Chapter 4 Congruent Triangles

 

Section 4.1 Classifying Triangles

Triangle—a three sided polygon 

 

Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is 180 degrees. 

 

 

Classification of Triangles by Angles:

            Acute Triangle—3 acute angles

            Right Triangle—1 right angle

            Obtuse Triangle—1 obtuse angle  

            Equiangular Triangle—3 congruent angles

 

Classification of Triangles by Sides:

            Equilateral Triangle—3 congruent sides

            Isosceles Triangle—at least 2 congruent sides

            Scalene Triangle—no congruent sides  

 

 

 

Section 4.2 Angles of  Triangles

Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is 180 degrees. 
What are the measures of the angles of an equilateral triangle?

 

 

Third Angle Theorem:  If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

 

 

 

Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

 

 

 

Corollary 4.1The acute angles of a right triangle are complementary. 

Corollary 4.2There can be at most one right or obtuse angle in a triangle.

 

Section 4.3 Congruent Triangles

Congruent Triangles:  Triangles that are the same size and shape.

Triangles have six measurable parts (3 angles and 3 sides)

 

If all six corresponding parts are congruent in the two triangles, then the triangles are congruent.

 

 

 

Definition of Congruent Triangles:
Two Triangles are congruent IFF their corresponding parts are congruent

CPCTCCorresponding Parts of Congruent Triangles are Congruent

 

 

 

Theorem 4.4:  Congruence of triangles is Reflexive, Symmetric, and Transitive.

 

 

Section 4.4 Proving Congruence: SSS, SAS

           

Postulate 4.1: (Side-Side-Side Congruence):  If the sides of one triangle are congruent to the corresponding sides of a second triangle, then the triangles are congruent.

 

 

 

Postulate 4.2: (Side-Angle-Side Congruence):  If two sides and the included angle of one triangle are congruent to the two corresponding sides and the included angle of a second triangle, then the triangles are congruent.

 

 

 

 

Section 4.5 Proving Congruence: ASA, AAS

Postulate 4.3: (Angle-Side-Angle Congruence):  If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

 

Postulate 4.4: (Angle-Angle-Side Congruence):  If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the triangles are congruent.

 

Parts of a Right Triangle

            Legs:  the two sides that form the right angle

            Hypotenuse:  the side opposite the right angle.

Right Triangle Congruence:

Theorem 4.6:  (Leg-Leg Congruence): same as______

Theorem 4.7:  (Hypotenuse-Angle Congruence): same as____

Theorem 4.8:  (Leg-Angle Congruence): same as_____

Postulate 4.4: (Hypotenuse-Leg Congruence): same as____

 

 

 

Section 4.6 Isosceles Triangles

 

Parts of an Isosceles Triangle

            Legs:  the two congruent sides 

            Base:  the non-congruent side. (Not necessarily on the bottom!!!)

            Vertex Angle:  the angle formed by the legs. 

            Base Angles:  the angle formed by the base and one leg.

 

 

 

 

Isosceles Triangle Theorem:  If two sides of a triangles are congruent, then the angles opposite those sides are congruent.
This is also know as the Base Angles Theorem

 

 

Converse of Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.
This is also know as the Converse of the Base Angles Theorem

 

 

 

 

Corollary 4.3:  A triangle is equilateral if and only if it is equiangular.

 

 

 

Corollary 4.4: Each angle of an equilateral triangle measures 60 degrees.

 

 

EXAMPLE: