Download Congruent and Similar Triangles (MASMTS408).notebook

Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Triangle Geometry
1) Naming angles Often we can use one letter (capitalised) to name an angle.
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When more than two lines meet at a vertex, then we must use three letters to name an angle.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Complementary and Supplementary Angles
Complementary Angles add up to 90°.
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Supplementary Angles add up to 180°.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
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Vertically Opposite angles are congruent.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Parallel Lines and a Transversal
Transversal
When a transversal cuts across two parallel lines, various combinations of congruent angles are created
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Corresponding Angles are congruent.
Alternate Interior Angles are congruent.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Alternate Exterior Angles are congruent.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Example: Find the value of x.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Example: Find the value of x.
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Statement
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Justification
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
If two lines are parallel and they are intersected by a
transversal, then the corresponding (or alt. interior or alt.
exterior) angles are congruent....
Then...
We can also say that if the corresponding (or alt.
interior or alt. exterior) angles created by a transversal
intersecting two lines are congruent, then the lines must be
parallel.
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50° F
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50°
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Since the two alternate
exterior angles are
congruent, the two lines
AB and CD must be
parallel.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Determine the measures of each angle. Justify your answers.
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Measure
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Justification
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Measure
Justification
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Congruent Triangles
Two triangles are congruent when all three corresponding sides and all three corresponding angles have the same measurements.
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To prove that two triangles are congruent, it is not necessary to show all six conditions. (Note: Isometric and congruent mean the same thing.)
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
There are minimum conditions for showing that triangles are congruent. These are known as Theorems of Congruence.
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1) If the three sides of one triangle are congruent to the three corresponding sides of another triangle, then the triangles are congruent. This is called the Side Side Side theorem (SSS).
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
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2) If two sides and the contained angle of one triangle are congruent to two corresponding sides and contained angle of another triangle, then the triangles are congruent. This is called the Side Angle Side theorem (SAS).
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
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3) If two angles and the contained side of one triangle are congruent to two corresponding angles and contained side of another triangle, then the triangles are congruent. This is called the Angle Side Angle theorem (ASA).
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Similar Triangles
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Two figures are similar when ...
All corresponding angles are congruent.
All corresponding sides are proportional.
Therefore, similar figures have the same shape, but are not necessarily the same size.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
There are also minimum conditions to prove that two triangles are similar.
1) Side Side Side (SSS) If the corresponding sides of two triangles are proportional in length, then the triangles are similar.
2) Angle Angle (AA)
If two triangles have one congruent angle contained between corresponding sides of proportional length, then the triangles are similar.
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3) Side Angle Side (SAS)
If two corresponding angles of two triangles are congruent, then the triangles are similar.
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Similar Triangles
Knowing that triangles are similar allows us to solve some geometric problems.
Example: Given that the triangles below are similar, solve triangle DEF. (To solve a triangle is to find all its measures.)
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Congruent and Similar Triangles (MASMTS408).notebook
January 06, 2014
Example:Determine the value of x.
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