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Transcript
Chapter 4: Congruent Triangles 4.1 Classifying OBJECTIVES: Students will be able to Triangles * identify and classify triangles by angles * identify and classify triangles by sides Why is important to learn Triangles in construction: about triangles? ___________________________________________________________ Triangle Label Sketch an example Types of Triangles: Acute – all angles are acute Classify by Angles There are 3 types of angles: Obtuse – one angle is obtuse Acute: Obtuse: Right – one angle is right Right: What is the symbol used to show that angles are congruent? Classify by Sides What is the symbol used to show that line segments are congruent? Equiangular – all angles are equal Scalene – no two sides are congruent Isosceles – at least two sides are congruent Equilateral –all sides are congruent Examples Identify the indicated type of triangles if AB AD BD DC , BE ED, AB BC and ED DC . Day 1 Homework: Assignment 4.1 page 180 (3-4, 9-10, 22-25, 48, 50) 1. right 2. obtuse 3. scalene 4. isosceles Use the B/D/A Strategy Find x and the measure of each side of the triangle. 5. Triangle FGH is equilateral. FG = x + 5, GH = 3x - 9, and FH = 2x - 2. Underline Draw Underline (Key Words & Numbers): Find Choose Draw (don’t forget to label): Find (what the problem is asking): Solve Choose (how are you going to solve? Set up to solve) & Solve: CHECK: does your answer make sense? Do you have everything you needed to answer the problem? 6. Triangle LMN is isosceles, L is the vertex angle, LM = 3x - 2, LN = 2x + 1, and MN = 5x - 2. Use the distance formula In problems 7 & 8, find the measures of the sides of triangle KPL and classify each triangle by its sides. 7. K(-3, 2) P(2, 1), L(-2, -3) Distance formula review: The distance, d, between points 𝑥�1�, 𝑦�1�� and 𝑥�2�, 𝑦�2�� is How to solve problems like this? 8. K(5, -3), P(3, 4), L(-1, 1) Day 2 Homework: Assignment 4.1 page 180 (26-29, 33-37 odd, 46, 49) 4.3 Congruent Triangles Objective: -Name and label corresponding parts of congruent triangles Why does order matter?: What would happen if you put your shoes on before your pants? Definition of Triangles CPCTC Naming corresponding a. parts **The vertices of the triangles correspond in the same order as the letters naming the triangles. A ______ B ______ F ______ AC ______ DE ______ FE ______ b. ∆ABD ∆CBD If two triangles are congruent, you can slide, flip, or turn one of the triangles and they will still be congruent. These are called congruence transformations because they do not change the size or shape of the figure. Assignment 4.3 page 195 #9-16, 29-32,38,39 4.4 and 4.5 Proving Objective: Congruence – SSS, SAS, -Use the SSS, SAS, ASA, and AAS Postulate to test for triangle ASA, AAS congruence. Postulates that Prove Triangles Congruent: (shortcuts to proving triangles congruent) Using The Postulates: 1. Suppose D is the midpoint of both GE and FH. Label the pictures first. Then decide, if you are given enough information to prove the triangles are congruent? Explain 2. How would you complete the following congruence statement? ▲FDE __________ 3. Suppose you are not told anything about D but instead you are told that GH║FE. Can you prove the triangles above are congruent? Explain. 4. This time, suppose you know that EF GH and EF║ GH. Can you prove the two triangles above are congruent? Explain. 5. In this diagram, D is the midpoint of AC and AB CB. First label the picture. Then decide if you can prove any triangles congruent? Explain. Proving Triangles Examples: 1. Given: X is the midpoint of FM ; OF AM; OX AX . Prove: ΔFOX ΔMAX 2. First label the pictures with the given information. Then ask your self if you have enough information to prove the triangles using SSS, SAS, ASA, AAS Statements Reasons Statements Reasons 3. Statements Reasons 4. Statements Assignment 4.4 & 4.5 Part 1: page 203 #7-11, Part 2:page 203 #6,16 page 210 #6,9,10,15 Reasons 4.6 Isosceles Triangles Objective:-use properties of isosceles triangles -use properties of equilateral triangles Properties of Isosceles Triangles Vertex Angle: ______ legs:_____________ Definition of an Isosceles Triangle:___ _________________________________ _________________________________ _________________________________ Base Angles:_____________ Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Example: Find x. Converse Isosceles Triangle If two angles of a triangle are congruent, then the sides opposite Theorem those angles are congruent. Example: Find x. Equilateral Triangles Draw an Equilateral Triangle Compare the isosceles and Equilateral Triangles List all properties True or false. All Equilateral triangles are Isosceles._______________________ Assignment A triangle is equilateral if and only if it is page 219 #6,8-28, 35,36 equiangular.__________ All Isosceles triangles are Equilateral___________________ Each angle of an equilateral triangle measures 60°.____________