Transcript Theorems - PBworks

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Theorem 1
Vertically opposite angles are equal in measure.
Theorem 2 The measure of the three angles of a triangle sum to 180o .
Theorem 3 An exterior angle of a triangle equals the sum of the two interior opposite
angles in measure.
Theorem 4
If two sides of a triangle are equal in measure, then the angles
opposite these sides are equal in measure.
Theorem 5
The opposite sides and opposite sides of a parallelogram
are respectively equal in measure.
Theorem 6
A diagonal bisects the area of a parallelogram
Theorem 7
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 8
A line through the centre of a circle perpendicular to a chord
bisects the chord.
Theorem 9
If two triangles are equiangular, the lengths of the corresponding
sides are in proportion.
Theorem 10
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 1:
Vertically opposite angles are equal in measure
1
4
2
3
1 = 3 and
To Prove:
2 = 4
1 + 2 = 1800
Proof:
…………..
2 + 3 = 1800

…………..
Straight line
Straight line
1 + 2 = 2 + 3
1 = 3
Similarly
2 = 4
Q.E.D.
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Theorem 2:
The measure of the three angles of a triangle sum to 1800 .
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 3:
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
…………..
Triangle.
1 + 2 = 2 + 3 + 4
1 = 3 + 4
Q.E.D.
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Theorem 4:
If two sides of a triangle are equal in measure, then the angles
opposite these sides are equal in measure.
a
3 4
Given:
Triangle abc with |ab| = |ac|
To Prove:
1 = 2
Construction: Construct ad the bisector of bac
Proof:
b
…………..
|ab| = |ac|
|ad| = |ad|
…………..
…………..
Construction
Given.
Common Side.
The triangle abd is congruent to the triangle adc

1 = 2
c
d
In the triangle abd and the triangle adc
3 = 4
2
1
………..
SAS = SAS.
Q.E.D.
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Theorem 5:
b
The opposite sides and opposite angles of a parallelogram
are respectively equal in measure.
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 6:
A diagonal bisects the area of a parallelogram
b
a
c
x
d
Given:
Parallelogram abcd
To Prove:
Area of the triangle abc = Area of the triangle adc
Construction:
Proof:
Draw perpendicular from b to ad
Area of triangle adc = ½ |ad| x |bx|
Area of triangle abc = ½ |bc| x |bx|
As |ad| = |bc| …… Theorem 5
Area of triangle adc = Area of triangle abc
The diagonal ac bisects the area of the parallelogram
Q.E.D
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Theorem 7: 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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Theorem 8: A line through the centre of a circle perpendicular to a chord
bisects the chord.
L
Given:
A circle with o as centre
and a line L perpendicular to ab.
To Prove:
o
| ar | = | rb |
Construction:
Proof:
a
Join a to o and o to b
r
90 o
In the triangles aor and the triangle orb
aro = orb
………….
90 o
|ao| = |ob|
…………..
Radii.
|or| = |or|
…………..
Common Side.
The triangle aor is congruent to the triangle orb

b
………
RSH = RSH.
|ar| = |rb|
Q.E.D
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Theorem 9:
If two triangles are equiangular, the lengths of the corresponding
sides are in proportion.
Given:
Two Triangles with equal angles
|ab|
To Prove:
=
|de|
|ac|
|df|
=
|bc|
|ef|
Construction: On ab mark off ax equal in length to de.
On ac mark off ay equal in length to df
x
4
a
d
2
2
5
y e
1
1 = 4
Proof:

3

f
[xy] is parallel to [bc]
|ab|
|ax|
=
|ab|
b
1
3
|de|
c
|ac|
|ay|
=
|ac|
|df|
As xy is parallel to bc
Similarly =
|bc|
|ef|
Q.E.D.
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Theorem 10:
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
3
4
c
a
Given:
Triangle abc
To Prove:
a2 + b2 = c2
Construction: Three right angled triangles as shown
b
Proof:
Area of large sq. = area of small sq. + 4(area D)
(a + b)2 = c2 + 4(½ab)
a2 + 2ab +b2 = c2 + 2ab
c
1
2
a
a2 + b2 = c2
Q.E.D.
b
Must prove that it is a square. i.e. Show that │∠1 │= 90o
│∠1│+ │∠2│ =│∠3│+│∠4│ (external angle…)
⇒│∠1│=│∠4│= 90o QED
But │∠2│=│∠3│ (Congruent triangles)
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