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Geometry Section 4.1 ________________________________________________ Start Thinking: What does it mean for figures to be congruent? Example 1: Give a congruence statement to describe that the figures are congruent. Then explain what their congruent parts are. Example 2: Triangle ABC is Congruent to Triangle MNK. Explain what side/ angle is congruent to the following. a.) <A is congruent to _______________ b.) <C is congruent to _________________________ c.) AB ___________________ d.) KM ____________________________ Example 3: Write a congruence statement for TWO sets of congruent triangles. Be careful with the order of the points. USING CONGRUENT TRIANGLES: EX1: Triangle ABC is congruent to Triangle ABD. If m<DAB=48 and m<c = 60 then what is the m<D? EX2: Triangle ABD is congruent to Triangle DEA. M<B = 50 m<A = 90. What is m<ADB? PROVING TRIANGLES ARE CONGRUENT To prove that two triangles are congruent you need to make sure that all corresponding __________________ are congruent and all corresponding ___________________________ are congruent. You can then justify that the triangles are congruent by the _____________________________________. Example 1: EXAMPLE 1: EXAMPLE 2: EXAMPLE 3: (do not use 3rd angle theorem) Given: Diagram to Right Prove: Triangle ACB is congruent to Triangle ECD WRAP UP: 1.) To Prove that two triangles are congruent you need to show that all __________________________________ __________________________ are congruent and all ______________________________________ are congruent. 2.) If two angles of a triangle are congruent to two angles of another then ________________________________ _________________________________________. 3.)When you write a congruence statement you must be careful with the _______________________ you list the points. Geometry Section 4.2 ______________________________________ So Far: In order to prove that two triangles are congruent you must show that all the corresponding ____________________ are congruent and that all corresponding ________________________________ are congruent. POSTULATE 4-1: Side – Side – Side (SSS) Postulate Postulate: If: Then: If the three sides of one triangle are congruent to three sides of another triangle then _________ __________________________. EXAMPLE 1: EXAMPLE 2: POSTULATE 4-2: Side – Angle – Side (SAS) Postulate Postulate: If: Then: If ________________ and the _______________ of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are _________________. EXAMPLES: 1.) 2.) EX: Determine if there is enough information to determine if the following triangles MUST be congruent. 1.) 4.) 2.) 3.) 5.) PROOFS: 1.) GIVEN: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷 and ̅̅̅̅ 𝐴𝐵 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 ̅̅̅̅ 𝐶𝐷 PROVE: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐶𝐵 ̅̅̅̅ and 𝐵𝐹 ̅̅̅̅ ̅̅̅̅ is perpendicular to 𝐶𝐹 2.) GIVEN: 𝐴𝐶 ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ and 𝐶𝐷 ≅ 𝐷𝐹 ̅̅̅̅ 𝐴𝐶 ≅ 𝐵𝐹 PROVE: ∆𝐷𝐴𝐶 ≅ ∆𝐷𝐵𝐹 3.)GIVEN: ̅̅̅̅ 𝐴𝐶 ≅ ̅̅̅̅ 𝐶𝐹 AND C is the midpoint of ̅̅̅̅ 𝐵𝐷 PROVE: ∆𝐶𝐴𝐵 ≅ ∆𝐶𝐹𝐷 4.) GIVEN: Figure ABGN is a square ̅̅̅̅̅ ≅ 𝐵𝐹 ̅̅̅̅ 𝑀𝑁 G is the midpoint of ̅̅̅̅̅ 𝑀𝐹 PROVE: ∆𝑀𝑁𝐺 ≅ ∆𝐹𝐵𝐺 WRAP UP You can prove that two triangles are congruent by showing that all corresponding sides are congruent and all corresponding angles are congruent OR you can use ______________________ or _______________________. If you are going to use SAS you must be sure that the angles is ____________________________ the sides. Geometry Section 4.4 ___________________________________ Start thinking: You are given the following information about Triangle ABC and Triangle DEF are congruent. You are also given that 𝑚∠𝐴 = 50 and 𝑚∠𝐷 = 2𝑥. What does x equal? How do you know this? Theorem 4.4: CPCTC Thm: 4.4: If two figures are _______________ then their ___________________ are congruent. Example: Example: Example: ̅̅̅̅ ≅ ̅̅̅̅ ̅̅̅̅ Given: 𝐴𝐵 𝐴𝐶 , M is the midpoint of 𝐵𝐶 Prove: AMB AMC Example: EX: EX: EX: Given: ̅̅̅̅ 𝐵𝐷 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐴𝐵𝐶 ∠𝐵𝐴𝐷 ≅ ∠𝐵𝐶𝐷 ̅̅̅̅ 𝐴𝐸 ≅ ̅̅̅̅ 𝐶𝐹 ̅̅̅̅ ̅̅̅̅ 𝐸𝐷 ≅ 𝐷𝐹 Prove: <E is congruent to <F Geometry Section 4.5 ________________________________________ Start Thinking: The following are all examples of isosceles triangles. How would you define an isosceles triangle? PARTS OF AN ISOSCELES TRIANGLE: Theorem 4.3: Isosceles Triangle Theorem Theorem: If two sides of a triangle are congruent, then ___________________________________. Example: Prove It: APPLICATIONS: 1.) 2.) 3.) 4.) Converse of Isosceles Triangle theorem If ___________________________________________ then______________________________. EXAMPLE: Proof: Practice: Use the diagram. Tell what theorem or corollary you used. a. If AE CE , then ______ ______ b. If DAE DEA, then _______ _______ c. If BDF DBF BFD, then ________ _______ _______ d. If AB BC AC , then ______ _______ ______ Find the value of x. 1) Corollary to the Isosceles Triangle Theorem IF all sides of a triangle are congruent then _____________________. Corollary to the CONVERSE OF Isosceles Triangle Theorem IF all angles of a triangle are congruent then _____________________. THEOREM 4.5 If the vertex angle of a isosceles triangle is bisected then the bisector is a_________________ of the base. Proof: Practice: 1.) ̅̅̅̅ is an angle bisector of the vertex angle of isosceles Triangle ACD. It 2.) 𝑨𝑩 intersects the base at B. If CD= 40 and CB=2x then what is x? What m<ABD? Geometry Section 4.6 _______________________________________________ Start thinking: The current methods that we have to prove that triangles are congruent are ____________, ________________, ___________________, and ___________________. We know that AAA does not prove congruent but the SSA has always been questionable. See if you can change the appearance of the triangle below without changing the measure of the angle or the length of the two sides. 3 KEY CONDITIONS NEEDED TO USE THE _____________________________ Conditions: Ex1: Decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem. Ex2: State the third congruence that must be given to prove congruence. IF: AC DF , A is a right angle and A D, ______ _______ Use HL Thm IF: J is a right angle and J D, _______ ________Use HL Thm Practice: Decide whether enough information is given to prove that the triangles are congruent. If there is enough information, state the congruence postulate or theorem you would use AND write a congruence statement. PROOF ME: 1.) Given: RT SU , RU RS Prove: ∆RUT ∆RST 2.) Given: AD bisects EB, AB DE; ECD, ACB are right angles. Prove: ∆ACB ∆DCE PRACTICE: 1) GIVEN: QS PR , PS RS , Q R RS PROVE: PRS QSR 2) GIVEN: OM LN , ML MN, PROVE: OML OMN R Statements Reasons 1. 1. Given 2. 2. If 2 angles are , then they form 4 right s. 3. 3. Right Angle Congruence Theorem 4. 4. Given 5. OM OM 6. OML OMN 5. 6. Geometry Section 4.7 ___________________________________________ Start Thinking: **The methods of proving that triangles are congruent are _____________, _____________, _____________, __________________, and ______________. **When proving angles or sides are congruent first you ____________________________ then use _____________________________________. DEALING WITH INTERLOCKING TRIANGLES: 1st : ______________________________________________________________________ 2nd: ______________________________________________________________________ Determining what are shared angles and sides: EX 1: YOU TRY: EX 2: Example 1: EX 2: EX 3: EX 4: EX 5 EX6 EX 7 EX 8 EX 9: