Transcript Theorems - PBworks

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Theorem 4 The measure of the three angles of a triangle sum to 180 degrees .
Theorem 6 An exterior angle of a triangle equals the sum of the two interior opposite
angles in measure.
Theorem 9
The opposite sides and opposite sides of a parallelogram
are respectively equal in measure.
Theorem 14
In a right-angled triangle, the square of the length of the side opposite to the right angle
is equal to the sum of the squares of the other two sides.
Theorem 19
The measure of the angle at the centre of the circle is twice the
measure of the angle at the circumference standing on the same arc.
Theorem 4:
The measure of the three angles of a triangle sum to 1800 .
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Given:
Triangle
To Prove:
1 + 2 + 3 = 1800
Construction: Draw line through 3 parallel to the base
4 3 5
3 + 4 + 5 = 1800
Proof:
Straight line
1 = 4 and 2 = 5 Alternate angles

1
2
3 + 1 + 2 = 1800
1 + 2 + 3 = 1800
Q.E.D.
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Theorem 6:
An exterior angle of a triangle equals the sum of the two interior
opposite angles in measure.
3
4
To Prove:
Proof:
1
2
1 = 3 + 4
1 + 2 = 1800
…………..
2 + 3 + 4 = 1800
Straight line
…………..
Theorem 2.
1 + 2 = 2 + 3 + 4
1 = 3 + 4
Q.E.D.
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Theorem 9:
The opposite sides and opposite sides of a parallelogram
are respectively equal in measure.
Use mouse clicks to see proof
b
c
3
Given:
Parallelogram abcd
To Prove:
|ab| = |cd| and |ad| = |bc|
4
and
Construction:
1
a
2
d
Proof:
abc = adc
Draw the diagonal |ac|
In the triangle abc and the triangle adc
1 = 4 …….. Alternate angles
2 = 3 ……… Alternate angles
|ac| = |ac| …… Common
The triangle abc is congruent to the triangle adc

………
ASA = ASA.
|ab| = |cd| and |ad| = |bc|
and
abc = adc
Q.E.D
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Theorem 14:
In a right-angled triangle, the square of the length of the side
opposite to the right angle is equal to the sum of the squares of
the other two sides.
b
a
a
c
c
b
Given:
Triangle abc
To Prove:
a2 + b2 = c2
Construction: Three right angled triangles as shown
Proof:
b
c
2
90o 1 4
a
c
3
Area of large sq. = area of small sq. + 4(area D)
(a + b)2 = c2 + 4(½ab)
a
a2 + 2ab +b2 = c2 + 2ab
b
a2 + b2 = c2
Note must show that the angles in small square are 90o
1 + 2 = 90o …. Complimentary angles
2 = 3 …. Similar triangles
1 + 3 = 90o
1 + 4 + 3 = 180o …. Straight angle
4 = 90o
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Theorem 19: The measure of the angle at the centre of the circle is twice the
measure of the angle at the circumference standing on the same arc.
a
To Prove:
| boc | = 2 | bac |
Construction:
Join a to o and extend to r
Proof:
2 5
o
In the triangle aob
3
| oa| = | ob | …… Radii

1 4
r
| 2 | = | 3 | …… Theorem 4
b
c
| 1 | = | 2 | + | 3 | …… Theorem 3
 | 1 | = | 2 | + | 2 |
 | 1 | = 2| 2 |
Similarly
| 4 | = 2| 5 |
 | boc | = 2 | bac |
Q.E.D
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