Section 4-1 Congruent Figures. Objectives: recognize congruent figures and their corresponding parts
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1 Section 4-1 Congruent Figures Objectives: recognize congruent figures and their corresponding parts
2 Congruent Polygons Congruent Polygons have congruent corresponding parts Congruent sides Congruent Angles Matching vertices are corresponding vertices always list corresponding parts in the same order Line AB line DE Line BC line EF Line CA line FD A D B E C F B D E A C F
3 Example ABC corresponding parts. QTJ. List the congruent List the corresponding vertices in the same order. Angles: A Q B T C J List the corresponding sides in the same order. Sides: AB QT BC TJ AC QJ
4 The fins of the space shuttle suggest congruent pentagons. Find the m B. What is the angle corresponding to B? Use the Polygon Angle Sum Thm: (n 2)180 = (5 2)180 = 540
5 XYZ KLM, m Y = 67, and m M = 48. Find m X. Use the Triangle Angle-Sum Theorem and the definition of congruent polygons to find m X. m X + m Y + m Z = 180 m Z = m M m Z = 48 m X = 180 m X = 180 m X = 65 Triangle Angle-Sum Theorem Corresponding angles of congruent triangles that are congruent Substitute 48 for m M. Substitute. Simplify. Subtract 115 from each side.
6 Are the Two Triangles Congruent? Two triangles are congruent if they have: 3 pairs of corresponding sides 3 pairs of corresponding angles Look at the diagram. What do you know by the markings? AC EC Given AB ED AB 3 ED BC DC BC 4 DC A E Given B D Right s are What is true about angles ACB and DCE? Explain. They are vertical angles, so they are congruent.
7 Proving Triangle Congruency Use the information in the diagram to give a reason why each statement is true. Statement PQ PS, QR SR PR PR Q S, QPR QRP SRP PQR PSR SPR Reason Given Reflexive Prop of Given Theorem 4-1 Def of triangles
8 Examples Given: CG DG, CN DN, C D, CNG and DNG are right angles. Prove: CNG DNG. Congruent triangles have three congruent corresponding sides and three congruent corresponding angles. Examine the diagram, and list the congruent corresponding parts for CNG and DNG. a. CG DG Given b. CN DN Given c. GN GN Reflexive Property of Congruence d. C D Given e. CNG DNG Right angles are congruent. f. CGN DGN If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. (Theorem 4-1.) g. CNG DNG Definition of triangles
9 Practice! Pg # 1-31 odd #44 turn in for 5 extra credit points!!
10 Proving Triangles Congruent: SSS and SAS Section 4-2
11 Side-Side-Side (SSS) Congruence Postulate If 3 sides of one triangle are congruent to 3 sides of another triangle, then the two triangles are congruent. If all 3 sides match, then ABC DEF
12 Side-Angle-Side (SAS) Congruence Postulate If 2 sides of one triangle and the angle they form are congruent to 2 sides of another triangle and the angle they form, then the triangles are congruent.
13 Decide if the triangles are congruent. If so, state the congruence postulate that applies. CEA and DEB are vertical angles. CE DE; AE BE CEA and DEB are congruent! SAS
14 Decide if the triangles are congruent. If so, state the congruence postulate that applies. MN LN is shared by both triangles. LN LN PL; NP congruent! LM LMN and NPL are SSS
15 Decide if the triangles are congruent. If so, state the congruence postulate that applies. NO!!!
16 Pg #1-39 odd Assignment
17 Section 4-3 Triangle congruence by ASA and AAS
18 Postulate 4-3: Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. C B E A F D
19 Theorem 4-2: Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the triangles are congruent. C B D E A F
20 Theorem 4-2: Angle-Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF Prove: ABC DEF B A E C F D
21 Theorem 4-2: Angle-Angle-Side (AAS) Congruence Theorem You are given that two angles of ABC are congruent to two angles of DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ABC DEF. C B D E A F
22 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. E F G J H
23 Ex. 1 Developing Proof A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that EFG JHG. E F G J H
24 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. M N P Q
25 Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. M N P Q
26 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ WX AND UW W 3 4 U 1 2 Z WX. X
27 Ex. 1 Developing Proof The two pairs of parallel sides can be used to show 1 3 and 2 4. Because the included side WZ is congruent to itself, WUZ ZXW by the ASA Congruence Postulate. W 3 4 U 1 2 Z X
28 Ex. 2 Proving Triangles are Congruent Given: AD EC, BD BC Prove: ABD EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC A B C. Use the fact that AD EC to identify a pair of congruent angles. D E
29 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC Reasons: 1. E
30 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC E Reasons: 1. Given
31 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC E Reasons: 1. Given 2. Given
32 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles
33 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem
34 A C Proof: B D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC E Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem 5. ASA Congruence Theorem
35 Note: You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D C and A E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.
36 Practice Pg # 1-41 odd
37 Using congruent Triangles: CPCTC Section 4-4
38 Match the Corresponding Parts A B C AB BC AC DF DE D EF E F
39 C.P.C.T.C. Corresponding Parts (of) Congruent Triangles (are) Congruent If two triangles are congruent, then their corresponding parts are also congruent..
40 Very important!!!! Before you use CPCTC you must prove or know that the two triangles congruent!!!
41 Using CPCTC LJK LMK JK MK, JL ML, KL KL JLK MLK, LJK LMK, J M
42 Using CPCTC ABC DEF With your partner write down all congruent parts. A D, B E, C F AB DE, BC EF, AC EF
43 Find the value of all the angles ΔJKL ΔMKL J = 3x + 2 JLK = 90 M = 5x - 32 J = JKL = JLK = MLK = MKL = M =
44 CPCTC in a Proof: T S Given: TI SN TN SI E Prove: T S I N
45 CPCTC in Proofs: A J M R Given: JM RM AM MI Prove: AJ RI I
46 Write a proof. Given: LM NP; MN PL Prove: LMN NPL STATEMENTS REASONS 1. LM NP 1. Given 2. MN PL 2. Given LN LN LMN NPL Reflexive Prop. SSS Congruence Post.
47 Write a proof. Given: Prove: Z is the midpoint of WY. XZY 90º WXZ YXZ Z is of WY. Given the midpoint WZ ZY Def. of midpt. XZY Given 90º XZW 90º Linear pair WXZ YXZ SAS Cong. Post. XZ Reflex. Prop. XZ
48 Assignment Exercises Pg #1-27 odd
49 Isosceles and Equilateral Triangles Section 4-5
50 Vocabulary legs of an isosceles triangle vertex angle base base angles
51 Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 3 is the vertex angle. 1 and 2 are the base angles.
52
53 Reading Math The Isosceles Triangle Theorem is sometimes stated as Base angles of an isosceles triangle are congruent.
54 Example 1: Astronomy Application The length of YX is 20 feet. Explain why the length of YZ is the same. The m YZX = , so m YZX = 40. Since YZX X, XYZ is isosceles by the Converse of the Isosceles Triangle Theorem. Thus YZ = YX = 20 ft.
55 Example 1 If the distance from Earth to a star in September is km, what is the distance from Earth to the star in March? Explain ; since there are 6 months between September and March, the angle measures will be approximately the same between Earth and the star. By the Converse of the Isosceles Triangle Theorem, the triangles created are isosceles, and the distance is the same.
56 Example 2A: Finding the Measure of an Angle Find m F. m F = m D = x Isosc. Thm. m F + m D + m A = 180 x + x + 22 = 180 2x = 158 Sum Thm. Substitute the given values. Simplify and subtract 22 from both sides. x = 79 Divide both sides by 2. Thus m F = 79
57 Example 2B: Finding the Measure of an Angle Find m G. m J = m G Isosc. Thm. (x + 44) = 3x 44 = 2x Substitute the given values. Simplify x from both sides. x = 22 Thus m G = = 66. Divide both sides by 2.
58 Example 2A Find m H. m H = m G = x Isosc. Thm. m H + m G + m F = 180 x + x + 48 = 180 2x = 132 Sum Thm. Substitute the given values. Simplify and subtract 48 from both sides. x = 66 Divide both sides by 2. Thus m H = 66
59 Check It Out! Example 2B Find m N. m P = m N Isosc. Thm. (8y 16) = 6y Substitute the given values. 2y = 16 Subtract 6y and add 16 to both sides. y = 8 Divide both sides by 2. Thus m N = 6(8) = 48.
60 The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.
61
62 Example 3A: Using Properties of Equilateral Triangles Find the value of x. LKM is equilateral. Equilateral equiangular (2x + 32) = 60 The measure of each of an equiangular is 60. 2x = 28 Subtract 32 both sides. x = 14 Divide both sides by 2.
63 Example 3B: Using Properties of Equilateral Triangles Find the value of y. NPO is equiangular. Equiangular equilateral 5y 6 = 4y + 12 Definition of equilateral. y = 18 Subtract 4y and add 6 to both sides.
64 Check It Out! Example 3 Find the value of JL. JKL is equiangular. Equiangular equilateral 4t 8 = 2t + 1 Definition of equilateral. 2t = 9 Subtract 4y and add 6 to both sides. t = 4.5 Divide both sides by 2. Thus JL = 2(4.5) + 1 = 10.
65 Remember! A coordinate proof may be easier if you place one side of the triangle along the x- axis and locate a vertex at the origin or on the y-axis.
66 Example 4: Using Coordinate Proof Prove that the segment joining the midpoints of two sides of an isosceles triangle is half the base. Given: In isosceles ABC, X is the mdpt. of AB, and Y is the mdpt. of AC. Prove: XY = AC. 1 2
67 Example 4 Continued Proof: Draw a diagram and place the coordinates as shown. By the Midpoint Formula, the coordinates of X are (a, b), and Y are (3a, b). By the Distance Formula, XY = 4a 2 = 2a, and AC = 4a. Therefore XY = AC. 1 2
68 Example 4 What if...? The coordinates of isosceles ABC are A(0, 2b), B(-2a, 0), and C(2a, 0). X is the midpoint of AB, and Y is the midpoint of AC. Prove XYZ is isosceles. Proof: Draw a diagram and place the coordinates as shown. A(0, 2b) y X Y Z x B( 2a, 0) C(2a, 0)
69 Example 4 Continued By the Midpoint Formula, the coordinates. of X are ( a, b), the coordinates. of Y are (a, b), and the coordinates of Z are (0, 0). By the Distance Formula, XZ = YZ = a 2 +b 2. So XZ YZ and XYZ is isosceles. y A(0, 2b) X Y Z x B( 2a, 0) C(2a, 0)
70 Lesson Quiz: Part I Find each angle measure. 1. m R 2. m P Find each value. 3. x y 5. x 26
71 Lesson Quiz: Part II 6. The vertex angle of an isosceles triangle measures (a + 15), and one of the base angles measures 7a. Find a and each angle measure. a = 11; 26 ; 77 ; 77
72 Warm Up 1. Find each angle measure. 60 ; 60 ; 60 True or False. If false explain. 2. Every equilateral triangle is isosceles. True 3. Every isosceles triangle is equilateral. False; an isosceles triangle can have only two congruent sides.
73 Congruence in Right Triangles Section 4-6
74 Right Triangles Hypotenuse: the longest side of a right triangle Legs: The sides of a right triangle that are not the hypotenuse
75 Theorem Theorem 4.6: Hypotenuse-Leg (HL) Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. To use the HL Theorem, you must show that three conditions are met: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs
76 Using the HL Theorem
77 Using the HL Theorem 1. Statements Reasons AD is the bisector of CE, 1. CD EA Defn. of 3. CBD & EBA are rt. s CB EB HL Thm.
78 Using the HL Theorem Statements Reasons 1. PRS and SP QR RPQ are rt s Defn. of rt s Refl. Prop.of 4. PRS RPQ 4.
79 Practice! Pg #1-23 odd
80
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