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similarity & proportionality
similarity & proportionality
Congruent Triangles
MPM2D: Principles of Mathematics
Two triangles are congruent if they have the same shape, and
the same size.
To meet both of these criteria, congruent triangles have
equal corresponding angles and equal corresponding sides.
Similar & Congruent Triangles
J. Garvin
There are three distinct conditions that will result in
congruent triangles.
J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
similarity & proportionality
Congruent Triangles
Congruent Triangles
Side-Side-Side Congruency (SSS)
Side-Angle-Side Congruency (SAS)
If |AB| = |DE |, |AC | = |DF | and |BC | = |EF |, then
∆ABC ∼
= ∆DEF .
If |AB| = |DE |, |AC | = |DF |, and ∠A = ∠D, then
∆ABC ∼
= ∆DEF .
No matter how the sides are arranged, it will always be
possible to find equal pairs of corresponding sides.
Two sides of a fixed length, containing a fixed angle, will
always result in a third size of a specific length.
J. Garvin — Similar & Congruent Triangles
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J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
similarity & proportionality
Congruent Triangles
Congruent Triangles
Angle-Side-Angle Congruency (ASA)
Example
If ∠A = ∠D, ∠B = ∠E and |AB| = |DE |, then
∆ABC ∼
= ∆DEF .
State why ∆ABC ∼
= ∆DEF .
Like SAS, two fixed angles at each end of a side will result in
two other sides of specific lengths.
Since |AB| = |DE | = 5, |AC | = |DF | = 7 and ∠A = ∠D,
∆ABC ∼
= ∆DEF due to SAS.
J. Garvin — Similar & Congruent Triangles
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J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
similarity & proportionality
Congruent Triangles
Similar Triangles
Example
Two triangles are similar if they have the same overall shape.
State why ∆ABC ∼
= ∆ADC .
Triangles with equal corresponding angles will have the same
shape, but may be different sizes.
Since |AB| = |AD| = 12, ∆ABD is isosceles. This means
that ∠B = ∠D.
Since ∠ACB = ∠ACD = 90◦ , then ∠BAC = ∠DAC .
AC is common to both ∆ABC and ∆ADC , so
∆ABC ∼
= ∆ADC due to SAS.
J. Garvin — Similar & Congruent Triangles
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There are three distinct conditions that will result in similar
triangles.
J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
similarity & proportionality
Similar Triangles
Similar Triangles
Angle-Angle Similarity (AA∼)
Side-Side-Side Similarity (SSS∼)
If ∠A = ∠D and ∠B = ∠E , then ∆ABC ∼ ∆DEF .
If |AB| = k|DE |, |AC | = k|DF |, and |BC | = k|EF |, then
∆ABC ∼ ∆DEF .
It is not necessary to specify that the remaining
corresponding angles are equal, since it will always be true.
Since each side has been scaled by the same amount, the
overall shape is preserved.
J. Garvin — Similar & Congruent Triangles
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J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
similarity & proportionality
Similar Triangles
Similar Triangles
Side-Angle-Side Similarity (SAS∼)
Example
If |AB| = k|DE |, |AC | = k|DF |, and ∠A = ∠D, then
∆ABC ∼ ∆DEF .
State why ∆ABC ∼ ∆ADE .
∠A is common to both triangles, and ∠ACB = ∠AED, since
they are corresponding angles (F pattern).
Since two sides containing the angle have been scaled by the
same amount, the third side will also by scaled by the same
amount as well.
Therefore, ∆ABC sin ∆ADE due to AA∼.
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J. Garvin — Similar & Congruent Triangles
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similarity & proportionality
Similar Triangles
similarity & proportionality
Questions?
Example
State why ∆ABC ∼ ∆EDC .
∠ACB = ∠ECD, since they are opposite angles.
∠B = ∠F , since they are alternate angles (Z pattern).
Therefore, ∆ABC sin ∆EDC due to AA∼.
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J. Garvin — Similar & Congruent Triangles
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