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Geometry
Chapter 4 Notes
4.1
Triangles and Angles
A triangle is a figure formed by three segments joining three noncollinear points.
Names of Triangles
Classified by Sides
Equilateral -
3 Congruent sides
Isosceles
-
At lease 2 congruent sides
Scalene
-
No congruent sides
Classified by Angles
Acute
-
3 acute angles
Equiangular -
3 congruent angles
Right
-
1 right Angle
Obtuse
-
1 obtuse angle
Labeling a triangle
Each of the 3 points jointing the sides of a triangle is a vertex
In a triangle, two sides sharing a common vertex are adjacent sides
In a right triangle the sides that form the right angle are the legs and the side
opposite the right angle is the hypotenuse.
Leg
hypotenuse
Leg
In an isosceles triangle, the congruent sides are the legs and the third side is the
base.
Leg
Leg
Base
Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°
m A + m B + m C = 180°
B
A
C
Theorem 4.2 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of
B
the two nonadjacent interior angles.
m 1 = m A + m B
1
A
Corollary
A corollary to a theorem is a statement that can be proved easily using the theorem
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary
m A + m B = 90°
A
B
4.2
Congruence and Triangles
Definition – When two figures are congruent, there corresponding angles are
congruent and their corresponding sides are congruent
Theorem 4.3 – Third angle Theorem
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the 3rd
angles are also congruent
B
E
A  D & B  E, then C  F
C
A
D
F
Theorem 4.4 – Properties of Congruent Triangles
4.3
Reflexive
Every triangle is congruent to itself
Symmetric
If ABC  DEF, then DEF  ABC
Transitive
If ABC  DEF & DEF  JKL, then ABC  JKL
Proving Triangles are Congruent (SSS and SAS)
Postulate 19 – Side-Side-Side (SSS) Congruence Postulate
If 3 sides of one triangle are congruent to 3 sides of a another triangle, then the 2
triangles are congruent
Side
Side
Side
BC  AC
BC  AC, and
BC  AC, then
ABC  DEF
E
B
A
C
D
F
Postulate 20 – Side-Angle-Side (SAS) Congruence Postulate
If 2 sides and the included angle of one triangle are congruent to 2 sides and the
included angle of a another triangle, then the 2 triangles are congruent
Side
BC  AC
Angle A  D , and
Side
BC  AC, then
ABC  DEF
4.4
B
E
C
A
F
D
Proving Triangles are Congruent (ASA and AAS)
Postulate 21 – Angle-Side-Angle (ASA) Congruence Postulate
If 2 angles and the included side of one triangle are congruent to 2 angles and the
included side of a another triangle, then the 2 triangles are congruent
Angle A  D
Side
BC  AC, and
Angle A  D , then
ABC  DEF
B
A
E
C
D
F
Theorem 4.5 – Angle-Angle-Side (AAS) Congruence Postulate
If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the
corresponding nonincluded side of a another triangle, then the 2 triangles are
congruent
Angle A  D
Angle A  D, and
Side
BC  AC, then
ABC  DEF
4.5
Using Congruent Triangles
B
A
E
C
D
F
4.6
Isosceles, Equilateral, and Right Triangles
Theorem 4.6 – Base Angles Theorem
If 2 sides of a triangle are congruent, then the angles opposite them are congruent.
B
If AB  AC, then B  C
A
C
Theorem 4.7 – Converse of the Base Angles Theorem
If 2 angles of a triangle are congruent, then the sides opposite them are congruent.
If B  C, then AB  AC
B
A
C
Corollary to Theorem 4.5
If a triangle is equilateral, then it is equiangular
Corollary to Theorem 4.6
If a triangle is equiangular, then it is equilateral,
Theorem 4.8 – Hypotenuse-Leg (HL)Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse
and a leg of a 2nd triangle, then the two triangles are congruent.
If BC  AC & AC  DF, then ABC  DEF
4.7
Triangles and Coordinate Proof