Theorems

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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.
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Theorem 4 The measure of the three angles of a
triangle sum to 1800 .
Use mouse clicks to see proof
Given Triangle
To Prove Ð1 Ð2 Ð3 1800
Construction Draw line through Ð3 parallel to
the base

• Proof Ð3 Ð4 Ð5 1800 Straight line
• Ð1 Ð4 and Ð2 Ð5 Alternate angles
• Þ Ð3 Ð1 Ð2 1800
• Ð1 Ð2 Ð3 1800
• Q.E.D.

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3
Theorem 6 An exterior angle of a triangle
equals the sum of the two interior
opposite angles in measure.
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.
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4
Theorem 9 The opposite sides and opposite sides
of a parallelogram
are respectively equal in measure.
Use mouse clicks to see proof
Given Parallelogram abcd
To Prove ab cd and ad bc
and Ðabc Ðadc
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Construction Draw the diagonal ac
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Proof In the triangle abc and the triangle adc
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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
and Ðabc Ðadc
Q.E.D
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5
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.
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
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90o
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3
Note must show that the angles in small square
are 90o
Ð1 Ð2 90o . Complimentary angles
gt Ð1 Ð3 90o
Ð2 Ð3 . Similar triangles
gt Ð4 90o
Ð1 Ð4 Ð3 180o . Straight angle
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6
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.
To Prove Ðboc 2 Ðbac
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Construction Join a to o and extend to r
Proof In the triangle aob
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1
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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
Þ Ðboc 2 Ðbac
Q.E.D
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