Download Geom ch. 4

4-1 Classifying Triangles
Acute triangle—a triangle in which all of the angles are acute angles
Obtuse triangle—a triangle with an obtuse angle
Right triangle—a triangle with a right angle
Equiangular triangle—a triangle with all angles congruent
Scalene triangle—a triangle with no two sides congruent
Isosceles triangle—a triangle with at least two sides congruent.
Equilateral triangle—a triangle with all sides congruent
4-2 Angles of Triangles
Exterior angle—an angle formed by one side of a triangle and the extension of another side
Remote interior angles—the angles of a triangle that are not adjacent to a given exterior angle
Flow proof—a proof that organizes statements in logical order, starting with the given
statements. Each statement is written in a box with the reason verifying the statement written
below the box. Arrows are used to indicate the order of the statements.
Corollary
Theorem 4.1
Angle Sum Theorem—the sum of the measures of the angles of a triangle is 180.
Theorem 4.2
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.
Theorem 4.3
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.
Corollaries
4.1—the acute angles of a right triangle are complementary
4.2—there can be at most one right or obtuse angle in a triangle.
4-3 Congruent Triangles
Congruent triangles—triangles that have their corresponding parts congruent
Congruence-Transformations—in a plane, a mapping for which each point has exactly one image point and
each image point has exactly one pre-image point.
Theorem 4.4
Congruence of triangles is reflexive, symmetric, and transitive.
4-4 Proving congruence---SSS, SAS
Included angle—in a triangle, the angle formed by two sides is the included angle for those two
sides.
Postulate 4.1
Side-Side-Side Congruence—If the sides of one triangle are congruent to the sides of a second
triangle, then the triangles are congruent.
Abbreviation: SSS
Postulate 4.2
Side-Angle-Side Congruence—if two sides and the included angle of one triangle are congruent
to two sides and the included angle of another triangle, then the triangles are congruent.
Abbreviation: SAS
4-5 Proving Congruence—ASA, AAS
Included side—the side of a triangle that is a side of each of two angles
Postulate 4.3
Angle-Angle-Side congruence—if two angles and the included side of one triangle are congruent
to two angles and the included side of another triangle, then the triangles are congruent.
Abbreviation: ASA
Theorem 4.5
Angle-Angle-Side Congruence
If two angles and a non-included side of one triangle are congruent to the corresponding two
angles and side of a second triangle, then the triangles are congruent.
4-6 Isosceles Triangles
Vertex angle—the angle in the middle of the two legs in an isosceles triangle
Base angles—the angles opposite the congruent legs in an isosceles triangle.
Theorem 4.9
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Theorem 4.10
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Corollaries
4.3 A triangle is equilateral if and only if it is equiangular.
4.4 Each angle of an equilateral triangle measures 60 degrees.
4-7 Triangles and Coordinate Proof
Coordinate proof—a proof that uses figures in the coordinate plane, and algebra to prove
geometric concepts.