Theorems - PowerPoint PPT Presentation

1 / 11
About This Presentation
Title:

Theorems

Description:

Menu 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 ... – PowerPoint PPT presentation

Number of Views:116
Avg rating:3.0/5.0
Slides: 12
Provided by: Cammi2
Category:

less

Transcript and Presenter's Notes

Title: Theorems


1
Menu
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.
2
Theorem 1 Vertically opposite angles are equal
in measure
1
2
4
3
To Prove Ð1 Ð3 and Ð2 Ð4
Proof Ð1 Ð2 1800 .. Straight line
Ð2 Ð3 1800 .. Straight
line
Þ Ð1 Ð2 Ð2 Ð3
Þ Ð1 Ð3
Similarly Ð2 Ð4
Q.E.D.
Menu
3
Theorem 2 The measure of the three angles of a
triangle sum to 1800 .
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.

Menu
4
Theorem 3 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 .. Triangle.
Þ Ð1 Ð2 Ð2 Ð3 Ð4
Þ Ð1 Ð3 Ð4
Q.E.D.
Menu
5
Theorem 4 If two sides of a triangle are equal
in measure, then the angles
opposite these sides are equal in measure.
4
3
Given Triangle abc with ab ac
To Prove Ð1 Ð2
2
1
Construction Construct ad the bisector of Ðbac
Proof In the triangle abd and the triangle
adc
Ð3 Ð4 .. Construction
ab ac .. Given.
ad ad .. Common Side.
Þ The triangle abd is congruent to the
triangle adc .. SAS SAS.
Þ Ð1 Ð2
Q.E.D.
Menu
6
Theorem 5 The opposite sides and opposite
angles of a parallelogram
are respectively equal in measure.
Given Parallelogram abcd
To Prove ab cd and ad bc
and Ðabc Ðadc
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
and Ðabc Ðadc
Q.E.D
Menu
7
Theorem 6 A diagonal bisects the area of a
parallelogram
Given Parallelogram abcd
To Prove Area of the triangle abc Area of the
triangle adc
Construction Draw perpendicular from b to ad
Proof 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
Þ The diagonal ac bisects the area of the
parallelogram
Menu
Q.E.D
8
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.
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
Þ Ðboc 2 Ðbac
Q.E.D
Menu
9
Theorem 8 A line through the centre of a circle
perpendicular to a chord
bisects the chord.
Given A circle with o as centre
and a line L perpendicular to ab.
To Prove ar rb
Construction Join a to o and o to b
Proof In the triangles aor and the triangle orb
Ðaro Ðorb . 90 o
ao ob .. Radii.
or or .. Common Side.
Þ The triangle aor is congruent to the
triangle orb RSH RSH.
Þ ar rb
Q.E.D
Menu
10
Theorem 9 If two triangles are equiangular, the
lengths of the corresponding
sides are in proportion.
Given Two Triangles with equal angles
Construction On ab mark off ax equal in length
to de. On ac mark off
ay equal in length to df
Proof Ð1 Ð4 Þ xy is
parallel to bc
Q.E.D.
Menu
11
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.
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.
3
4
1
2
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)
Menu
Write a Comment
User Comments (0)
About PowerShow.com