§3.1 Triangles

The student will learn about:

proof of congruency, and

congruent triangles,

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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

Definition – Angle Bisector

If D is in the interior of BAC, and BAD is congruent to DAC then bisects BAC, and is called the bisector of BAC.

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AD AD

C

B

A

D

Definition – Special TrianglesA 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.

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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.

Theorem - Isosceles Triangle Theorem

The base angles of an Isosceles triangle are congruent.

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Proof is a homework assignment.

Theorem – Converse of the Isosceles Triangle Theorem

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

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Proof is a homework assignment.

Definition – Right TrianglesA triangle with one right angle is a right triangle.

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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.

Assignment: §3.1