Acute triangle- all angle measures are less than 90 degrees Obtuse triangle- one angle measure is greater than 90 degrees Right triangle- one angle measures 90 degrees Equiangular triangle- an acute triangle with all angles congruent Scalene triangle- no two sides are congruent Isosceles triangle- at least two sides are congruent Equilateral triangle- all of the sides are congruent Exterior angle- formed by one side of a triangle and the extension of another side Remote interior angles- the interior angles of the triangle no adjacent to a given exterior angle. Flow proof- organizes a series of statements in logical order, starting with the given statements Corollary- a statement that can be easily proved using a theorem Congruent triangles- triangles that are the same size and shape Congruence transformations- slide, flip, or turning a triangle Included angle- a unique triangle Included side- two angles of a triangle and the side between them Vertex angle- angle formed by the congruent sides Base angles- the two angles formed by the base and one congruent side Coordinate proof- uses figures in the coordinate plane and algebra to prove geometric concepts
Chapter 4 Section 1 - Identify and classify triangles by angles and sides. Explain how to classify triangles by o Angles o Sides Classifying triangles by angleso In an acute triangle, all of the angles are acute.
o o All angle measures are less than 90 degrees o In an obtuse triangle, one angle is obtuse.
o o One angle measure is greater than 90 degrees. o In a right triangle, one angle is right.
o o One angle measure equals 90 degrees. Classifying triangles by sideso No two sides of a scalene triangle are congruent.
o o At least two sides of an isosceles triangle are congruent.
o o All of the sides of an equilateral triangle are congruent.
o
Example- find x, JM, MN, and JN if triangle JMN is an isosceles triangle with segment JM congruent to segment MN. 2x-5=3x-9: -5=x-9: 4=x 2(4)-5: 8-5: 3
3(4)-9: 12-9: 3 3-2: 1
Chapter 4 Section 2 -Classify Triangles - Apply the angle and sum theorem, apply the exterior angle sum theorem. THE SUM OF THE MEASURES OF THE ANGLES OF A TRIANGLE IS 180! o Third angle theorem If two angles of a triangle are congruent to two angles of another triangle, then the third pair of angles is congruent. o Exterior angle sum theorem The measure of an exterior angle is the sum of its two remote interior angles. o Examples of corollaries In any triangle, there can be at most one right angle or one obtuse angle. The acute angles of a right triangle are complementary.
Chapter 4 Section 3 -
Solve problems involving congruent triangles. o Congruent triangles Triangles are congruent if and only if their corresponding parts are congruent.
 o Example-
Chapter 4 Section 4 -
Prove triangles are congruent using SSS and SAS postulates o Side-Side-Side postulate SSS If 3 sides of a triangle are congruent to 3 sides of another triangle, then the triangles are congruent by SSS.
o Side-Angle-Side postulate SAS If two sides of a triangle are congruent to corresponding sides of another triangle and the included angles are congruent, then the triangles are congruent.
Chapter 4 Section 5 -
Prove triangles are congruent using ASA and AAS o Angle-Side-Angle postulate ASA If 2 angles of a triangle and the included side are congruent to 2 angles and the included side of another triangle, then the triangles are congruent.
o Angle-Angle-Side postulate AAS If 2 angles and their non-included side of a triangle are congruent to 2 angles and their non-included side of another triangle, then the triangles are congruent.
Chapter 4 Section 6 -
Prove right angles are congruent o Hypotenuse-Leg Theorem
Triangle ABC is congruent to Triangle XYZ
o Leg-Angle Theorem o Leg-Leg Theorem
o Hypotenuse-Angle Theorem
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Chapter 4 Section 7 -
Solve problems involving isosceles triangles Solve coordinate proofs o Isosceles triangle has two congruent sides
 o Isosceles triangle theorem- angles opposite of the legs of an isosceles triangle are congruent (base angles are congruent) Steps for a coordinate proof1. Use the origin as a vertex of the triangle 2. Place at lease one side of the triangle on an axis 3. Keep the triangle in the first quadrant, whenever possible 4. Use coordinates that make computations as simple as possible.