3 ……… Alternate angles

|ac| = |ac| …… Common

Þ The triangle abc is congruent to the triangle adc……… ASA = ASA.

Þ |ab| = |cd| and |ad| = |bc|

Q.E.D

Constructions

Quit

a

a

c

b

c

c

c

a

b

b

a

Theorem 14:Theorem of Pythagoras : In a right angle triangle, the square of the hypotenuse is the sum of the squares of the other two sides

Use mouse clicks to see proof

Given: Triangle abc

To Prove:a2 + b2 = c2

Construction: Three right angled triangles as shown

Proof: ** Area of large sq. = area of small sq. + 4(area D) (a + b)2 = c2 + 4(½ab)

a2 + 2ab +b2 = c2 + 2ab

a2 + b2 = c2

Q.E.D.

Constructions

Quit

o

r

c

b

Theorem 19:The angle at the centre of the circle standing on a given arc is twice the angle at any point of the circle standing on the same arc.

Use mouse clicks to see proof

To Prove:| Ðboc | = 2 | Ðbac |

5

2

Construction:Join a to o and extend to r

Proof: In the triangle aob

4

1

3

| oa| = | ob | …… Radii

Þ | Ð2 | = | Ð3 | …… Theorem 4

| Ð1 | = | Ð2 | + | Ð3 | …… Theorem 3

Þ | Ð1 | = | Ð2 | + | Ð2 |

Þ | Ð1 | = 2| Ð2 |

Similarly| Ð4 | = 2| Ð5 |

Q.E.D

Þ | Ðboc | = 2 | Ðbac |

Constructions