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§3.1 Triangles The student will learn about: proof of congruency, and congruent triangles, 1 some special triangles.
§3.1 Triangles
The student will learn about:
proof of congruency, and
congruent triangles,
1
some special triangles.
§3.1 Congruent Triangles
The topic of congruent triangles is perhaps the most used and important in plane geometry. More theorems are proven using congruent triangles than any other method.
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Triangle DefinitionA triangle is the union of three segments
(called its sides), whose end points (called its vertices) are taken, in pairs, from a set of three noncollinear points. Thus, if the vertices of a triangle are A, B, and C, then its sides are , and , and the triangle is then the set defined by
, denoted ΔABC. The angles of ΔABC are A BAC, B ABC, and C ACB.
AB BC AC
AB BC AC
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EuclidEuclid’s idea of congruency involved the act of
placing one triangle precisely on top of another. This has been called superposition.
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CONGRUENCY
Definitions
Angles are congruent if they have the same measure.
Segments are congruent if they have the same length.
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Definition
One concern should be how much of this information do we really need to know in order to prove two triangles congruent.
Two triangles are congruent iff the six parts of one triangle are congruent to the corresponding six parts of the other triangle.
Congruency is an equivalence relation – reflexive, symmetric, and transitive.
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We know that corresponding parts of congruent triangles are congruent. We abbreviate this fact as CPCTC and find it quite useful in proofs.
Properties of Congruent Triangles
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Important Note When we write Δ ABC Δ DEF we are
implying the following: A DB EC F• AB DE • BC EF• AC DF
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Order in the statement, Δ ABC Δ DEF, is important.
We Will Use CPCTE To Establish Three Types of Conclusions
1. Proving triangles congruent, like
Δ ABC and Δ DEF.
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2. Proving corresponding parts of congruent triangles congruent, likeAB DE
3. Establishing a further relationship, like
A B.
Some Postulate
Postulate 12. The SAS Postulate
Every SAS correspondence is a congruency.
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Postulate 13. The ASA Postulate
Every ASA correspondence is a congruency.
Postulate 12. The SSS Postulate
Every SSS correspondence is a congruency.
Marking Drawings
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A
B
C
D
AB CD
AC BD
A C
CBD BCA
AC BD
Suggestions for proofs that involve congruent triangles: Mark the figures systematically, using:A. A square in the opening of a right
triangle;
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Mark the figures systematically, using:A. A square in the opening of a right
triangle;B. The same number of dashes on congruent
sides; And.
Mark the figures systematically, using:A. A square in the opening of a right
triangle;B. The same number of dashes on congruent
sides; And.C. The same number of arcs on congruent
angles.
Mark the figures systematically, using:A. A square in the opening of a right
triangle;B. The same number of dashes on congruent
sides; And.C. The same number of arcs on congruent
angles.D. Use coloring to accomplish the above.
Mark the figures systematically, using:A. A square in the opening of a right
triangle;B. The same number of dashes on congruent
sides; And.C. The same number of arcs on congruent
angles.D. Use coloring to accomplish the above.F. If the triangles overlap, draw them
separately.
Example Proof
Statement Reason
1. AR and BH bisect each other. Given
Given: AR and BH bisect each other at F
Prove: AB RH
3. AFB = RFH Vertical Angle Theorem
4. ∆AFB = ∆RFH ASA
5. AB = RH CPCTE
2. AF = FR and BF = FH Definition of bisect.
6. QED
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B
A
F
H
R
