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4-1 Classifying Triangles A. Definitions 1. A________ is a three sided polygon. 2. A____________ is a closed figure, in a plane, made up of segments, called ________, that intersect only at their endpoints, called ___________. 3. Triangle ABC, written ABC, has the following parts. sides: AB, BC,AC vertices: A, B, C angles: <ABC , <ACB , <BAC A B C B. Classifying triangles 1. By sides: a. __________ – all 3 sides are different lengths. b. __________ – at least two of the sides are the same length. c. ____________ – All 3 sides are the same length 2. By angles a. Acute triangle – All the angles are less than 90. b. Right triangle – One of the angles is exactly 90. c. Obtuse triangle – One of the angles is greater than 90. C. Special Triangles 1. Right triangles Hypotenuse Leg Leg 2. Isosceles triangles vertex angle Vertex Angle Leg Leg Base Angle Base Angle Base Ex 1: ABC is an equilateral triangle. Find x and the measure of each side if AB = 4x -3 and BC = 3x + 4 Ex 2: Given with DAR vertices D(2, 6), A(4, -5), R(-3,0), use the distance formula to show DAR is scalene. Geometry 4-2 Measuring Angles in Triangles A. Theorems 1. Theorem 4-1 – __________ _________ ____________ – The sum of the measures of a triangle is 180. Example 1: Find x in the triangle. x 68o 33o 2. Theorem 4-2 – ______________ _____________ ________________ – If two angles of one triangle are congruent to two angles of a second triangle, then the 3rd angles of the triangles are congruent. 3. Theorem 4-3 – _________________ _____________ _______________ -The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. B. Definitions 1. ______________ ____________ -<BCD is an exterior angle 2. _____________ ________________ ______________ - <A and <B are two remote interior angles. B A C C. Explanation of Exterior Angle Theorem Imagine <BCA = 70o, how big is <BCD =_______ Now, take a look at ABC , what is going to be the sum of <A and <B ?=_______ D Example 2: Find the measure of each numbered angle. 65o 46o 82o 1 3 2 142o 4 5 3. A ___________ _____________ organizes a series of statements in logical order, starting with the given statement. Each statement is written in a box with the reason verifying the statement written below the box. Example 3: Write a flow proof of the Exterior Angle Theorem. ABC Given <CBD and <ABC form a Linear Pair m<A + m<ABC + m<C = 180 <CBD and <ABC are Supplementary m<A + m<ABC + m<C = m<CBD + m<ABC m<CBD + m<ABC = 180 m<A + m<C = m<CBD 4. A statement that can be easily proved using a theorem is called a ________________. a.) Corollary 4.1 -The acute angles of a right triangle are complimentary. Example: m<A + m<C = ___ b.) Corollary 4.2 -There can be at most one right or obtuse angle in a triangle. Geometry 4-3 Congruent Triangles A. Corresponding Parts of Congruent Triangles 1. Triangles that have the same size and shape are called ______________ ______________. ABC XYZ -the order matters! Vertex A corresponds to vertex X 2. Definition of Congruent Triangles (CPCTC) -Two triangles are congruent iff their corresponding parts are congruent. (CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent it is the converse of the definition). Example 1: If the two triangles are congruent. Name all the congruent angles. ________ ________ ________ ________ ________ ________ Name all the congruent sides. ________ ________ ________ ________ ________ ________ 3. Theorem 4.4 -Congruence of triangles is reflexive, symmetric, and transitive. -meaning ABC ABC for reflexive. -meaning if ABC DEF, then DEF ABC B. Identity Congruence Transformations B A Y C X Z 1. In the picture, ABC XYZ , if you turn, flip, or slide, affect the congruence of the triangle. 2. These changes are called congruence transformations. ABC DEF XYZ , it does not Example 2: The vertices of RST are at R(-3, 0), S(-1, 5), and T(1, 1), and the vertices of R’S’T’ are at R’(3, 0), T’(-1, -1), S’(1, -5). a.) Prove that RST R’S’T’ b.) Name the congruence transformation for RST and R’S’T’ Geometry 4-4 Proving Triangles Congruent A. Postulate 4-1 SSS Postulate (Side-Side-Side Postulate) -If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Ex 1: Given STU with vertices S(0, 5), T(0, 0), and U(-2, 0), and XYZ with vertices X (4, 8), Y(4, 3) and Z(6, 3), determine if STU XYZ . ST=_____ TU=_____ SU=_____ XY=_____ YZ=_____ XZ=_____ B. Postulate 4-2 SAS Postulate (Side-Angle-Side Postulate) -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. Ex 2: Given: X is the midpoint of BD. X is the midpoint of AC. Prove: DXC BXA D A X C B Method 1 Two-Column Proof Statements Reasons 1. X is the midpoint of BD X is the mid point of AC 1.) Given 2.) DX XB AX XD 2.) 3.) <DXC <BXA 3.) 4.) DXC BXA 4.) Method 2 – Flow Proof X is midpoint of BD X is midpoint of AC <DXC <BXA DX BX CX AX DXC BXA Geometry 4-5 More Congruent Triangles A. Current Congruence 1. We know three ways to prove triangles congruent so far. They are ______, ________, _________. B. Postulate 4-3 ASA Postulate (Angle-Side-Angle Postulate) -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. Ex 1: Given: BE bisects AD, and <A D Prove: AB DC Statements 1) BE bisects AD, <A <D 2) Reasons 1) Given 2) Vertical <’s are 3) AE ED 3) 4) AEB DEC 4) 5) AB DC C. Theorem 4-5 AAS Postulate (Angle-Angle-Side Postulate) -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 two triangles are congruent. Ex 2: Write a proof Given: <PSU <PTR SU TR Prove: SP TP S T R U P Statements Reasons 1) < PSU <PTR, SU TR 2) 1) Given 2) Reflexive 3) 3) 4) SP TP Ex 3: Some of the measurements of ABC and determine if the two triangles are congruent? DEF are given. Can you Explore: Since we know <A <D, and we know AB DE, and CB FE ; we know that there are two congruent sides and one non-included angle that are congruent. Our five methods of proving triangles congruent are: 1.) definition of congruent 2.) SSS congruence 3.) SAS congruence 4.) ASA congruence 5.) AAS congruence ’s (all 3 sides and all 3 angles) -since none of these work, the triangles must not be congruent. Counterexample: here is a case where the triangles are obviously not congruent, but the same conditions hold true. D. HL (Hypotenuse-Leg) – A special case of SSA that only works when trying to prove 2 or more right triangles are congruent. 4-6 Analyzing Isosceles Triangles A. Isosceles Triangles 1. Theorem 4-6 Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite those sides are congruent Ex 1: Proof of Theorem 4-6 Given: PQR, PQ QR Prove: <P <R Statements Reasons 1) Let S be the midpoint of PR 2) Draw segment QS 3) PS RS 1) Every segment has exactly one midpoint 2) Through any 2 points there is 1 line 3) 4) QS QS 4) 5) PQ QR 5) 6) PQS RQS 7) <P <R 6) 7) Ex 2: In isosceles ISO with base SO. If m<S = 2x + 40 , and m<F = 3x + 22, find the measure of each angle of the triangle. 2. Theorem 4-7 -If two angles of a triangle are congruent, then the sides opposite those angles are congruent 3. Corollary 4-3 -A triangle is equilateral if and only if it is equiangular. 4. Each angle of an equilateral triangle measures 60o .