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Congruent Triangles
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Polygons MNOL and ZYXW are congruent
∆ABC and ∆DEF are congruent Rectangles ABCD and EFGH are not congruent A D Y Z E H ∆ZXY and ∆JLP are not congruent L X B C G F J P
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4-1 Congruent Figures Objective: To recognize congruent figures and their corresponding parts
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Vocabulary/ Key Concept
Congruent polygons- two polygons are congruent if their corresponding sides and angles are congruent
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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
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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
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Each pair of polygons are congruent
Each pair of polygons are congruent. Find the measure of each numbered angle M1 = 110 m 2 = 120 M3 = 90 m 4 = 135
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We know: óB óF óA óE Then we can conclude: óC óD Key Concept: If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent. Pose question: WHAT IS THIS CALLED BASED ON OUR INVESTIGATION? AAA Can we call these congruent? NO! WARNING: This is only true for ANGLES not side lengths!
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Concept Check! How do we know if two triangles are congruent?
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To prove two triangles are congruent using SSS and SAS Postulates
Objective: To prove two triangles are congruent using SSS and SAS Postulates
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Key Concepts SSS – Side-side-side corresponding congruence.
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)
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Example 1: State if 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 #1
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Example 2: State if the two triangles are congruent
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
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Key Concepts SAS – Side-Angle-Side corresponding Congruence.
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
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Example 1: State if 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
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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 Answer: GH congruent to RQ OR angle H congruent to angle S
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Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. AB CB --CONGRUENCE MARKING BD BD – REFLEXIVE PROPERTY OF CONGRUENCE ABD CBD –CONGRUENCE MARKING ABD CBD by SAS
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óB óE Example: If we know:
What other information must we know in order to prove ∆ABC ∆DEF using SAS?
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WARM UP (will be collected):
Name the three pairs of corresponding sides Name the three pairs of corresponding angles Do we have enough information to conclude that the two triangles are congruent? Explain your reasoning. *CORRESPONDING DOES NOT MEAN THEY ARE CONGRUENT!
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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
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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:
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To prove two triangles are congruent using ASA, AAS, and HL Postulates
Objective: To prove two triangles are congruent using ASA, AAS, and HL Postulates
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Key Concepts ASA – Two angles and an included side.
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.
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Key Concepts AAS – Two angles and a non-included side.
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.
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Determine if you can use ASA or AAS to prove two triangles are congruent. Write the congruence statement.
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Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement. Explain:
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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
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Congruence that works: Congruence that does not work:
ASS SSA AAA SSS SAS AAS ASA *Remember, we don’t swear in math (not even backwards). And no screaming!
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What did you learn today?
What are the five ways (one for right triangles) to prove triangles are congruent?
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óABE óDEB Example 1: Complete the 2 column proof:
Given: óBAE óEDB, óABE óDEB Prove: óABE óDEB Statements Reasons
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CPCTC So what do we know about the parts of congruent triangles?
Corresponding Parts of Congruent Triangles are Congruent CPCTC Hence, *Remember, you can only use CPCTC, AFTER you have proven two triangles to be congruent!
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Write a Proof Statement FJ GH JFH GHF HF FH JFH GHF
FG JH Reasons Given Reflexive property of congruence SAS CPCTC
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TRY ONE: Write a Proof Statement AC CD, óBAC óCDE ACB ECD
Given: óBAC óCDE, AC CD Prove: óB óE Statement AC CD, óBAC óCDE ACB ECD DEC ABC B E Reasons Given Vertical angles ASA CPCTC
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What did you learn today?
What does CPCTC mean and when do we use it?
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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
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Mini Lab
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Similar Polygons: Two polygons are similar if:
Corresponding angles are congruent Corresponding sides are proportional
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Determine whether rectangle HJKL is similar to rectangle MNPQ.
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Transformations
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Reflection over the x-axis
Preserves congruence
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Rotation 90ô about the origin
(x,y) (-y,x) Ex: D(6,3)D’(-3,6) 180ô (x,y)(-x,-y) Preserves congruence
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Translation 3 units right, 4 units down Preserves congruence
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Dilation Scale factor (length of image/length of original) :2/1
Does not preserve congruence BUT, are they similar?
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Description:
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Description:
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1) Which transformations preserve congruence?
2) What criteria needs to be met for triangles to be similar?
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Proving Triangles Similar
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There are 3 postulates that we can use to prove triangles are similar…
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Angle-Angle Similarity (AA ~) Postulate:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
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Explain why the triangles are similar. Write a similarity statement.
Statements Reasons Equal measures Vertical angles are AA ~ postulate óR óV óRSW óVSB ∆RSW ~ ∆VSB
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Side-Angle-Side Similarity (SAS ~) Postulate:
If an angle of one triangle is congruent to an angle of a second and the sides including the two angles are proportional, then the two triangles are similar.
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Explain why the triangles are similar. Write a similarity statement.
Statements Reasons Both right angles SAS ~ Postulate óR óY And and Therefore ∆RSW ~ ∆VSB
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Side-Side-Side Similarity (SSS ~) Postulate:
If the corresponding sides of two triangles are proportional then the two triangles are similar.
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Explain why the triangles are similar. Write a similarity statement.
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Determine if the two triangles are similar
Determine if the two triangles are similar. If they are, write a similarity statement and find DE. Yes, by SAS~… ABC ~ EBD…DE = 15
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Solve Proportions.
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6/x = 7/40.5
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In sunlight, a cactus casts a 9-ft shadow
In sunlight, a cactus casts a 9-ft shadow. At the same time a person 6 ft. tall casts a 4 ft. shadow. Use similar triangles to find the height of the cactus. 6/x = 4/9
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Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides then it divides those sides proportionally. Test Taking Tip: Whenever you see a line that passes through a triangle and is parallel to one side, the Side Splitter Theorem may apply! x/16 = 5/10
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Triangle Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. 5/8 = 6/x
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Name the three postulates that prove two triangles are similar.
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