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## Chapter 4 Congruent Triangles

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**Chapter 4Congruent Triangles**• Identify the corresponding parts of congruent figures • Prove two triangles are congruent • Apply the theorems and corollaries about isosceles triangles**4.1 Congruent Figures**Objectives • Identify the corresponding parts of congruent figures**What we already know…**• Congruent Segments • Same length • Congruent Angles • Same degree measure**F**E Congruent Figures Exactly the same size and shape. Don’t ASSume ! C B A D**Two figures are congruent if corresponding vertices can be**matched up so that: 1. All corresponding sides are congruent 2. All corresponding angles are congruent. Definition of Congruency**What does corresponding mean again?**• Matching • In the same position**Volunteer**• Draw a large scalene triangle (with a ruler) • Cut out two congruent triangles that are the same • Label the Vertices A, B, C and D, E, F**You can slide and rotate the triangles around so that they**MATCH up perfectly. ABC DEF A E C B F D**The order in which you name the triangles matters !**ABC DEF A E C B F D**Based on the definition of congruency….**• Three pairs of corresponding sides • Three pairs of corresponding angles 1. A D 1. AB DE 2. B E 2. BC EF 3. C F 3. CA FD**It is not practical to cut out and move the triangles around**** ABC XYZ**• Means that the letters X and A, which appear first, name corresponding vertices and that • X A. • The letters Y and B come next, so • Y B and • XY AB**CAUTION !!**Don't ASSume • If the diagram doesn’t show the markings or • You don’t have a reason • Shared sides, shared angles, vertical angles, parallel lines**White Boards**• Suppose TIM BER IM ___**White Boards**• Suppose TIM BER IM ER , Why ?**White Boards**• Corresponding Parts of Congruent Triangles are Congruent**White Boards**• Suppose TIM BER ___ R**White Boards**• Suppose TIM BER M R, Why?**White Boards**• Corresponding Parts of Congruent Triangles are Congruent**White Boards**• Suppose TIM BER MTI ____**White Boards**• Suppose TIM BER MTI RBE**White Boards**• If ABC XYZ m B = 80 m C = 50 Name four congruent angles**White Boards**• If ABC XYZ m B = 80 m C = 50 A, C , X, Z**White Boards**• If ABC XYZ Write six congruences that must be correct**White Boards**• If ABC XYZ 1. A X 1. AB XY 2. B Y 2. BC YZ 3. C Z 3. CA ZX**Remote time**• Always • Sometimes • Never • I don’t know**A. AlwaysB. SometimesC. NeverD. I don’t know**• An acute triangle is __________ congruent to an obtuse triangle.**A. AlwaysB. SometimesC. NeverD. I don’t know**• A polygon is __________ congruent to itself.**A. AlwaysB. SometimesC. NeverD. I don’t know**• A right triangle is ___________ congruent to another right triangle.**A. AlwaysB. SometimesC. NeverD. I don’t know**• If ABC XYZ, A is ____________ congruent to Y.**A. AlwaysB. SometimesC. NeverD. I don’t know**• If ABC XYZ, B is ____________ congruent to Y.**A. AlwaysB. SometimesC. NeverD. I don’t know**• If ABC XYZ, AB is ____________ congruent to ZY.**4.2 Some Ways to Prove Triangles Congruent**Objectives • Learn about ways to prove triangles are congruent**Don’t ASSume**• Triangles cannot be assumed to be congruent because they “look” congruent. and • It’s not practical to cut them out and match them up so,**We must show 6 congruent pairs**• 3 angle pairs and • 3 pairs of sides**WOW**• That’s a lot of work Isn't there a shortcut ?**Spaghetti Experiment**• Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle. • Be careful, IT’S SPAGHETTI, and it will break.**Compare your spaghetti triangle to your neighbors**• Compare your spaghetti triangle to my spaghetti triangle. What do you notice ?**We are lucky…..**• There is a shortcut • We don’t have to show • ALL pairs of angles are congruent and • ALL pairs of sides are congruent**If three sides of one triangle are congruent to the**corresponding parts of another triangle, then the triangles are congruent. SSS Postulate E B C D A F**Patty Paper Practice**5 inches 3 inches 2.5 inches**If two sides and the included angle are congruent to the**corresponding parts of another triangle, then the triangles are congruent. SAS Postulate E B C D F**If two angles and the included side of one triangle are**congruent to the corresponding parts of another triangle, then the triangles are congruent. ASA Postulate E B C D A F**The order of the letters MEAN something**• Is SAS the same as SSA or A$$ ?**Construction 2**Given an angle, construct a congruent angle. Given: Construct: Steps:**Construction 3**Given an angle, construct the bisector of the angle Given: Construct: Steps:**CAUTION !!**Don't ASSume • If the diagram doesn’t show the markings or • You don’t have a reason • Shared sides, shared angles, vertical angles, parallel lines**Remote Time**Can the two triangles be proved congruent? If so, what postulate can be used? A. SSS Postulate • SAS Postulate • ASA Postulate • Cannot be proved congruent • I don’t know**A. SSS PostulateB. SAS PostulateC. ASA PostulateD.**Cannot be proved congruentE. I don’t know