Congruent Triangles

Polygons MNOL and ZYXW are congruent ∆ABC and ∆DEF are congruent Rectangles ABCD and EFGH are not congruent ∆ZXY and ∆JLP are not congruent A C D B H G E F Y Z X P J L

4-1 Congruent Figures Objective: To recognize congruent figures and their corresponding parts

Vocabulary/ Key Concept Congruent polygons- two polygons are congruent if their corresponding sides and angles are congruent

Naming Congruent Figures Ang Legs Triangle: Construct two triangles with the following sides-1 red, 1 blue, 1 yellow ∆ABC and ∆DEF óA óB óC

Warm Up: WXYZ  JKLM. List 4 pairs of congruent sides and angles. WX  JK XY  KL YZ  LM ZW  MJ  W   J  K   X  Y   L  Z   M

Each pair of polygons are congruent. Find the measure of each numbered angle M  1 = 110 m  2 = 120 M  3 = 90 m  4 = 135

We know: óBóF óAóE Then we can conclude: óCóD Key Concept: If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent. WARNING: This is only true for ANGLES not side lengths!

How do we know if two triangles are congruent? Concept Check!

Objective: To prove two triangles are congruent using SSS and SAS Postulates

Key Concepts SSS – Side-side-side corresponding congruence. SSS If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent (all corresponding sides are equal)

Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #1

Example 2: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #2

Key Concepts SAS – Side-Angle-Side corresponding Congruence. SAS ANGLE MUST BE IN BETWEEN THE TWO SIDES (INCLUDED ANGLE) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #3

Example 2: Can you use SAS to prove these two triangles are congruent? If no, what information would you need in order to use SAS to prove these triangles are congruent? Student Slide #4

Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement.  ABD   CBD by SAS AB  CB --CONGRUENCE MARKING BD  BD – REFLEXIVE PROPERTY OF CONGRUENCE  ABD   CBD –CONGRUENCE MARKING

óBóE If we know: What other information must we know in order to prove ∆ABC ∆DEF using SAS? Example:

WARM UP (will be collected): a)Name the three pairs of corresponding sides b)Name the three pairs of corresponding angles c)Do we have enough information to conclude that the two triangles are congruent? Explain your reasoning. *CORRESPONDING DOES NOT MEAN THEY ARE CONGRUENT!

WUP#1: Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. What do you know? NP QP -- CONGRUENT MARKS NR QR -- CONGRUENT MARKS RP RP -- REFLEXIVE PROPERTY OF   PRN   PRQ by SSS

WUP #2: What one piece of additional information must we know in order to prove the triangles are congruent using SAS. Explain your reasoning and then write a congruence statement. Explanation: Statement:

Objective: To prove two triangles are congruent using ASA, AAS, and HL Postulates

Key Concepts ASA – Two angles and an included side. ASA SIDE IS IN BETWEEN THE ANGLES If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

AAS – Two angles and a non-included side. AAS Key Concepts If two angles and the non-included side of a triangle are congruent to two angles and the non- included side of another triangle, then the two triangles are congruent.

Determine if you can use ASA or AAS to prove two triangles are congruent. Write the congruence statement.

Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement. Explain:

Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement. Explain: TRY ONE

Congruence that works: Congruence that does not work:  SSS SAS AAS ASA ASS SSA AAA *Remember, we don’t swear in math (not even backwards). And no screaming!

What did you learn today? What are the five ways (one for right triangles) to prove triangles are congruent?

Example 1: Complete the 2 column proof: Given: óBAE óEDB, óABE óDEB Prove: óABE óDEB StatementsReasons

So what do we know about the parts of congruent triangles? Corresponding Parts of Congruent Triangles are Congruent Hence, *Remember, you can only use CPCTC, AFTER you have proven two triangles to be congruent!

Write a Proof Statement 1.FJ  GH  JFH   GHF 2.HF  FH 3.  JFH   GHF 4.FG  JH Reasons 1.Given 2.Reflexive property of congruence 3.SAS 4.CPCTC

TRY ONE: Write a Proof Statement 1.AC  CD, óBAC  óCDE 2.  ACB   ECD 3.  DEC   ABC 4.  B   E Reasons 1.Given 2.Vertical angles 3.ASA 4.CPCTC Given : óBAC  óCDE, AC  CD Prove: óB  óE

What did you learn today? What does CPCTC mean and when do we use it?

CPCTC Song (sung to the tune of “YMCA” by the Village People) Author of lyrics: Eagler Young man, there's no need to feel down I said, young man, pick yourself off the ground I said, young man, 'cause there's a new thing I've found There's no need to be unhappy Young man, there's this thing you can do I said, young man, it's so easy to prove You can use it, and I'm sure you will see Many ways to show congruency It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C Barely takes any time, uses only one line It's the easiest thing you'll find It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C If you don't have a clue, it's so simple to do Write five letters and you'll be through

Mini Lab

Similar Polygons: Two polygons are similar if: 1)Corresponding angles are congruent 2)Corresponding sides are proportional

Determine whether rectangle HJKL is similar to rectangle MNPQ.

Transformations

Reflection over the x-axis Preserves congruence

Rotation 90 ô about the origin Preserves congruence 90 ô (x,y)  (-y,x) Ex: D(6,3)  D’(-3,6) 180 ô (x,y)  (-x,-y)

Translation Preserves congruence 3 units right, 4 units down

Dilation Does not preserve congruence Scale factor (length of image/length of original) :2/1 BUT, are they similar?

Description:

1) Which transformations preserve congruence? 2) What criteria needs to be met for triangles to be similar?