Chapter 4: Congruent Triangles

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1 Name : Date Block # Chapter 4: Congruent Triangles Day Topic ssignment all dates are subject to change 1 1 Triangle ngle-sum Theorem pg 221 # even Congruent Figures pg 228 #5-11,26 2 Quiz 3-5 Triangle Congruence pg 236 # 5-8, 9, 18,24, 26 SSS, SS, S, S and HL pg 243 #3,4, 9-11, 20-22, 35,36 3 Continue previous lesson with more Practice pg 252 #3-5, 8, 31, 33, 34 4 Quiz 6 Using Congruent pg 259 # 3-8, Triangles CPCTC 5 7 Isosceles and pg # 267 # , 26, 46 Equilateral Triangles Congruence 6 Using CPCTC and overlapping triangles 7 Review 8 Test

2 Name: Date Block # Triangle Sum Properties Triangles can be classified by the number of congruent sides that they have. Zero Congruent Sides Two+ Congruent Sides ll 3 Congruent Sides Triangles can also be classified by their angles 3 cute ngles One Obtuse ngle One Right ngle ll ngles lways, sometimes, never a. right triangle is equiangular: b. n acute triangle is equilateral: c. n obtuse triangle is isosceles: d. right triangle is scalene: The Triangle Sum Theorem ( Thm.) a. The sum of the measures of the angles of a triangle is m 1 m 2 m

3 Proof: Given: BC with angles 1, 2, and 3 Prove: m 1 m 2 m C t 1 2 B Statements Reasons BC with angles 1, 2, and 3 Given Line t is parallel to B m 4 m 1 5. m 4 m 3 m Substitution What if the triangle is a right triangle, what must the two acute angles always add up to? 6. The ratio of the angles of a triangle is 3:6:9. Classify the triangle by its angles.

4 7. Exterior angles of a triangle formed by one of the sides. What angle is supplementary to that angle? 8. Each exterior angle has two interior angles 9. Find the degree measure of the exterior angles below Exterior ngles Theorem: The measure of an exterior angle is equal to 3 1 2

5 Directions: Find the value of each variable. Directions: Find the measure of each numbered angle Closure: Compare and contrast the triangle sum theorem and triangle exterior angle thm.

7 Name: ) BC is an isosceles triangle such that each leg is 4cm less than twice the base. If the perimeter of the triangle is 17cm, find the length of each side of the triangle. ) PQR is an equilateral triangle. One side measures 2x + 5 and another side measures x Find the length of each side. ) The perimeter of BC is 18m. If the second side is three times larger than the first side, and the third side is three more than the first side, what is the measure of each side of the triangle? ) BCD is isosceles with C as the vertex angle. Find x and the measure of each side if BC = 2x + 4, BD = x + 2, and CD = ) HKT is equilateral. Find x and the measure of each side if HK = x + 7 and HT = 4x 8. 6.) BC is isosceles with as the vertex angle. C is five less than two times a number. B is three more than the number. BC is one less than the number. Find the measure of each side. Use the distance formula to classify each triangle by the measures of its sides. 7.) BC with vertices (0, 6), B(3, 6), and C(3,0) 8.) PST with vertices P(4, 0), S(-2, 0), and T(1, 5)

8 Name: Date Block # Corresponding Parts of Congruence & Triangle Congruence Warmup: Determine if each pair of objects is congruent or not. Explain your choice! 1) 2) 3) 4) 5) 6) 7) 8)

9 Reminder: Congruent figures have the same &. a. Each ( matching ) side and angle of congruent figures will also be! B W V C X Example: E D Z Y Congruent ngles Congruent Sides Naming Congruent Figures a) Points can be named in any consecutive order b) Each corresponding vertex must be in the same order for each figure Example #2: Given the fact that BCD EFGH, complete the following. a. Rewrite the congruence statement in a different way. b. Name all congruent angles c. Name all congruent sides

10 This chapter will deal with congruent triangles. Formal Definition: Congruent Triangles a. Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles & sides) of the triangles are congruent. It means that if you put the two triangles on top of each other, they would match up perfectly Triangle congruence works the same as it did for the pentagons, and for all polygons. Given HIJ MNO. Name all congruent sides and all congruent angles. Write the triangles congruent in two other ways. 5. If BC XYZ, and m 3x 12, m B 3x 1 m X x 44, find x. 6. The Third ngle Theorem: If two angles of one triangle are congruent to two angles of another triangle, the third angles are 7. What if we want to prove that polygons are congruent? What do we need to do?! 8. Because triangles only have three sides, we can take some shortcuts

11 9. If all three sides are given, we call this. a. SSS Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. 10. If two sides and the angle BETWEEN those sides are given, we call this. a. SS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. * Included: 1 Using the postulates in proofs Given the figure below, prove: YX YBX X B Y

12 Key concept: ny time a side is shared, always think property!!! Example

13 Given: IE GH, EF HF and F is the midpoint of GI Prove: EFI HFG E I F G H Given:PQ SR ; PQ RS Prove: PQS RSQ

14 Name: Date Block # S &S Triangle congruence Warmup: Define the postulate below. lso, mark the triangles appropriately. ngle Side ngle (S) Postulate: ngle ngle Side (S) Theorem a. If two angles and the non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. Flow chart proof of S Theorem (remember: theorems are proven, postulates are accepted!) Given: X, B Y, and BC YZ Prove: BC XYZ

15 Name: Date Block # HL Right Triangle Congruence We ve already briefly looked at the S theorem. nother method is the HL theorem. a. H stands for. *** Only applies to right triangles! b. L stands for. Quick refresher: the hypotenuse is the side the right angle. Both of the other two sides are called legs. The HL Theorem a. If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. E M D F L N b. Write a congruence statement: * Review! c. Identify six congruent parts in the space below. For what values of x and y can the triangles be proven congruent by the HL theorem. x x +3 3y y +1

16 5. Determine another side or angle that would have to be congruent to use the B a. HL: b. SS: D C Q 6. Given: m O m P 90, MN QR, OM PQ O R Prove: MOR QPN Statements M N Reasons P Closure: nswer the following. 1) Compare and contrast the HL theorem to the S theorem 2) When do you use the SS postulate for a right triangle?

17 Name: Date Block # Ways to Prove Congruent Triangles SSS SS S S HL Triangle Classifications that do NOT prove Congruent Triangles SS Understanding the term INCLUDED for Triangles Given 2 Sides and 1 ngle Given 2 ngles and 1 Side

19 With a partner, describe the process for proving that two triangles are congruent. Be thorough in your explanation. Imagine that you were explaining it to someone that was absent the last couple of days Use drawings, labels, or whatever you need in your explanation.

20 Name: Block: Complete each proof Given: B DC BC D Prove: ΔBC Proof: Δ CD Statement 2 B 1 3 C 4 D Reasons Triangle Proofs R Given: l m, PT QS, Q is the midpoint of PR Prove:Δ PQT Δ QRS Proof Statement Reasons P Q 2 1 T S l m

21 Y X Given: YZ bisects WX at P. WX bisects YZ at P. WY ZX P Prove: ΔWYP ΔXZP Statement Reasons W Z Given: X XY, BY XY, Z is the midpoint of XY, 1 2 Prove: ΔXZ ΔBYZ 1 2 B 3 4 Statement Reasons X Z Y

22 D Given: M = BM, Cm = DM Prove: ΔCM ΔBDM M C B Statement Reasons S R T Given: RS= RQ; ST = QT Prove: ΔRST ΔRQT Statement Reasons Q P R Given: M R, O is the midpoint of MR Prove: ΔMON ΔROP Statement Reasons O M N 5. 5.

23 C Given: C BC, m is the midpoint Prove: ΔCM ΔBCM Statement Reasons B M D B 2 4 C Given: B is the midpoint of C, 3 & 4 are right angles Prove: ΔBE Δ CBD E Statement Reasons

24 Name: Date Block # Corresponding Parts of Congruence & Triangle Congruence Complete the proof below. Given: C CE, BC CD Prove: CB ECD B C D E Now that you know CB ECD (hopefully you proved it!), list all of the congruent sides and congruent angles. Think back to the first day of congruence if it will help you! Look back at the triangles at the top of the page. What if the problem asked us to prove B DE instead of asking us to prove CB ECD? How could we do that? Well, we could use the same strategy as we used for #2! 5. Once the triangles themselves are congruent, then all parts of the triangles are also congruent! a. We can use this as a in our proofs. We call it

25 B 6. Example using CPCTC Given: B BC & BD is the angle bisector of BC Prove: D CD D C Game plan: First, prove that BD CBD (we can use SSS, SS, S, or S) Next, conclude that D BC using CPCTC! Statements Reasons B BC Given Definition of an bisector 5. BD CBD D CD 7. Challenge: What does CPCTC stand for? a. Say it 3 times fast 8. Wrap Up: Explain what CPCTC means. When do you use it? Why is it helpful!?

26 Name: Date Block # Corresponding Parts of Congruence & Triangle Congruence Definition: triangles that has congruent sides. Vocabulary Terms The Isosceles Triangle Theorem: a. If two sides of a triangle are congruent, then the angles opposite those sides (the base angles) are also congruent (proof on page 229). The Converse of the Isosceles Triangle Theorem a. If

27 More Examples: Example #1: Solve for x. The figure shown is a regular hexagon. x Example #2: Triangle BC is isosceles with base C. m 2x 8 m B 4x 20 What type of triangle is BC, acute, obtuse, right, or equiangular? Closure: Compare and contrast isosceles triangles with equilateral & equiangular triangles.

28 Name: Date Block # Using CPCTC for Overlapping Triangles Given: B DE & BD and DE are rt 's Prove: DB DE D STTEMENTS RESONS B E Sometimes it is easier to prove triangles are congruent when you separate them D D Overlapping Triangles will sometimes have shared sides or angles. B E Make sure to state that they are congruent using the property Examples:

29 Formal Proof: Redraw the triangles, add arc and tick marks and complete the proof. Given: BD CDB and CBD DB Prove: B CD E C B D

30 a. Outline Δ SWU. b. outline Δ MIU with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. U a. Outline Δ MUW. b. outline Δ SUI with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. U S W I M S W I M Which part appear to be corresponding? Complete the statement below sw and S SU and SUW and Which part appear to be corresponding? Complete the statement below si and S UW and SWU and SU and SIU and IU and SUI and 1B) Outline the Δ s again!! U 2B) Outline the Δ s again!! U S W I M If ΔUWI is isosceles, what pair(s) of corresponding parts are congruent? Why? S W I M If ΔUWI is isosceles, what pair(s) of corresponding parts are congruent? Why?

31 Outline the Δ s again!! U Outline the Δ s again!! U S W I M S W I M If ΔSUM is isosceles, what pair(s) of corresponding parts are congruent? Why? If ΔSUM is isosceles, what pair(s) of corresponding parts are congruent? Why? a. Outline Δ BO. b. outline Δ ST with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. O T a. Outline Δ BSO. b. outline Δ SBT with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. O T B S B S Which part appear to be corresponding? Complete the statement below BO and O Which part appear to be corresponding? Complete the statement below BS and O B and OB and BO and OBS and O and OB and SO and BSO and If ΔBS is isosceles, what pair(s) of corresponding parts are congruent? Why? If ΔBS is isosceles, what pair(s) of corresponding parts are congruent? Why?

32 5. a. Outline Δ HOS. b. outline Δ DNS with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. U 6. a. Outline Δ HUN. b. outline Δ DUO with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. U O N O N H D H D Which part appear to be corresponding? Complete the statement below HO and H Which part appear to be corresponding? Complete the statement below HU and U HS and HOS and UN and H and OS and OSH and HN and HNU and Extra Practice Given:HL HR, H HD H Prove: L R D L R Given: LD ID, I IN Prove: 1 2 L I N D

33 Given: Fm FN, Dm HN, EF GF, DE HG E G Prove: ΔDEN ΔHGM F D M N H Given: WY XZ WZ ZY, XY ZY Prove ΔWYZ ΔXZY W X Z Y Closure: Confidence Meter for the upcoming test? Name the good and the bad

Unit 3 Syllabus: Congruent Triangles

Date Period Unit 3 Syllabus: Congruent Triangles Day Topic 1 4.1 Congruent Figures 4.2 Triangle Congruence SSS and SAS 2 4.3 Triangle Congruence ASA and AAS 3 4.4 Using Congruent Triangles CPCTC 4 Quiz