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Chapter 22 Polygons - Triangles - Coggle Diagram
Chapter 22
Polygons - Triangles
22.1
Examples of polygons
Triangle (3-sides)
Quadrilateral (4-sides)
Pentagon ( 5-sides)
Polygons were named by their number of sides, 12 sides is called a 12-gon whereas 25 sides is called a 25-gon.
Diagonal
The number of diagonals of n-gon = ½ [ ( n-3 ) × n ]
Example
The number of diagonals of a hexagon.
= ½ [ ( n - 3 ) × n ]
= ½ [ ( 6 - 3 ) × 6 ]
= ½ ( 18 )
= 9
22.4
Congruency Test
SSS Property
If the three sides of one triangle (SSS) are equal to the other three sides of the other triangle, it is congruent.
Example
In the diagram, given that AB = AC
D is the mid-point of BC. Prove that ∆ABD is congruent to ∆ACD
AB = AC (given)
AD is the common side
BD = DC ( D is the mid-poi
22.2
Triangles
Interior Angle of the Triangle
The sum of the interior angles of a triangle is always 180°
Example
x + 50° + 70° = 180°
x = 180° - 50° - 70°
x = 60°
Interior Angle of Triangles
The sum of the interior angles of a triangle is 180°
Example
x + 50° + 70° = 180°
x = 180° - 50° - 70°
x = 60°
Exterior Angles of the Triangle
The exterior angle is always sum of the interior opposite angle
Example
x = 70° + 50°
= 120°
22.3
Congruent Triangles
Two triangles are congruent if they have the same shape, and all three sides and three angles of them are equal.
SAS Property
If two sides and the included angle of one triangle (SAS) are equal to the other two sides and the included angle of the other triangle, they are congruent.
Example
Prove that ∆ABC congruent to ∆CDA
CB = AD (given)
Angle DAC = Angle BCA
AC is the common side,
= ∆ABC congruent to ∆CDA (SAS)
ASA Property
If two angles and one side of one triangle (ASA) are equal to the other two angles and one side of another triangle, they are congruent.
Example
In the diagram, AB = CD and AB//CD
Prove that ∆ABE congruent to ∆DCE
AB//CD (given)
= Angle A = Angle D
Angle B = Angle C
AB = CD (given)
= ∆ABE congruent ∆DCE (ASA)
RHS Property
If the hypotenuse and one side of one right-angled triangle (RHS) are equal to the other hypotenuse and one side of another right-angled triangle, they are congruent.
Example
In the figure, OA = OB, Angle A = Angle B= 90°
Prove that Angle AOC = Angle BOC
OA=OB (given)
Angle A = Angle B = 90° (given)
OC is the common side
= ∆AOC congruent to ∆BOC (RHS)
= Angle AOC = Angle BOC
22.5
Types of Triangles
Equilateral Triangle
Three equal sides
(Each angle is 60°)
Isosceles Triangle
Two equal sides
(The two base angles are equal)
Scalene Triangle
No sides equal
(All angles are different in size)
Acute-angled Triangle
Three acute angles
Right-angled Triangle
One right angle
Obtuse-angled Triangle
One obtuse angle
22.6
Relationship Among the Three Sides of Triangles
In a triangle of ABC, with sides namely abc, a + b > c
b + c > a
a + c > b
Using this three segments, if three of them fits the formula, we can decide wether the 3 sides could form a triangle.
Example
4cm,5cm,6cm
4+5 > 6
5+6 > 4
6+4 > 5
These three segments can form a
3cm,4cm,8cm
Cannot form one, as 3+4 < 8
From the above, we know that a + b > c , a > c - b
22.7
Relation Between Sides and Angles of a Triangle
If Angle C > Angle B,
then AB > AC
If AB > AC,
Angle C > Angle B
Example
In the digram, AB > AC , show that Angle ACB > Angle B
Draw CD such that AD = BC
AD = BC
Angle ADC = Angle ACD ( base angle )
Angle ACB > Angle ACD
Angle ACB > Angle ADC
Angle ADC = Angle B + Angle BCD
Angle ACB > Angle B + Angle BCD
=Angle ACB > Angle B