Download similar

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Euclidean geometry wikipedia, lookup

Pythagorean theorem wikipedia, lookup

History of geometry wikipedia, lookup

Rational trigonometry wikipedia, lookup

Integer triangle wikipedia, lookup

Trigonometric functions wikipedia, lookup

Multilateration wikipedia, lookup

Tessellation wikipedia, lookup

Regular polytope wikipedia, lookup

Technical drawing wikipedia, lookup

History of trigonometry wikipedia, lookup

Dessin d'enfant wikipedia, lookup

Golden ratio wikipedia, lookup

Euler angles wikipedia, lookup

Reuleaux triangle wikipedia, lookup

Penrose tiling wikipedia, lookup

List of regular polytopes and compounds wikipedia, lookup

Apollonian network wikipedia, lookup

Transcript
Math 2 Geometry
Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein
5.2
Similar Triangles and Polygons
Informal Definition
• When two figures have the same shape,
they are said to be similar.
Informal Definition
• When two figures have the same shape,
they are said to be similar.
• When two figures have the same shape,
and all corresponding parts have equal
measures, the two figures are congruent.
Symbols
• Equal
=
Symbols
• Equal
=
• Similar ~
Symbols
• Equal
=
• Similar ~
• Congruent

Symbols
• Equal
=
• Similar ~
• Congruent

• The Congruent symbol is a combination of
the symbols for Equal and Similar
Intuitive Definition
• Two figures are similar if one is an
enlargement of the other.
Definition
Two polygons are similar if and only if two
conditions are satisfied:
1.
2.
Definition
Two polygons are similar if and only if two
conditions are satisfied:
1. All pairs of corresponding angles are
congruent.
2.
Definition
Two polygons are similar if and only if two
conditions are satisfied:
1. All pairs of corresponding angles are
congruent.
2. All pairs of corresponding sides are
proportional.
Which Figures must be Similar?
• Any two isosceles triangles.
Which Figures must be Similar?
• Any two isosceles triangles.
• Any two regular pentagons.
Which Figures must be Similar?
• Any two isosceles triangles.
• Any two regular pentagons.
• Any two rectangles.
Which Figures must be Similar?
•
•
•
•
Any two isosceles triangles.
Any two regular pentagons.
Any two rectangles.
Any two squares.
Which Figures must be Similar?
•
•
•
•
•
Any two isosceles triangles.
Any two regular pentagons.
Any two rectangles.
Any two squares.
Any two rhombuses.
Example
Given ABC ~DEF, with indicated measures.
Find the measures of the remaining parts of
each triangle.
F
C
12
3
53
A
5
B
D
E
Postulate 15 (AAA)
If the three angles of one triangle are
congruent to the three angles of a second
triangles, then the triangles are similar.
Corollary 5.2.1 (AA)
If two angles of a triangle are congruent to
two angles of another triangle, then the
triangles are similar.
CSSTP
• What could these letters stand for?
CSSTP
Corresponding sides of similar triangles
are proportional.
Theorem 5.2.2
The lengths of the corresponding altitudes
of similar triangles have the same ratio of
any pair of corresponding sides.
F
C
A
B
E
F
What would be an outline for proving this?
Theorem 5.2.3 (SAS~)
If an angle of one triangle is congruent to
an angle of a second triangle and the pairs
of sides that form the angles are
proportional, then the triangles are similar.
Theorem 5.2.4 (SSS~)
If the three sides of one triangle are
proportional to the three corresponding
sides of a second triangle, then the
triangles are similar.