T
R
I
A
N
G
L
E
S
profile
MATHEMATICS PROJECT
Janice Poh (3) ,
Katherine Marie Lee (7) ,
Lin Lei (8) ,
Tan Qiu Xuan (11).
CLASS 3D.
WE ♥ MATHS.
CREDITS :Blogger
Youtube
Discovering Mathematics 3A
yourteacher.com
math.pppst.com/congruent.html
http://www.absorblearning.com/mathematics/demo/units/KCA035.html
CONGRUENCY
Introduction to congruent triangles:
When two shapes are said to be congruent, it means that they have the same shape and size.
All shapes can be congruent; this property is not limited to triangles alone.
Rotating congruent shapes do not alter their congruity, as all their sides and angles retain their size.
This also means that a reflection of a shape, despite being a mirror image, is congruent to the original.
∠A = ∠X,
∠B = ∠Y,
∠C = ∠Z,
AB = XY,
BC = YZ,
CA = ZX.
Therefore, triangle ABC is congruent to Triangle XYZ.
this sentence can be rewritten as :
ΔABC ≡ ΔXYZ
( ≡ is read as 'is congruent to' ) .
The equal pairs of angles are known as the corresponding angles (eg ∠A is corresponding to ∠X) ,
and the equal pairs of sides are known as corresponding sides (eg AB is corresponding to XY) .
When two triangles are congruent, each pair of corresponding angles and sides are equal.
( Abbreviation: corr. parts of ≡ Δs )
CONDITIONS OF CONGRUENT TRIANGLES
There are three conditions in which two triangles are identified as congruent triangles.
- SSS (Side-Side-Side) Congruence
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle) Congruence
- RHS (Right angle-Hypotenuse-Side) Congruence
1. SSS (Side-Side-Side) Congruence
First, find the corresponding sides.
PQ = LM
QR = MN
RQ = NL
If all the sides are of equal length, ΔPQR ≡ ΔLMN.
2. SAS (Side-Angle-Side) Congruence
If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, then the two triangles are congruent. This is also known as the Side-Angle-Side test.
In Δs PQR & LMN,
PR = LN,
∠PRQ = ∠LNM,
QR = MN.
Therefore, ΔPQR ≡ ΔLMN based on SAS (Side-Angle-Side) Congruency.
3. ASA (Angle-Side-Angle) Congruence
If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, then the two triangles are congruent. This is also known as the Angle-Side-Angle test.
In Δs PQR & LMN,
∠PRQ = ∠LNM,
QR = MN,
∠RPQ = ∠NLM.
Therefore, ΔPQR ≡ ΔLMN based on ASA (Angle-Side-Angle) Congruency.
4. RHS (Right angle-Hypotenuse-Side) Congruence
If the hypotenus and one side of a right-angled triangle are equal to the corresponding parts of another right-angled triangle, then both triangles are congruent. This is also known as the Right Angle-Hypotenus-Side test.
In Δs PQR & LMN,
∠QPR = ∠MLN, = 90° ,
PR = LN,
QR = MN.
Therefore, ΔPQR ≡ ΔLMN based on RHS (Right angle-Hypotenuse-Side) Congruency.
SIMILARITY
Introduction to similar triangles:
Similar triangles have the same shape, but may not necessarily have the same size.
If two shapes are similar, one is an enlargement of the other. This means that the two shapes will have the same angles and their sides will be in the same proportion.
Two triangles are similar if:
[1] 3 angles of a triangle are the same as the 3 corresponding angles of the other.
[2] 3 pairs of corresponding sides are in the same ratio.
CONDITIONS OF SIMILAR TRIANGLES
There are three conditions in which two triangles are identified as similar triangles.
- Angle-Angle-Angle Similarity
- Side-Side-Side Similarity
- Side-Angle-Side Similarity
1. Angle-Angle-Angle Similarity
First, find the corresponding angles.
Angle A = Angle D
Angle B = Angle E
Angle C = Angle F
Follow up by substituting the values in to prove that the angles are the same.
60° = 60°
50° = 50°
70° = 70°
-> Therefore, it is proven that the two triangles are similar based on Angle-Angle-Angle Similarity.
2. Side-Side-Side Similarity
First, find the corresponding sides.
Side AB/Side DE = Side BC/Side EF = Side CA/Side FD
Secondly, substitute the values in.
2cm/4cm = 3cm/6cm = 4cm/8cm
Finally, simplify the equation to make sure that it is equal.
2cm = 2cm = 2cm
-> Therefore, it is proven that the two triangles are similar based on Side-Side-Side Similarity.
3. Side-Angle-Side Similarity
Firstly, find the corresponding sides that join at a point.
Side AB = Side DE
Side BC = Side EF
Substitute the values in to make sure that the ratio is proportional.
2cm = 4cm
3cm = 6cm
Thirdly, find the angle that lies on where the two lines meet.
Angle B = Angle E
End off by substituting the values in to make sure that it is equal.
50° = 50°
-> Therefore, it is proven that the two triangles are similar based on Side-Angle-Side Similarity.
Important: Make sure that the angle you are finding is the angle between the two sides.
APPLICATION TO EVERYDAY LIFE
Real Life Situations whereby
Congruent/Similar Triangles are used :
You might be wondering, 'Why do I need to learn all these for?'
Therefore to answer that burning question and to make sure that your mindset of 'wasting time to learn this' is changed, here are some examples.
1. Construction of Buildings
Here is a functioning example of a 'triangle building' :
Source: X
2. Art pieces
Source: X
Many congruent shapes put into one, producing a magnificient and beautiful art piece.
3. Measurements of tall buildings
Task:
Sometimes we might want to measure something that is quite tall such as a building or a goalpost.
To do this, we can use what we know about similar triangles and shadows to help us determine the height.
The only criteria needed : A sunny day.
We can take a meter stick (or other object of a known length) and place it near the object we want to measure.
Then, we can measure the shadow of the tall item and the measuring device.
From these measurements, a ratio can be created as shown in the diagram above.
TEST YOUR UNDERSTANDING!
attempt these few questions :
ON CONGRUENCY :[ Q1 ] ABCD is a parallelogram and BEFC is a square. Show that triangles ABE and DCF are congruent.
SOLUTION :
In the parallelogram ABCD,
BA is equal to CD.
In the square BEFC,
EB is equal to FC.
By using Pythagoras’ theorem,
AB is equal to DF.
Hence by using SSS, triangle ABE is congruent to triangle DCF.
[ Q2 ] ABC is a triangle and M is the midpoint of AC. I and J are points on BM such that AI and CJ are perpendicular to BM. Show that triangles AIM and CJM are congruent.
SOLUTION :
Since M is the midpoint of AC,
AM is equal to MC.
Since AI and CJ are perpendicular to the line BM,
Angle AIM is equal to angle CJM.
Angle AMI is equal to CMJ,
When the angles are vertically opposite.
Hence, by using AAS, triangles AIM and CJM are congruent.
ON SIMILARITY :
[ Q1 ] A research team wishes to determine the altitude of a mountain as follows:
They use a light source at L, mounted on a structure of height 2 meters, to shine a beam of light through the top of a pole P' through the top of the mountain M'.
The height of the pole is 20 meters. The distance between the altitude of the mountain and the pole is 1000 meters. The distance between the pole and the laser is 10 meters.
We assume that the light source mount, the pole and the altitude of the mountain are in the same plane. Find the altitude h of the mountain.
SOLUTION :
Triangle LPP’ is similar to Triangle LMM’.
Proved using Angle-Angle-Angle similarity.
1010/10 = (h-2)/(20-2)
h -2 = 1818
h = 1820m
The altitude of the mountain is 1820 metres.
[ Q2 ] In the triangle ABC shown below, A'C' is parallel to AC. Find the length y of BC' and the length x of A'A.
SOLUTION :
Triangle ABC is similar to Triangle A’BC’
Use the proportionality of the lengths of the side to write equations that help in solving for x and y.
(30 + x) / 30 = 22 / 14 = (y + 15) / y
(30 + x) / 30 = 22 / 14
420 + 14 x = 660
x = 17.1 (rounded to one decimal place).
22 / 14 = (y + 15) / y
y = 26.25
INTERACTIVE !
ELEMENTRY LEVEL GAME
Description :
Decide which shape are congruent, similar, or neither one.
BASICS ON CONGRUENCY
Description :
Interactive pieces help explain more about congruent shapes.
GET TO KNOW MORE ABOUT SIMILAR TRIANGLES
Description :
Move the corners and see how these triangles stay similar at different sizes.
helpful websites :
Interactive Mathematics Activities
Congruent & Similar Shapes
Practice Worksheets
video lessons :