Analytic Geometry Unit 1: Similarity, Congruence & Proofs Lesson 6. Name Date Period

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Name Date Period Topic: 2-Column Proofs and Rigid Motion Class Website: msgiwa1.weebly.com Transformations seen in Coordinate Algebra Translation: A translation is sometimes called a slide.

Reflection: A reflection creates a mirror image of the original figure over a reflection line. Rotation: A rotation moves all points of a figure along a circular arc about a point.

Looking at the triangles in the above examples, does the size or shape ever change after performing each transformation? Each of these transformations is known as a rigid motion, or isometry.

1 Name Date Period Topic: 2-Column Proofs and Rigid Motion Class Website: msgiwa1.weebly.com Transformations seen in Coordinate Algebra Translation: A translation is sometimes called a slide. In a translation, the figure is moved horizontally and/or vertically. For example, triangle ABC is translated by 2 units to the right. Reflection: A reflection creates a mirror image of the original figure over a reflection line. Rotation: A rotation moves all points of a figure along a circular arc about a point. Rotations are sometimes called turns. For example, triangle ABC is rotated about O through 90º in an anticlockwise direction. Looking at the triangles in the above examples, does the size or shape ever change after performing each transformation? Each of these transformations is known as a rigid motion, or isometry. A rigid motion is a transformation done to a figure that maintains the figure s shape and size or its segment lengths and angle measures. Hence, congruent triangles are formed. Next week, we will look at dilations and similar triangles where the shape stays the same, but the size changes. These are called non-rigid motions. Problem 1: DA B C A ' B ' C ' is the image of DABC ABC. Write the translation rule. a) b)

Problem 2: Find the equation of the line of

Problem 3: Identify the type transformation(s) that

Then, determine if it is a rigid transformation,

a) b) Problem 4: A truss is a structure used in

The bridge truss pictured below is made up of 5

2 Problem 2: Find the equation of the line of reflection between the pre-image and the image. a. b. c. Problem 3: Identify the type transformation(s) that have taken place. Then, determine if it is a rigid transformation, meaning the 2 triangles are congruent. a) b) Problem 4: A truss is a structure used in building bridges. The bridge truss pictured below is made up of 5 triangles. Describe the transformations that have taken place, and determine whether the triangles are congruent in terms of rigid and non-rigid motions.

3 Matching: Use the choices listed at the bottom in the box for problems #1 4 Problem 1: 1. LM LO MN ON LN LN ΔLMN Δ LON 4. Problem 2: Problem 3: Problem 4: 1. QS! RT R S QT QT ΔQST Δ TRQ GI KI HI JI GIH KIJ ΔGIH Δ KIJ AC PBD, AB P CD , AD AD ΔADC Δ DAB 4.

4 Fill in the blank proofs: Problem 5: 1. I K IHJ KJH HJ HJ ΔHJK Δ JHI 4. Problem 6: 1. MLN ONL OLN Reflexive Property 4. ΔLNO Δ NLM 4. Problem 7: 1. PQ QS PQT RQS ΔPQT Δ SQR 4. Problem 8: 1. UV UX Right Angle Congruence Reflexive Property 5. ΔUWV Δ UWX 5. Problem 9: 1. Y C Vertical Angles 4. ΔYZA Δ CBA 4. Problem 10:

5 1. BAC DCA ΔABC Δ CDA 4. Problem 11: 1. F I ΔEFG Δ HIJ 4. Problem 12: 1. M KLO ΔKLO Δ NLM K N 5. CPCTC Problem 13: 1. P Reflexive 4. ΔPQS Δ RSQ 4. Problem 14: 1. AC P BD CAD BDA Reflexive Property 5. ΔACD Δ 5.

6 Problem 1: : WX YX, Z is the midpoint of WY Prove: WXZ YXZ Place the following items in as one of the reasons below: Definition of a midpoint Reflexive Property SSS Congruence Postulate s WX YX Z is the midpoint of WY WZ ZY XZ XZ 5) VWXZ V YXZ s Problem 2: : B is the midpoint of AE, B is the midpoint of CD Prove: VABD V EBC Place the following items in as one of the statements or reasons below: AB EB BD BC Vertical Angles are Congruent SAS Congruence Postulate s s B is the midpoint of AE Definition of Midpoint B is the midpoint of CD Definition of Midpoint 5) ABD EBC 6) VABD V EBC

7 Other Types of Proofs Proofs with Angles: Problem 1: : 1 and 2 are supplementary. 2 3 Prove: = 180 s s Problem 2: : The top line is running parallel to the base of the triangle. Prove: = 180 s s Problem 3: : NLM LNO and OLN MNL Prove: M O s 5) s O L N M Problem 4: : AFB is complementary to BFC. EFD is complementary to DFC. BFC DFC Prove: AFB EFD s 5) 6) 7) 8) s

Test for the unit is 8/21 Name:

Test for the unit is 8/21 Name: Angles, Triangles, Transformations and Proofs Packet 1 Notes and some practice are included Homework will be assigned on a daily basis Topics Covered: Vocabulary Angle relationships Parallel Lines & Transversals