Monday, July 20, 2009 @ 1:53 AM

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Congruent Triangles

Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

Rotation

One triangle can be rotated, but as long as they are otherwise identical, the triangles are still congruent.

Reflection

One triangle can be a mirror image of the other, but the triangles can still be congruent if the corresponding sides and angles have the same measure. It can be reflected in any direction, up down, left, right or anything in between.

They can have common parts

Congruent triangles can also have a common side or vertex that is shared by both triangles.

Imagine the triangles are cardboard

One way to think about triangle congruence is to imagine they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in various ways so they make a pile where they exactly fit over each other.

How to tell if triangles are congruent

Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent if:

SSS (side side side)
All three corresponding sides are equal in length.
SAS (side angle side)
A pair of corresponding sides and the included angle are equal.
ASA (angle side angle)
A pair of corresponding angles and the included side are equal.
AAS (angle angle side)
A pair of corresponding angles and a non-included side are equal.
HL (hypotenuse leg of a right triangle)
Two right triangles are congruent if the hypotenuse and one leg are equal.

AAA does not work.

Two triangles that have the same shape, but one is larger than the other If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see Why AAA doesn't work.

They are called similar triangles (See Similar Triangles).

SSA does not work.

Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence. See Why SSA doesn't work.

Constructions

Another way to think about the above is to ask if it is possible to construct a unique triangle given what you know. For example, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw. If you drew two of them, they would be the same shape and size - the definition of congruent. For more on constructions, see Introduction to Constructions

Properties of Congruent Triangles

If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.

To remember this important idea, some find it helpful to use the acronym CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent".

In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.

P.S: taken from http://www.mathopenref.com/congruenttriangles.html

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@ 1:45 AM

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REAL LIFE EXAMPLE!!!! ^^
This is the egyptian pyramid...
They are triangles. They are similar too!
Taken frm:
http://www.1uptravel.com/sevenwonders
/pyramid/pyramid.jpg

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Sunday, July 19, 2009 @ 5:13 AM

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url resources : www.youtube.com

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@ 4:41 AM

♥Similar triangles and their ratios

Two triangles ABC and A'B'C' are similar if

1. the three angles of the first triangle are congruent to the corresponding three angles of the second triangle

2. the lengths of their corresponding sides are proportional as follows.
AB / A'B' = BC / B'C' = CA / C'A'
Theorem
Angle-Angle (AA) Similarity:

If two angles in a triangle are congruent to the two corresponding angles in a second triangle, then the two triangles are similar.

Example 1: Let ABC be a triangle and A'C' a segment parallel to AC. What can you say about triangles ABC and A'BC'? Explain your answer.

Solution to Example 1:
Since A'C' is parallel to AC, angles BA'C' and BAC are congruent.

Also angles BC'A' and BCA are congruent.

Since the two triangles have two corresponding congruent angles, they are similar.

Side-Side-Side (SSS) Similarity:

If the three sides of a triangle are proportional to the corresponding sides of a second triangle, then the triangles are similar.

Example 2:

Let the vertices of triangles ABC and PQR defined by the coordinates:

A(-2,0), B(0,4), C(2,0), P(-1,1), Q(0,3), and R(1,1).

Show that the two triangles are similar.

Solution to Example 2:
Let us first plot the vertices and draw the triangles.
Since we know the coordinates of the vertices, we can find the length of the sides of the two triangles.

AB = sqrt [ 4 2 + 2 2 ] = 2 sqrt(5)

BC = sqrt [ (-4) 2 + 2 2 ] = 2 sqrt(5)

CA = sqrt [ 4 2 ] = 4

PQ = sqrt [ 2 2 + 1 2 ] = sqrt(5)

QR = sqrt [ (-2) 2 + 1 2 ] = sqrt(5)

RP = sqrt [ 2 2 ] = 2
We now calculate the ratios of the lengths of the corresponding sides.

AB / PQ = 2 , BC / QR = 2 and CA / RP = 2
We can now write.

AB / PQ = BC / QR = CA / RP = 2
The lengths of the corresponding sides are proportional and therefore the two triangles are similar. (shown)

Side-Angle-Side (SAS) Similarity:

If an angle of a triangle is congruent to the corresponding angle of a second triangle,

and the lengths of the two sides including the angle in one triangle are proportional to the lengths of the corresponding two sides in the second triangle,

then the two triangles are similar.

Example 3: Show that triangles ABC and A'BC', in the figure below, are similar.

Solution to Example 3:

Angles ABC and A'BC' are congruent.
Since the lengths of the sides including the congruent angles are given,

let us calculate the ratios of the lengths of the corresponding sides. BA / BA' = 10 / 4 = 5 / 2 BC / BC' = 5 / 2
The two triangles have two sides whose lengths are proportional and a congruent angle included between the two sides.

The two triangles are similar. (shown)

For two similar plane figures,
if their ratio of their corresponding sides is
a:b,
then the ratio of their ares is
a (square) : b (square)
and,
if the ratio of their volume would be
a(cube) : b(cube)

url resources :
http://images.google.com.sg/imgres?imgurl=http://www.analyzemath.com/Geometry/similar_1.gif&imgrefurl=http://www.analyzemath.com/Geometry/similar_triangles.html&usg=__0mEj7gAD5qN2LyhPNvvaIbVvsPM=&h=216&w=432&sz=3&hl=en&start=30&um=1&tbnid=6ObHK-M1BuvLEM:&tbnh=63&tbnw=126&prev=/images%3Fq%3Dsimilar%2Btriangles%26ndsp%3D20%26hl%3Den%26rlz%3D1R2MOOI_enSG330%26sa%3DN%26start%3D20%26um%3D1
http://www.analyzemath.com

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@ 3:05 AM

♥Congruent triangles

Consider this two triangles in the above diagrams

To determine if two triangles are indeed congruence,we must look at its sides and it angles.

In the above diagram,we could see that,

AB=DE,
BC=EF,

AC=DF

Angle ABC=Angle DEF

Angle CBA = Angle FED

Angle BAC = Angle EDF

Therefore,we could say that Triangle ABC is congruence to Triangle DEF

Next,if we need to prove that two triangles are congruent, we have five different methods:

SSS (side side side) =

If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

SAS (side angle side) =

If two sides and the angle in between are congruent to the corresponding parts of another triangle, the triangles are congruent.

ASA (angle side angle) =

If two angles and the side in between are congruent to the corresponding parts of another triangle, the triangles are congruent.

AAS (angle angle side) =

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

RHS (Right-angle hypotenuse side) =

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. Remember that if we know two sides of a right triangle we know the third side anyway, so this is really just SSS.In addition,to find the side of any triangles,we have to apply the pythagoras theorem.

NOTE 1: AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to show congruence. You can draw 2 equilateral triangles that are the same shape but not the same size.

NOTE 2: The Angle Side Side Theorem (yes, we all know what it spells) does NOT necessarily work.

For example,

Is triangle ABC congruent to triangle DEF?
In the pictures we have:

angle A = angle D.

angle B = angle E.

side AC = side DF.

Conclusion: triangle ABC triangle DEF (AAS)

That's congruency! (:

url sources:

http://images.google.com.sg/imgres?imgurl=http://www.analyzemath.com/Geometry/congruence_sss.gif&imgrefurl=http://www.analyzemath.com/Geometry/congruent_triangles.html&usg=__1EmLlhHpIoo43vkoc9naXuofbCY=&h=172&w=361&sz=3&hl=en&start=6&um=1&tbnid=YmMl2NnfFhyJeM:&tbnh=58&tbnw=121&prev=/images%3Fq%3DSSS%2Bcongruent%2Btriangles%26hl%3Den%26rlz%3D1R2MOOI_enSG330%26um%3D1

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Friday, July 17, 2009 @ 1:01 AM

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Two shapes are congruent if they are the same (shape and size)- in other words, if the lengths of the sides and the angles are the same. It is often useful to know when two triangles are congruent.

Two triangles are congruent if any one of the following is true:

All three sides of one triangle are the same length as all three sides of the other triangle (i.e. a = d, b = f and c = e below). If we know that this is true we write "SSS".
Two of the angles and a side of one triangle are equal to the corresponding two angles and side of the other triangle (e.g. A = D, C = E and a = d) . For this we write "ASA".
An angle between two sides of a triangle is equal to the corresponding angle in the other triangle and the sides in question are equal (e.g. C = E, b = f, a = d). We write "SAS".
Two right angled triangles have the same hypotenuse and one other equal side

P.S Taken from http://www.mathsrevision.net/gcse/pages.php?page=20

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