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Menu. Theorem 1 Vertically opposite angles are equal in measure. Theorem 2 The measure of the three angles of a triangle sum to 180 o. Theorem 3 An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. - PowerPoint PPT PresentationTRANSCRIPT
MenuTheorem 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.
12
34
0
180
90
45
135
0
18090
45
135
Theorem 1:Theorem 1: Vertically opposite angles are equal in measure Vertically opposite angles are equal in measure
To Prove:To Prove: 1 = 3 and 2 = 4
Proof:Proof: 1 + 1 + 2 = 1802 = 1800 …………..0 ………….. Straight lineStraight line
2 + 2 + 3 = 1803 = 1800 ………….. 0 ………….. Straight lineStraight line
1 + 1 + 2 = 2 = 2 + 2 + 33
1 = 1 = 33
Similarly Similarly 2 = 2 = 44
Q.E.D.Q.E.D.
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Proof:Proof: 3 + 3 + 4 + 4 + 5 = 1805 = 18000Straight lineStraight line
1 = 1 = 4 and 4 and 2 = 2 = 55Alternate anglesAlternate angles
3 + 3 + 1 + 1 + 2 = 1802 = 18000
1 + 1 + 2 + 2 + 3 = 1803 = 18000
Q.E.D.Q.E.D.
4 5
Given:Given: Triangle
1 2
3Construction:Construction: Draw line through 3 parallel to the base
Theorem 2:Theorem 2: The measure of the three angles of a triangle sum to 180The measure of the three angles of a triangle sum to 18000 . .
To Prove:To Prove: 1 + 2 + 3 = 1800
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0 180
90
45 135
Theorem 3:Theorem 3: An exterior angle of a triangle equals the sum of the two An exterior angle of a triangle equals the sum of the two interior interior opposite angles in measure.opposite angles in measure.
To Prove:To Prove: 1 = 3 + 4
Proof:Proof: 1 + 1 + 2 = 1802 = 1800 …………..0 ………….. Straight lineStraight line
2 + 2 + 3 + 3 + 4 = 1804 = 1800 ………….. 0 ………….. Triangle.Triangle.
1 + 1 + 2 = 2 = 2 + 2 + 3 + 3 + 4 4
1 = 1 = 3 + 3 + 4 4
Q.E.D.Q.E.D.
12
3
4
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Theorem 4:Theorem 4: If two sides of a triangle are equal in measure, then the If two sides of a triangle are equal in measure, then the anglesangles opposite these sides are equal in measure. opposite these sides are equal in measure.
To Prove:To Prove: 1 = 2
Proof:Proof: In the triangle abd and the triangle adc In the triangle abd and the triangle adc
|ab| = |ac||ab| = |ac| ………….. ………….. Given.Given.
The triangle abd is congruent to the triangle adc……….. ……….. SAS = SAS.SAS = SAS.
Q.E.D.Q.E.D.
1 2
3 4
a
b c
Given:Given: Triangle abc with |ab| = |ac||ab| = |ac|
Construction:Construction: Construct ad the bisector of bacbac
d
3 = 3 = 4 4 …………..………….. ConstructionConstruction
|ad| = |ad||ad| = |ad| ………….. ………….. Common Side.Common Side.
1 = 2
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2
3
1
4
Given:Given: Parallelogram abcdcb
a d
Construction:Construction: Draw the diagonal |ac|
Theorem 5:Theorem 5: The opposite sides and opposite angles of a parallelogram The opposite sides and opposite angles of a parallelogram are respectively equal in measure. are respectively equal in measure.
To Prove:To Prove: |ab| = |cd| and |ad| = |bc|
and abc = abc = adcadc
Proof:Proof: In the triangle abc and the triangle adcIn the triangle abc and the triangle adc
1 = 1 = 4 4 …….. …….. Alternate anglesAlternate angles
|ac| = |ac| …… |ac| = |ac| …… CommonCommon
2 = 2 = 3 ……… 3 ……… Alternate anglesAlternate angles
The triangle abc is congruent to the triangle adc……… ……… ASA = ASA.ASA = ASA.
|ab| = |cd| and |ad| = |bc||ab| = |cd| and |ad| = |bc|
and abc = abc = adcadc
Q.E.DQ.E.D
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Given:Given: Parallelogram abcd
Construction:Construction: Draw perpendicular from b to ad
Theorem 6:Theorem 6: A diagonal bisects the area of a parallelogram A diagonal bisects the area of a parallelogram
To Prove:To Prove: Area of the triangle abc = Area of the triangle adc
Proof:Proof: Area of triangle adc = Area of triangle adc = ½ ½ |ad| x |bx||ad| x |bx|
Area of triangle abc = Area of triangle abc = ½ ½ |bc| x |bx||bc| x |bx|
As |ad| = |bc| …… As |ad| = |bc| …… Theorem 5Theorem 5
The diagonal ac bisects the area of the parallelogram
Q.E.DQ.E.D
b c
a dx
Area of triangle adc = Area of triangle abc Area of triangle adc = Area of triangle abc
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Theorem 7:Theorem 7: The measure of the angle at the centre of the circle is twice the The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference standing on the measure of the angle at the circumference standing on the same arc. same arc.
To Prove:To Prove: | | boc | = 2 | boc | = 2 | bac | bac |
Construction:Construction: Join a to o and extend to r
r
Proof:Proof: In the triangle aobIn the triangle aob
a
b c
o
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2
4
5
| oa| = | ob | …… | oa| = | ob | …… RadiiRadii
| | 2 | = | 2 | = | 3 | …… 3 | …… Theorem 4Theorem 4
| | 1 | = | 1 | = | 2 | + | 2 | + | 3 | …… 3 | …… Theorem 3Theorem 3
| | 1 | = | 1 | = | 2 | + | 2 | + | 2 |2 |
| | 1 | = 2| 1 | = 2| 2 |2 |
SimilarlySimilarly | | 4 | = 2| 4 | = 2| 5 |5 |
| | boc | = 2 | boc | = 2 | bac | bac | Q.E.DQ.E.D
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Theorem 8:Theorem 8: A line through the centre of a circle perpendicular to a chord A line through the centre of a circle perpendicular to a chord bisects the chord.bisects the chord.
Given:Given: A circle with o as centre
and a line L perpendicular to ab.
To Prove:To Prove: | ar | = | rb | | ar | = | rb |
L
90 o
oa
b
r
Proof:Proof: In the triangles aor and the triangle orbIn the triangles aor and the triangle orb
Construction:Construction: Join a to o and o to b
|ao| = |ob||ao| = |ob| ………….. ………….. Radii.Radii.
aro = aro = orb orb ………….…………. 90 90 oo
|or| = |or||or| = |or| ………….. ………….. Common Side.Common Side.
The triangle aor is congruent to the triangle orb……… ……… RSH = RSH.RSH = RSH.
|ar| = |rb||ar| = |rb|Q.E.DQ.E.D
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Given:Given: Two Triangles with equal angles
Proof:Proof: 1 = 1 = 44
[xy] is parallel to [bc][xy] is parallel to [bc]
Construction:Construction: On ab mark off ax equal in length to de.
On ac mark off ay equal in length to df
|df|
|ac|=
|de|
|ab|Similarly
|ef|
|bc|=
Q.E.D.
a
cb
d
fe1
2
3
1 3
2
x y4 5 |ay|
|ac|=
|ax|
|ab| As xy is parallel to bcAs xy is parallel to bc
Theorem 9:Theorem 9: If two triangles are equiangular, the lengths of the If two triangles are equiangular, the lengths of the correspondingcorresponding sides are in proportion. sides are in proportion.
To Prove:To Prove:|df|
|ac|=
|de|
|ab|
|ef|
|bc|=
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Given:Given: Triangle abc
Proof:Proof: Area of large sq. = area of small sq. + 4(area Area of large sq. = area of small sq. + 4(area ))
(a + b)(a + b)22 = c = c2 2 + 4(½ab)+ 4(½ab)
aa22 + 2ab +b + 2ab +b22 = c = c2 2 + 2ab+ 2ab
aa22 + b + b22 = c = c22 Q.E.D.Q.E.D.
a
b
c
a
bc
a
b
c
a
b c
Construction:Construction: Three right angled triangles as shown
Theorem 10:Theorem 10: In a right-angled triangle, the square of the length of the side 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 opposite to the right angle is equal to the sum of the
squares of squares of the other two sides.the other two sides.
To Prove:To Prove: a2 + b2 = c2
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Must prove that it is a square. i.e. Show that │∠1 │= 90o
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3
2
│∠1│+ │∠2│ =│∠3│+│∠4│ (external angle…)But │∠2│=│∠3│ (Congruent triangles) ⇒│∠1│=│∠4│= 90o QED