§3.1 Triangles

The student will learn about: congruent triangles, proof of congruency, and some special triangles.

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§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 Definition

A 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 triangle is then the set defined by

AB  BC  AC

, denoted ΔABC . The angles of ΔABC are



A

 

BAC,



B

 

ABC, and



C

 

ACB. 3

Euclid

Euclid’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

Two triangles are congruent iff the six parts of one triangle are congruent to the corresponding six parts of the other triangle.

One concern should be how much of this information do we really need to know in order to prove two triangles congruent.

Congruency is an equivalence relation – reflexive, symmetric, and transitive.

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Properties of Congruent Triangles

We know that corresponding parts of congruent triangles are congruent. We abbreviate this fact as CPCTC and find it quite useful in proofs. 7

Important Note

When we writeΔ ABC



Δ DEF we are implying the following:

• 

A

 

D

• • • • • 

B

 

E



C

 

F AB



DE BC



AC



EF DF Order in the statement, Δ ABC



Δ DEF, is important.

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We Will Use CPCTE To Establish Three Types of Conclusions

1. Proving triangles congruent, like Δ ABC and Δ DEF. 2. Proving corresponding parts of congruent triangles congruent, like AB



DE 3.

Establishing a further relationship, like A B.

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Some Postulate

Postulate 12. The SAS Postulate Every SAS correspondence is a congruency.

Postulate 13. The ASA Postulate Every ASA correspondence is a congruency.

Postulate 12. The SSS Postulate Every SSS correspondence is a congruency.

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B

Marking Drawings

D A

AB



CD AC



BD AC



BD

C 

A

 

C



CBD

  B

CA 11

Suggestions for proofs that involve congruent triangles:

F. If the triangles overlap, draw them separately.

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Example Proof

Given: AR and BH bisect each other at F Prove: AB



RH A B H F Statement 1. AR and BH bisect each other.

2. AF = FR and BF = FH 3.



AFB =



RFH 4. ∆AFB = ∆RFH 5. AB = RH 6. QED Reason Given Definition of bisect.

Vertical Angle Theorem ASA CPCTE R 13

Definition – Angle Bisector

If D is in the interior of



BAC, and



BAD is congruent to AD



DAC then bisects is called the bisector of



BAC.



BAC, and B D A C 14

Definition – Special Triangles

A triangle with two congruent sides is called isosceles. The remaining side is the base. The two angles that include the base are base angles. The angle opposite the base is the vertex angle.

A triangle whose three sides are congruent is called equilateral.

A triangle no two of whose sides are congruent is called scalene.

A triangle is equiangular if all three angles are congruent.

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Theorem - Isosceles Triangle Theorem

The base angles of an Isosceles triangle are congruent.

Proof is a homework assignment.

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Theorem – Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.

Proof is a homework assignment.

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Definition – Right Triangles

A triangle with one right angle is a right triangle.

Because two right triangles automatically have one angle congruent (the right angle), congruency of two right triangles reduces to two cases: 1) HA which is equivalent to ASA since all angles are known, and 2) HL which is equivalent to SSS since all three sides are know.

We are assuming knowledge of angle sums and Pythagoras.

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Assignment: §3.1