Download 1.2 Congruent Figures The Idea: two things are called congruent if

1.2 Congruent Figures
The Idea: two things are called congruent if they are
essentially the same, but are just sitting in a different
position in space.
• Two angles are congruent if they have the same
measure.
• Two line segments are congruent if they have the
same length.
• Two circles are congruent if they have the same
radius.
• Two triangles are congruent if all corresponding
sides and angles are the same size.
• All rays are congruent.
• All lines are congruent.
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Theorem 1.2.1: Vertically opposite angles are congruent.
Proof:
We use the symbol ≡ to denote congruence. Thus
4ABC ≡ 4DEF if and only if all six of the following
hold:
∠A ≡ ∠D
∠B ≡ ∠E
∠C ≡ ∠F
AB ≡ DE
BC ≡ EF
AC ≡ DF.
Note that 4ABC ≡ 4DEF and 4ABC ≡ 4EF D are not
the same.
To check to see if two triangles are equivalent is it necessary to check all 6 of the above equivalences?
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Three frequently used triangle congruence checks:
Axiom 1.2.2, SAS Congruency: Two triangles are
congruent if two sides and the included angle of one are
congruent to two sides and the included angle of the
the other.
Theorem 1.2.3, SSS Congruency: Two triangles are
congruent if the three sides of one are congruent to the
three sides of the other.
Theorem 1.2.4 ASA Congruency: Two triangles are
congruent if two angles and the included side are congruent to two angles and the included side of the other.
Are their others?
Why SAS is an axiom and the others are theorems:
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Axiom 1.2.5, The Triangle Inequality: The sum of
the lengths of two sides of a triangle is always greater
than the length of the remaining side.
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Definition. An isosceles triangle is a triangle with
two sides of equal length.
Theorem 1.2.6, The Isosceles Triangle Theorem:
In 4ABC, if AB = AC then ∠ABC = ∠ACB.
Proof:
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Theorem 1.2.7, (Converse of The Isosceles Triangle Theorem):
Proof:
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Theorem 1.2.9, The Angle-Side Inequality: In 4ABC
if ∠ABC is larger than ∠ACB, then AC is longer than
AB.
Proof:
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Theorem 1.2.10, (Converse of The Angle-Side Inequality:)
Proof:
Summary:
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