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Transcript
904 L Geometry 904 Date: Name: Geometry 904 Table of Contents Objectives ................................................................................................................................. 2 Vocabulary: ............................................................................................................................... 2 Character Trait: ......................................................................................................................... 2 Chapter 1: Proving Triangles are Congruent ............................................................................ 3 Section 1: Corresponding Parts of Triangles. ....................................................................... 3 Section 2: Congruent Triangles ............................................................................................ 5 Section 3: Postulates about Angles and Sides of Congruent Triangles ................................ 6 Chapter 1 Review ................................................................................................................ 12 Chapter 2: More Methods of Proving Congruence ................................................................. 14 Section 1: Overlapping Triangles ....................................................................................... 14 Section 2: AAS and HY-LEG ............................................................................................. 16 Chapter 2 Review .................................................................Error! Bookmark not defined. Character Lesson:.....................................................................Error! Bookmark not defined. Chapter 3: Applications of Congruent Triangles .....................Error! Bookmark not defined. Section 1: Proving Segments and Angles are Congruent ....Error! Bookmark not defined. Section 2: Proving Bisectors and Perpendicular Lines with Congruent Triangles. ..... Error! Bookmark not defined. Chapter 3 Review .................................................................Error! Bookmark not defined. Chapter 4: Area .......................................................................Error! Bookmark not defined. Section 1: Areas of Polygons ...............................................Error! Bookmark not defined. Chapter 4 Review .................................................................Error! Bookmark not defined. Unit Review .............................................................................Error! Bookmark not defined. Appendix A: Theorem reference list.......................................Error! Bookmark not defined. Appendix B: .............................................................................Error! Bookmark not defined. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 1 Objectives Upon the successful completion of this unit you will be able to: Explain what is required for triangles to be congruent. Use shortcut methods for proving triangles are congruent. Prove overlapping triangles are congruent Use AAS and Hy-Leg methods of proving congruence of triangles Use congruent triangles to prove segments and angles are congruent. Use congruent triangles to prove special properties of lines. Find the areas of triangles, trapezoids, polygons. Define and give examples of TACTFUL Vocabulary: Correspondence Agreement between particular things. SSS Side-Side-Side method of proving congruence when three sides of one triangle are congruent to the corresponding sides of another triangle. SAS Side-Angle-Side method of proving congruence when two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle. ASA Angle-Side-Angle method of proving congruence when two angles and the included side are congruent to the corresponding parts of another triangle. AAS Angle-Angle-Side method of proving congruence when two angles and an opposite side are congruent to the corresponding parts of another triangle. Hy-Leg Hypotenuse-Leg method of proving the congruence of right triangles when the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another right triangle. Included Side The side of a triangle between two of the vertices. Included Angle An angle of a triangle that is between two of the sides. Character Trait: Tactful: Choosing to be kind with my words. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 2 Chapter 1: Proving Triangles are Congruent Section 1: Corresponding Parts of Triangles. Correspondence is visible in many of the everyday things in the world about us. You can see correspondence between two cars on the road. Each car may be a different make, model, color, and age. Yet, each will have four wheels, a windshield and an engine. The agreement between the parts of cars indicates correspondence. Examine a typical human hand, and you will see five fingers. Examine another, and you will see five fingers. The fingers on one typical human hand correspond to the fingers on another. Y X Corresponding parts can be used for comparison. B The parts of two triangles can be ‘paired’ to establish corresponding parts. Paired of corresponding vertices are seen below. A C The diagram shows the following correspondence: vertex X corresponds to vertex A vertex Y corresponds to vertex B vertex Z corresponds to vertex C Z Pairing Corresponding Vertices Relationships of correspondence can be written with mathematical symbols. The symbol is read “corresponds to.” AX BY CZ The order of the vertices in the triangle name is important in establishing a correspondence between triangles. We can note the correspondence between triangles as follows: ΔABC ΔXYZ (triangle ABC corresponds to triangle XYZ) The first vertex listed in the first triangle corresponds to the first vertex listed in the second triangle. Examine this statement. ΔDEF ΔLMN (triangle DEF corresponds to triangle LMN) There are three pairs of corresponding vertices. You must list the corresponding vertices in the same order that they are listed in the statement of triangle correspondence. DL ΔDEF EM Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 3 FN Example 1: Determine the corresponding vertices when ΔGHI ΔRST You know that both triangles correspond and that the corresponding vertices are written in the same order. From the statement of triangle correspondence you can identify three pairs of corresponding vertices. You can write each pair as a corresponding vertices statement. Your answer has three statements: GR HS I T Corresponding vertices determine the corresponding sides of a triangle. You can use logic to deduce these relationships. AX BY It follows that and are corresponding line segments. Notice how this is shown in the diagram below. We have used capital letters to indicate points, lines, and line segments while lowercase letters have been used to indicate interior angles. We will now use capital letters to indicate vertices as well as the angle that occurs inside the vertex of a closed figure. Lowercase letters will still be used to indicate angles that occur with intersecting lines and line segments. Y X B A C Z Pairing Corresponding Sides Examine the corresponding vertices C and Z. Because the vertices correspond, we can say that C Z. Notice that the sides opposite the corresponding angles are corresponding sides. Given that ΔABC ΔXYZ , let’s review the corresponding parts between these two triangles. Angle Correspondence A X B Y C Z Side Correspondence Y B A X Y C Z Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 4 Establishing angle and side correspondence does not establish that sides and angles congruency. Correspondence is used to compare two figures and their parts. We will examine the conditions necessary for congruency in the next section. Given, the conditions below, identify the corresponding parts of ΔABC ΔUVW: W A U 1. AU BC _______ 2. ΔABC ΔUVW AB _______ 3. ΔABC ΔUVW BC _______ 4. T OR F B V C ΔABC ΔUVW and ΔBAC ΔVUW 5. T OR F ΔABC ΔUVW and ΔCAB ΔWUV Score the above questions. Correct wrong answers. Rescore. Teacher’s Initials:_____ Section 2: Congruent Triangles Congruent Parts: To establish that triangles are congruent requires more than correspondence. Congruence requires that the triangles meet the following definition: Two triangles are congruent when all corresponding parts are congruent. Congruence of corresponding sides and angles establish congruent triangles. Every triangle has three sides. In order for two triangles to be congruent, all three pairs of corresponding sides must be congruent. Every triangle has three angles. In order for two triangles to be congruent, all three pairs of corresponding angles must be congruent. Six points of agreement (3 sides + 3 angles) must be congruent in order to know that the triangles are congruent. C If: XA ≅ X ≅ A ≅ Y ≅ B ≅ Z ≅ C 54° X 51° 75° 51° 75° ≅ Y B 54° Copyright 2002, Starline Press, Inc. www.starlinepress.com Z Page 5 A Then: ΔABC ≅ ΔXYZ Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 6 Included Parts: In any triangle such as ΔQRS at the right, the terms ‘included side’ and ‘included angle’ are commonly used to describe the parts. An angle is included between two sides. Between R and S is the included side . You can also have an angle included between two sides. R is an included angle between and . R Q Circle the correct answer using Figure 1: 1. True False M ΔDEF ≅ ΔKLM ≅ D ≅ K ≅ E ≅ L ≅ F ≅ M S F D L Figure 1 K E Using Figure 2, answer the following questions: A 2. The included side of A and Figure 2 3. The included angle of side AC Score the above questions. 4. Side BC is the included 5. Side AC is the included Correct wrong answers. C Rescore. B Teacher’s Initials:_____ Section 3: Postulates about Angles and We will begin examining relationships between Investigation 1: Follow the steps below. 1. On a separate sheet of paper use a ruler, to draw a triangle. Note the measurements of each side. 2. Draw another triangle, making the sides the same length as the sides on the first triangle. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 7 Cut out the second triangle with scissors. 3. Lay the triangle you cut out on top of the first triangle you drew. (You may need to rotate it.) The two triangles should match perfectly both in the lengths of the sides and the measurements of the angles. 1 2 Teacher’s Show your work to your teacher. Initials:_____ You did not need to measure the angles in order to create congruent triangles. The angles are congruent because all of the sides of the triangles are congruent. This is the basis of the Side-Side-Side postulate: This is postulate is often referred to by the letters SSS Postulate. Side-Side-Side Postulate If the vertices of one triangle can be paired with the vertices of another triangle so that the sides of the first triangle are congruent to the corresponding sides of the second triangle, then the two triangles are congruent. This postulate is helpful in proving that triangles Using this postulate only the side congruency M F D L Figure 1 E Copyright 2002, Starline Press, Inc. www.starlinepress.com K Page 8 Other relationships between corresponding parts of triangles can be used to prove that two triangles are congruent. Examine the following postulate. This is known as the Angle-Side-Angle (ASA) Postulate. Angle-Side-Angle Postulate If the vertices of one triangle can be paired with the vertices of another triangle so that two angles and the included side of the first triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent. The arc marks on figure 2 show: DK EL F D The hash marks show: . M L Figure 2 E K Using this information it can be determined DEF KLM Side-Angle-Side Postulate If the vertices of one triangle can be paired with the vertices of another triangle such that two sides and the included angle of one triangle are congruent to the corresponding parts of a second triangle, then the two triangles are congruent. This is known as the Side-Angle-Side (SAS) Postulate Having two sides and the included angle of the first triangle congruent to the corresponding parts of the second triangle also proves that the triangles are congruent. M F D L E Copyright 2002, Starline Press, Inc. www.starlinepress.com K Page 9 Example 1: Construct a proof. R Given: Figure 3 Figure 3 Prove: ΔRST ≅ ΔRPT S P SOLUTION: These two triangles share a common side, . The hash marks indicate that . There are two sides and an included angle that are congruent. T This allows us to use the SAS Postulate to prove the triangles are congruent. PROOF Statements Reasons 1. ΔRTS ≅ ΔRTP 1. GIVEN ≅ ≅ 2. GIVEN 3. Reflexive Property 4. ΔRST ≅ ΔRPT 4. SAS Postulate 2. 3. Figure 4 Example 2: Construct a Proof Given: A Figure 4 AB ∥ DC ADX BCY X Prove: A B T D U X B Y D Figure 5 Y C Copyright 2002, Starline Press, Inc. www.starlinepress.com C First, create two triangles by drawing line segments perpendicular to the parallel lines and through the vertices D and C forming right triangles (as shown in figure 5). Label the points where the perpendicular line segments cross as T and U. Page 10 The segments you added create two triangles Δ ADT and ΔBCU that have corresponding parts for comparison. Each triangle includes a segment that you want to prove is congruent to the other. ΔADT contains AD, and ΔBCU contains BC. This proof will use the definition of complementary angles and the definition of parallel lines to show that two angles and an included side are congruent. The ASA Postulate can then be used to prove the triangles are congruent. When the triangles are proven to be congruent, then the corresponding sides, AD and BC will be proven as congruent. PROOF Statements Reasons 1. GIVEN 2. Definition of complementary angles 3. Transitive Property 4. Substitution 5. Subtraction Property of Equality 6. Definition of Parallel Lines 7. m ATD = 90° = m∠BUC 7. Definition of Perpendicular Lines 8. ATD ≅ BUC 8. Transitive Property 9. Δ ATD ≅ Δ BUC 9. ASA Postulate 1. ADX ≅ ∠BCY 2. mADX + mADT= 90° mBCY + mBCU = 90° 3. mADX + mADT = mBCY + mBCU 4. mBCY + mADT= m∠BCY + m∠BCU 5. ADT ≅ BCU 6. 10. Copyright 2002, Starline Press, Inc. www.starlinepress.com 10. Corresponding parts of congruent triangles Page 11 Answer the following questions: 1. Is ΔA ΔB? Y N B Explain Your answer: 2. Is ΔA ΔB? A Y N 2.5 Explain Your answer: 2.5 1.5 A 1.5 B 2 2 3. Is ΔX ΔY? Y N Explain Your answer: X 55° 55° 55° 55° Y 4. Is Δ1 Δ2 ? Y N (Given: This is a parallelogram) 2 Explain Your answer: 1 Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 12 5. Is Δ4 Δ3 ? Y N 4 (Given: This is a regular pentagon) Explain Your answer: 6. 3 5 Are Δ5, Δ6, Δ7, & Δ8 congruent triangles? Y 6 N (Given: This is a regular Octagon) Explain Your answer: 8 7 7. What is the shape in the middle of the octagon in problem 6? ___________ How do you know? ____________________________________________ Score the above questions. Correct wrong answers. Rescore. Teacher’s Initials:_____ Chapter 1 Review Identify the corresponding parts using figure 1: 1. ∠A ∠U _______ 2. ΔABC ΔUVW _______ 3. ΔABC ΔUVW _______ 4. T or F ΔABC ΔUYW and ΔBAC ΔYUW 5. T or F ΔABC ΔUYW and ΔCAB ΔWUY Y U B A C Copyright 2002, Starline Press, Inc. www.starlinepress.com W Figure 1 Page 13 Circle the correct answer: 6. True False M ΔDEF ≅ ΔKLM ≅ D ≅ K ≅ E ≅ L ≅ F ≅ M F L E K E Using this triangle answer the following questions: A B C 7. Side BC is the included side of B and C. True False 8. Side AC is the included side of A and B. True False Fill in the missing statements in the following proof: 9. Complete the proof: R Given: ST PT Prove: ΔRST ≅ ΔRPT S P SOLUTION: Because both these T triangles share a common side, , then there are two sides and an included angle that are congruent. This allows us to use the SAS Postulate to prove the triangles are congruent. PROOF Statements Reasons 1. ______________________ 2. GIVEN 3. RT RT 3. Reflexive Property 4. ΔRST ≅ ΔRPT 4. _______________________ 1. RTS ≅ RTP 2. ST PT Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 14 Is ΔABC ΔADE ? 10. Y N B (Given: This is a regular pentagon) A C Explain Your answer: 2 D E Score the above questions. Correct wrong answers. Rescore. Teacher’s Initials:_____ Chapter 2: More Methods of Proving Congruence Section 1: Overlapping Triangles It is not always easy to distinguish between triangles in a complex figure. Overlapping triangles make it tricky to identifying corresponding sides and angles. Examine figure 1 below. Can you see the five triangles? How about the twelve angles? Remember relationships: These sides are the same side in this example! You can use different strategies to identify corresponding triangles, sides, and angles. Separate and reposition the triangles by redrawing them to help keep them organized: Use colored pencils or pens to outline triangles. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 15 Marking Triangles with colors: Redraw Triangles REMEMBER Same Angles: Mark one pair of corresponding triangles in each figure using colored pens or pencils: 1. 2. 3. 4. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 16 5. 6. Score the above questions. Correct wrong answers. Rescore. Teacher’s Initials:_____ Section 2: AAS and HY-LEG AAS Theorem: In the following triangles, ΔFGH & ΔCDE, there are two pairs of corresponding angles and one pair of corresponding sides that are congruent. The corresponding sides are not included sides. Can we determine if the triangles are congruent? The postulates we’ve learned about in this unit do not fit this example. We cannot use the SSS Postulate, ASA Postulate, or the SAS Postulate to establish that the triangles are congruent. G D Figure 1 F H C E We can use a corollary. Corresponding and Congruent Angles Corollary If two angles of a triangle are congruent to two angles of another triangle then the remaining angles are congruent. For Figure 1 we are can examine the arc marks and hash marks to state the givens: F C GD By applying the corollary we determine that E and H are congruent. Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 17 Angle-Angle-Side (AAS) Theorem If two triangles have two corresponding angles that are congruent and the side opposite one of the angles in the first triangle is congruent to a corresponding side in the second triangle then the two triangles are congruent. Example 1: Construct a Proof Given: A D • ABE ≅ DEB • ≅ EDF ≅ CAB •F Prove: ∆DEF ≅ ∆ABC E •C B SOLUTION: Since ABC and DEF are supplementary to congruent angles, they are congruent. With two angles congruent and an opposite side congruent, the AAS Theorem can be applied to prove the triangles are congruent. PROOF Statements 1. ABE ≅ DEB Reasons 1. Given 2. Given 3. EDF ≅ CAB 3. Given 4. ABC & ABE are supplementary 4. Definition of supplementary angles 5. DEF & DEB are supplementary 5. Definition of supplementary angles 6. ABC ≅ DEF 6. 7. ∆DEF ≅ ∆ABC 7. Supplementary angles of congruent angles are congruent. AAS Theorem 2. ≅ Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 18 Copyright 2002, Starline Press, Inc. www.starlinepress.com Page 19