
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