Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6

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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. 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)

Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6 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.

Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6

Matching: Use the choices listed at the bottom in the box for problems #1 – 4 Problem 1:

Statement Reason 1. LM LO≅ 1.

2. MN ON≅ 2.

3. LN LN≅ 3. 4. LMN LONΔ ≅ Δ 4.

Problem 2:

Statement Reason 1. QS ! RT 1.

2. R S∠ ≅ ∠ 2. 3. 1 2∠ ≅ ∠ 3. 4. QT QT≅ 4. 5. QST TRQΔ ≅ Δ 5.

Problem 3:

Statement Reason 1. GI KI≅ 1.

2. HI JI≅ 2. 3. GIH KIJ∠ ≅ ∠ 3. 4. GIH KIJΔ ≅ Δ 4.

Problem 4:

Statement Reason 1. ,AC BD AB CDP P 1. 2. 1 4, 2 3∠ ≅ ∠ ∠ ≅ ∠ 2. 3. AD AD≅ 3. 4. ADC DABΔ ≅ Δ 4.

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Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6 Fill in the blank proofs: Problem 5:

Statement Reason 1. I K∠ ≅ ∠ 1. 2. IHJ KJH∠ ≅ ∠ 2. 3. HJ HJ≅ 3. 4. HJK JHIΔ ≅ Δ 4.

Problem 6:

Statement Reason 1. MLN ONL∠ ≅ ∠ 1. 2. _____OLN∠ ≅∠ 2. Given 3. 3. Reflexive Property 4. LNO NLMΔ ≅ Δ 4.

Problem 7:

Statement Reason 1. PQ QS≅ 1. 2. 2. Given 3. PQT RQS∠ ≅∠ 3. 4. PQT SQRΔ ≅ Δ 4.

Problem 8:

Statement Reason 1. UV UX≅ 1. 2. 2. Right Angle Congruence 3. 3. Reflexive Property 5. UWV UWXΔ ≅ Δ 5.

Problem 9:

Statement Reason 1. Y C∠ ≅ ∠ 1. 2. 2. Given 3. 3. Vertical Angles 4. YZA CBAΔ ≅ Δ 4.

Problem 10:

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Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6 Statement Reason

1. BAC DCA∠ ≅ ∠ 1. Given 2. 2. Given 3. 3. 4. ABC CDAΔ ≅ Δ 4.

Problem 11:

Statement Reason 1. F I∠ ≅ ∠ 1. 2. ___ ___∠ ≅∠ 2. 3. 3. 4. EFG HIJΔ ≅ Δ 4.

Problem 12:

Statement Reason 1. ___ M∠ ≅ ∠ 1. Given 2. 2. Given 3. ____KLO∠ ≅ ∠ 3. 4. KLO NLMΔ ≅ Δ 4. 5. K N∠ ≅ ∠ 5. CPCTC

Problem 13:

Statement Reason 1. ___P∠ ≅ ∠ 1. 2. 2. 3. 3. Reflexive 4. PQS RSQΔ ≅ Δ 4.

Problem 14:

Statement Reason 1. AC BDP 1. 2. 2. Given 3. CAD BDA∠ ≅ ∠ 3. 4. 4. Reflexive Property 5. ______ACDΔ ≅ Δ 5.

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Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6

Problem 1: Given: WX YX≅ , Z is the midpoint of WY Prove: △WXZ ≅△YXZ Place the following items in as one of the reasons below: Given Definition of a midpoint Given Reflexive Property SSS Congruence Postulate

Problem 2: Given: B is the midpoint of AE , B is the midpoint of CD Prove: ABD EBC≅V V Place the following items in as one of the statements or reasons below: Given AB EB≅ Given BD BC≅ Vertical Angles are Congruent SAS Congruence Postulate

Statements Reasons 1) WX YX≅

2) Z is the midpoint of WY

3) WZ ZY≅

4) XZ XZ≅ 5) WXZ YXZ≅V V

Statements Reasons 1) B is the midpoint of AE

2) Definition of Midpoint

3) B is the midpoint of CD

4) Definition of Midpoint 5) ABD EBC∠ ≅ ∠ 6) ABD EBC≅V V

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Analytic Geometry | Unit 1: Similarity, Congruence & Proofs Lesson 6

Other Types of Proofs

Proofs with Angles:

Problem 1: Given: ∠1 and∠2 are supplementary. ∠2 ≅ ∠3 Prove: ∠1 + ∠3 = 180°

Problem 2: Given: The top line is running parallel to the base of the triangle. Prove: ∠1 + ∠2 + ∠3 = 180°

Problem 3: Given: NLM LNO∠ ≅ ∠ and OLN MNL∠ ≅ ∠ Prove: ∠M ≅ ∠O

Problem 4: Given: ∠AFB is complementary to ∠BFC. ∠EFD is complementary to ∠DFC. ∠BFC ≅ ∠DFC Prove: ∠AFB ≅ ∠EFD

Statements Reasons 1) 2) 3) 4)

Statements Reasons 1) 2) 3) 4)

Statements Reasons 1) 2) 3) 4) 5)

Statements Reasons 1) 2) 3) 4) 5) 6) 7) 8)

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