There are 3 type of Similarity proof postulates which are:
- AA
- SAS
- SSS
| AA | To show two triangles are similar, it is sufficient to show that two angles of one triangle are congruent (equal) to two angles of the other triangle. | ||||||||||
| |||||||||||
Monday, June 11, 2012
How do we complete similarity proofs?
is marked to be parallel
to 
, we know that we have
<BDE congruent to <DAC
(by corresponding angles). <B is shared by both
triangles, so the two triangles are similar by AA.
How do we use right triangle simliarity?
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
The sides measuring 8 and x are the longest legs on the right triangles.
Sides measuring 10 and 5 are the hypotenuse of the right triangles.
To find x, create a proportion.
8 x
--- = ---
10 5
Cross multiply and solve for x.
10x = 40
Answer: x = 4
Guided Practice:
1) Find X
Citations:
- http://mahigulshan97.blogspot.com/
- http://www.nexuslearning.net/books/ML-Geometry/Chapter9/ML%20Geometry%209-1%20Similar%20Right%20Triangles.pdf
- http://www.stopdown.net/education.htm
When are triangles similar?
Triangles are similar if they have the same shape, but can be different sizes.
(They are still similar even if one is rotated, or one is a mirror image of the other).
If two of their angles are equal, then the third angle must also be equal, because angles of a triangle always add to make 180°.
In this case the missing angle is 180° - (72° + 35°) = 83°.
So AA could also be called AAA.
In this example we can see that:
Now let us check the ratio of those two sides:
In this example, the ratios of sides are:
Guided Questions:
1) Which Postulate is correct to find the similarities of the triangle below
Citations:
http://www.mathsisfun.com/geometry/triangles-similar-finding.html
(They are still similar even if one is rotated, or one is a mirror image of the other).
AA
AA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar.
For example, these two triangles are similar:In this case the missing angle is 180° - (72° + 35°) = 83°.
So AA could also be called AAA.
SAS
SAS stands for "side, angle, side" and means that we have two triangles where:- the ratio between two sides is the same as the ratio between another two sides
- and we we also know the included angles are equal.
If two triangles have two pairs of sides in the same ratio and the
included angles are also equal, then the triangles are similar.
For example:- one pair of sides is in the ratio of 21 : 14 = 3 : 2
- another pair of sides is in the ratio of 15 : 10 = 3 : 2
- there is a matching angle of 75° in between them
Using Trigonometry
We could also use Trigonometry to calculate the other two sides using the Law of Cosines:| In Triangle ABC: | a2 | = b2 + c2 - 2bc cos A = 212 + 152 - 2 × 21 × 15 × Cos75° = 441 + 225 - 630 × 0.2588... = 666 - 163.055... = 502.944... |
|
| Therefore a = √502.94 = 22.426... | |||
| In Triangle XYZ: | x2 | = y2 + z2 - 2yz cos X = 142 + 102 - 2 × 14 × 10 × Cos75° = 196 + 100 - 280 × 0.2588... = 296 - 72.469... = 223.530... |
|
| Therefore x = √223.530... = 14.950... | |||
a : x = 22.426... : 14.950... = 3 : 2
the same ratio as before!
Note: you could also use the Law of Sines to show that the other two angles are equal.SSS
SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
For example:- a: x = 6 : 7.5 = 12 : 15 = 4 : 5
- b: y = 8 : 10 = 4 : 5
- c: z = 4 : 5
Guided Questions:
1) Which Postulate is correct to find the similarities of the triangle below
Citations:
http://www.mathsisfun.com/geometry/triangles-similar-finding.html
How do we solve similarity problems?
Similar
figures are figures that are the same shape, but not necessarily
the same size.
If these figures should also be the same size, the figures are called congruent.
If these figures should also be the same size, the figures are called congruent.
Ex)
 |
| Similar triangles wit scale factor of 2 |
Solving for Similariteis:
1) Solve for X and Y
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:
- BA is a
transversal that intersects the two parallel lines A'C' and AC, hence
the corresponding angles BA'C' and BAC are congruent. BC is also a
transversal to the two parallel lines A'C' and AC and therefore angles
BC'A' and BCA are congruent. These two triangles have two congruent
angles are therefore similar and the lengths of their sides are
proportional. Let us separate the two triangles as shown below.
- We now 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
- An equation in x may be written as follows.
(30 + x) / 30 = 22 / 14
- Solve the above for x.
420 + 14 x = 660
x = 17.1 (rounded to one decimal place).
- An equation in y may be written as follows.
22 / 14 = (y + 15) / y
- Solve the above for y to obtain.
y = 26.25
1)Find the length of the missing sides
Citation:
http://www.analyzemath.com/Geometry/similar_triangle_problems.html
http://www.mathwarehouse.com/geometry/similar/triangles/index.html
What is similarity?
Similarities: Two polygons are called similar if they have the same shape but can have different sizes.
Ex)
| We can perform any transformation on the figure including dilation, and the figure will still be the same |
Guided Questions:
Citations:
http://www.education.com/study-help/article/ratio-proportion-similarity-answers/
Sunday, March 4, 2012
How do we solve logic problems using conditionals?
In the logic unit, we are not looking at one statement at a time. If "and", "or" is in a statement it is true if one is true and vice versa. Conditionals are the most frequently used statements in the construction of an argument.
Ex)
Inverse: formed by negating the hypothesis & conclusion
- If it is cold, then i will take my coat
- If it is not cold, then i will not take my coat
Converse: is just switching the hypothesis and conclusion
- If today is Friday, then tomorrow is Saturday
- If tomorrow is Saturday, then today is Friday
Contrapositive: contra- prefix meaning "against" or "opposite"
- If i am tired, then i will go to sleep
- If i am not asleep, then I am not tired
Bi-conditional: when the conditional and the converse are both true
- If my shoes are untied, then I will tie them
- I will tie my shoes if and only if they are untied
Question:
Make a conditional using this phrase:
If today is Monday, then yesterday was Sunday.
What is a mathematical statement?
A mathematical statement is a statement that can be judged to be true or false. For example she is wearing a blue shirt is a mathematical statement because it can be proven true or false.
Ex.
He is holding an ax in his hand.
Questions:
Which one of these are NOT mathematical statements?
a) 3+5
b) Nobody is home
c) The dog is wagging its tail
d) 3 = 5
Ex.
He is holding an ax in his hand.
Questions:
Which one of these are NOT mathematical statements?
a) 3+5
b) Nobody is home
c) The dog is wagging its tail
d) 3 = 5
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