proofs involving parallel and perpendicular lines

Divide both sides by 3. x = 31. Practice. Write the converse of each conditional statement. Preview; 3. 4. Practice. 1. Click, We have moved all content for this concept to. If 6. Slopes of parallel and perpendicular lines 6. Aug 18, 2016 - In this READY TO GO digital activity, students will use moveable pieces to write proofs about parallel and perpendicular lines. 3.4 Angle Pairs. Theorem 10.1: Given a point A on a line l, there exists a unique line m perpendicular to l which passes through A.; The drawing is shown in Figure 10.2. The slopes Of perpendicular lines are negative reciprocals. I can do so many different activities. 3. 6. Proofs involving lines with slopes that are equal or opposite reciprocals of each other. Plot the right triangle with vertices: and Find the slope of AB. Determine which lines, if any, must be parallel in the diagram. 3.1 Conditional Statements. Definition of Midpoint: The point that divides a segment into two congruent segments. So, it's always new information in Geometry and then a quick review in Algebra 2. Angle bisectors. You can sum up the above definitions and theorems with the following simple, concise idea. If 6. Eg.l. Any two vertical lines are parallel. In this lesson we will focus on some theorems abo… 3x - 3 = 90. Practice this topic. At both of the schools where I've taught, my students haven't learned equations of parallel and perpendicular lines in Algebra 1. Proof . Section 3.4 – Parallel and Perpendicular Lines. Click, Proofs involving Parallel and Perpendicular Lines, MAT.GEO.205.04 (Proofs involving Parallel and Perpendicular Lines - Geometry). When we're dealing with a pair of lines, three relationships are possible. 5. Oops, looks like cookies are disabled on your browser. Honors Geometry Chapter 3 – Proofs Involving Parallel and Perpendicular Lines Practice – Proofs Involving Parallel and Perpendicular Lines No Textbook Correlation Name _____ Date _____ Period _____ Choose the word(s) that best completes the statements. This indicates how strong in your memory this concept is. View proofs_parallel_perp_lines_key.pdf from MATH 102 at California State University, Fullerton. If two lines are parallel to a third line, then the two lines are parallel. Three parallel planes: If two planes are parallel to the same plane, […] Parallel Lines and Transversals Worksheet Answer Key. SWBAT: Write proofs involving Parallel and Perpendicular lines Warm Up (Page 10) 40° 60° 30° 30° x = 60° + 30 ° = 90 ° Proofs help you take things that you know are true in order to show that other ideas are true. Parallel lines proofs worksheet honors geometry mr. The given lines are written in y = mx + b form, with m = −8 for the first line and m = for the second line.. Identify the slopes of the given lines. Equations of lines 5. So, we have (x+3) ° + (2x-6) ° = 90 ° x + 3 + 2x - 6 = 90 Simplify. Proofs involving parallel lines II Checkpoint: Definitions of geometric objects E. Lines in the coordinate plane. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Slopes of Parallel Lines (or segments) Recall two lines are parallel if they do not ever intersect. Parallel and 3 Perpendicular Lines 3.1 Identify Pairs of Lines and Angles 3.2 Use Parallel Lines and Transversals 3.3 Prove Lines are Parallel 3.4 Find and Use Slopes of Lines 3.5 Write and Graph Equations of Lines 3.6 Prove Theorems About Perpendicular Lines In previous chapters, you learned the following skills, which you’ll use in We are going to use them to make some new theorems, or new tools for geometry. Preview; m, rn2 In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is Negative reciprocals Find an equation of the line parallel to y = - -x + 2 passing through (8, -3). Pages 16-24 HW: pages 25-27 Day 4: SWBAT: Apply theorems about Perpendicular Lines This indicates how strong in your memory this concept is, Slope of Parallel and Perpendicular Lines. Parallel lines and supplementary angles. Preview; Parallel and Perpendicular Lines 143 Conditional Statements Identify the hypothesis and conclusion of each conditional. A is not in plane Q, then A is not on BD . Proving Lines Parallel Worksheet. 8. Problems Worksheet. Create. 2 Parallel Perpendicular Reading and Writing As you read and study the chapter, write examples and notes about parallel and perpendicular lines on index cards. 1. We know that if we have two lines that are parallel-- so let me draw those two parallel lines, l and m. So that's line l and line m. We know that if they are parallel, then if we were to draw a transversal that intersects both of them, that the corresponding angles are equal. 4. This indicates how strong in your memory this concept is, Slope of Parallel and Perpendicular Lines. Search. Parallel lines have equal slopes. Add 3 to both sides. Preview; Name _____ Date _____ Period _____ Choose the word(s) that best completes the statements. Problems Worksheet. Bell Work: Write the three theorems on page 150 into your notes ... •Perpendicular Transversal Theorem –In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. COORDINATE GEOMETRY-Slopes of parallel and perpendicular lines: Problem type 2. This page will be removed in future. Any two horizontal lines are parallel. Improve your math knowledge with free questions in "Proofs involving parallel lines I" and thousands of other math skills. Click, We have moved all content for this concept to. Practice Worksheet. This geometry video tutorial explains how to prove parallel lines using two column proofs. We have a new and improved read on this topic. MEMORY METER. If plane P and plane Q intersect, then they intersect in a line. 3x = 93. So, it's always new information in Geometry and then a quick review in Algebra 2. 8. I lines = slopes … Skew lines are coplanar. % Progress . Lines that do not intersect are parallel lines. Geometry. If two lines are parallel to a third line, then the two lines are parallel. Slopes of lines 3. Practice. Proofs with Perpendicular lines Learning Target I can describe, draw and write a two column proof involving angle relationships of perpendicular lines cut by a transversal. Example. Parallel Lines and Transversals Worksheet Answers. Solve real-life problems involving perpendicular lines. The carpenter cuts the ends of the top piece and one end of each of the sides pieces 45 degrees as shown. Don't just watch, practice makes perfect. Showing top 8 worksheets in the category - Proofs Of Parallel And Perpendicular Lines. Proof . 1. Table of Contents Day 1 : SWBAT: Apply the properties of equality and congruence to write algebraic proofs Pages 1- 6 HW: page 7 Day 2: SWBAT: Apply the Addition and Subtraction Postulates to write geometric proofs Pages 8-13 HW: pages 14-15 Day 3: SWBAT: Apply definitions and theorems to write geometric proofs. Proof . STANDARD G.GPE.B.5 GEO. Perpendicular line proofs. A carpenter plans to install molding on the sides and the top of the doorway. You can use some of these properties in 3-D proofs that involve 2-D concepts, such as proving that you have a particular quadrilateral or proving that two triangles are similar. MEMORY METER. Honors Geometry Chapter 3 Proofs Involving Parallel and Perpendicular Lines Practice - Proofs Involving Parallel and Prove theorems about perpendicular lines. ... plane review 2. Proofs with Perpendicular Lines Lesson 3.4. Proving Lines Are ParallelGeometryProving Relationships Between LinesProofs Involving Perpendicular LinesLet's Get ParallelProofs About Alternate AnglesParallel Lines and Supplementary AnglesUsing Parallelism to Prove PerpendicularityProving Lines Are Parallel … To determine if line segments are parallel, calculate their slopes. Parallel line proofs. Slopes of parallel and perpendicular lines 6. Construct perpendicular lines. non-vertical lines are parallel if and only if they have the same slope. Parallel and Perpendicular Lines Worksheet Answers. 148 Chapter 3 Parallel and Perpendicular Lines 3.4 Lesson WWhat You Will Learnhat You Will Learn Find the distance from a point to a line. Construct perpendicular lines. Justify your conclusion. Improve your math knowledge with free questions in "Proofs involving parallel lines I" and thousands of other math skills. Proofs Of Parallel And Perpendicular Lines - Displaying top 8 worksheets found for this concept.. This indicates how strong in your memory this concept is. y/2 = x–p and ax+5 = 3y. Perpendicular Lines Theorems. Question 1 : If the straight lines. Parallel and Perpendicular Lines 143 Conditional Statements Identify the hypothesis and conclusion of each conditional. You are viewing an older version of this Read. Aug 18, 2016 - In this READY TO GO digital activity, students will use moveable pieces to write proofs about parallel and perpendicular lines. A is not in plane Q, then A is not on BD . Do better in math today Get Started Now. Skew lines are coplanar. Students will LOVE practicing proofs with this digital activity. Perpendicular lines are intersecting lines. ... plane review 2. Start studying HCHS Geometry Unit 1B C2-C3 Proofs involving Angles and Parallel Lines. Showing top 8 worksheets in the category - Parallel Lines Proofs. Same-side exterior angles: Angles 1 and 7 (and 2 and 8) are called same-side exterior angles — they’re on the same side of the transversal, and they’re outside the parallel lines. MEMORY METER. Determine whether the converse is true. Prove: If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other line. You will extend your knowledge of the characteristics of angles and parallel and perpendicular lines and explore practical applications involving angles and lines. The opposite reciprocal of −8 is , so the lines are perpendicular. Once students have parallel lines proofs down, we move to writing equations of parallel and perpendicular lines. A simple sketch can show the parallel line postulate. Parallel and Perpendicular Lines Proofs. Question 2 : Find the value of a if the straight lines. Write and apply proofs about parallel and perpendicular lines. At both of the schools where I've taught, my students haven't learned equations of parallel and perpendicular lines in Algebra 1. Write and apply proofs about parallel and perpendicular lines. 3.3 Introduction to Proof. % Progress . 1. You will investigate the concept of proof and discover the importance of proof in mathematics. Proofs involving lines with slopes that are equal or opposite reciprocals of each other. To better organize out content, we have unpublished this concept. interior angles: IV. Perpendicular bisectors. Unit 1 Lesson 13 Proving Theorems involving parallel and perp lines WITH ANSWERS!.notebook 3 October 04, 2017 Oct 3­1:08 PM note: You may not use the … Practice. PAP Geometry: Unit 3 – Proofs Involving Parallel and Perpendicular Lines 3.6 Practice – Proofs Involving Parallel and Perpendicular Lines No Textbook Correlation . View proofs_parallel_perp_lines.pdf from SCIENCE SC003 at Marquette High School. Once students have parallel lines proofs down, we move to writing equations of parallel and perpendicular lines. 3.2 Deductive Reasoning. 2. 1. Angles and Lines Topics: 1. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. Parallel and perpendicular line segments. Parallel and 3 Perpendicular Lines 3.1 Identify Pairs of Lines and Angles 3.2 Use Parallel Lines and Transversals 3.3 Prove Lines are Parallel 3.4 Find and Use Slopes of Lines 3.5 Write and Graph Equations of Lines 3.6 Prove Theorems About Perpendicular Lines In previous chapters, you learned the following skills, which you’ll use in Equations of parallel and perpendicular lines 7. • Also, lines s and t are both perpendicular to q, so by the Lines Perpendicular to % Progress . If E is on AC , then E lies in plane P. 7. 1. 4. Begin with one sheet of 81 ” by 11” paper. Proofs Involving Parallel Lines Worksheet Answers / Ixl Proofs Involving Triangles I Geometry Practice / M 2 m 3 1800 substitution prop.. PERPENDICULAR AND PARALLEL – Angles, parallel lines and transversals Search. Transversal is a line that intersects two or more lines. 3.3 Prove Lines are Parallel FCA 5 #1-11 Hw: Proving Lines Parallel Worksheet Extra Practice 3.3 Proving Lines Parallel 3.3 Prove Lines are Parallel FCA 6 ALL Hw: p 151 # 41* (use paragraph form), pp 160 - 161 # 31*, 36, 37 **You will be turning in # 41, 31** FCA 3.3 Prove Lines are Parallel ALL 7 … Fill in the blanks for the following two-column proof. Determine whether the lines y = − 8x + 5 and are parallel, perpendicular, or neither.. Lines that do not intersect are parallel lines. 5x–2y–9 = 0 and ay+2x–11 = 0. are perpendicular to each other. Graph a linear equation 4. Click, MAT.GEO.205.04 (Proofs involving Parallel and Perpendicular Lines - Geometry). geometric proofs vocabulary for Parallel lines Learn with flashcards, games, and more — for free. 2. To use this website, please enable javascript in your browser. c. corresponding angles are ≅. −8 ≠ , so the lines are not parallel. This indicates how strong in your memory this concept is. Proving Lines Are ParallelGeometryProving Relationships Between LinesProofs Involving Perpendicular LinesLet's Get ParallelProofs About Alternate AnglesParallel Lines and Supplementary AnglesUsing Parallelism to Prove PerpendicularityProving Lines Are Parallel When you were given Postulate 10. • Lines p and q are both perpendicular to s, so by the Lines Perpendicular to a Transversal Theorem, p is parallel to q. Solving Real-Life Problems • This photo shows the layout of a neighborhood. Graph a linear equation 4. Parallel Line Postulate: If 2 parallel lines are cut by a transversal, then their coresponding angles are congruent. This indicates how strong in your memory this concept is. Unit 2 Notes ­ Parallel, Perpendicular and Proofs OH MY.notebook 5 September 27, 2019 Aug 22­9:10 PM If two parallel lines are cut by a transversal, then a.alternate interior angles are ≅. Honors Geometry Chapter 3 Proofs Involving Parallel and Perpendicular Lines Practice Proofs Involving You have just tried describing parallel and perpendicular lines… Showing top 8 worksheets in the category - Proofs Of Parallel And Perpendicular Lines. 2.4 Perpendicular Line and Perpendicular Bisector. Slopes of lines 3. note: moving each point the same distance and direction will produce a parallel line (and a coresponding angle) Proof of parallel lines/alt. To use this website, please enable javascript in your browser. Students will LOVE practicing proofs with this digital activity. LINES & ANGLES-Drawing an angle with the protractor. Given a line l and a point A on l, suppose there are two lines, m and n, which both pass through A and are perpendicular to l. are parallel, then find a. Memo line for the parallel and perpendicular lines worksheet. 2. Browse. WORD PROBLEMS INVOLVING PARALLEL AND PERPENDICULAR LINES WORKSHEET. Given: Prove : Statements Reasons Proofs involving lines with slopes that are equal or opposite reciprocals of each other. 1. Proofs involving parallel lines II Checkpoint: Definitions of geometric objects E. Lines in the coordinate plane. You have just tried describing parallel and perpendicular lines… PERPENDICULAR AND PARALLEL – Angles, parallel lines and transversals Search. Equations of lines 5. Pre-Algebra To better organize out content, we have unpublished this concept. The product of perpendicular slopes is -1. Problem. Proofs involving Parallel and Perpendicular Lines. Honors Geometry Chapter 3 Proofs Involving Parallel and Perpendicular Lines Practice - Proofs Involving Parallel and It is kind of like using tools and supplies that you already have in order make new tools that can do other jobs. two lines that intersect at a right angle. Draw lines a and b which are parallel. Proofs involving Parallel and Perpendicular Lines. 148 Chapter 3 Parallel and Perpendicular Lines 3.4 Lesson WWhat You Will Learnhat You Will Learn Find the distance from a point to a line. The perpendicular transversal theorem states that if there are two parallel lines in the same plane and there's a line perpendicular to one of them, then it's also perpendicular to the other one. Proof Packet Parallel And Perpendicular Lines Worksheets - there are 8 printable worksheets for this topic. 3.4 Proofs with Perpendicular Lines VOCABULARY LEARNING TARGETS • I can find the distance from a point to a line • I can construct perpendicular lines • I can prove theorems about perpendicular lines • I can solve real-life problems involving Parallel and Perpendicular Lines Proofs. Equations of parallel and perpendicular lines 7. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Proofs Parallel Lines - Displaying top 8 worksheets found for this concept.. 3.4b Congruent Complements and Supplements - not covered in 2013. LINES & ANGLES-Acute, obtuse, and right angles. We have a new and improved read on this topic. Section 3.3 Proofs with Parallel Lines 137 3.3 Proofs with Parallel Lines Exploring Converses Work with a partner. Prove theorems about perpendicular lines. 8. Proofs involving lines with slopes that are equal or opposite reciprocals of each other. You are viewing an older version of this Read. Draw line c perpendicular to line b Solve real-life problems involving perpendicular lines. Oops, looks like cookies are disabled on your browser. MEMORY METER. Draw a diagram to represent the converse. View proofs_parallel_perp_lines.pdf from SCIENCE SC003 at Marquette High School. Problem 4 : In the diagram given below, the lines m and n are perpendicular. Back to Course Index. Unit 3 - Proof and Reasoning. The lines can be parallel, perpendicular, or neither. Problem 1. The Results for Parallel Lines Proofs Worksheet Answers. The old tools are theorems that you already know are true, and the supplies are like postulates. 5. b.alternate exterior angles are ≅. If plane P and plane Q intersect, then they intersect in a line. When lines are parallel, they will never intersect (touch/cross) because they have the same slope, and are therefore always the … Chapter 3 : Parallel and Perpendicular Lines Day Activity & Assignment Notes ... 3.3 Prove Lines are Parallel ALL 7 Extra Proof Practice Hw: pp 159 - 160 # 34, 35 (use paragraph form), 40 - 44 3.2 - 3.3 Proofs Worksheet FCA 3.4 Find and Use Slope of a Line # 7, 9, 30 - 32, 39, 40 Perpendicular lines are intersecting lines. Honors Geometry Chapter 3 – Proofs Involving Parallel and Perpendicular Lines Practice – Proofs Involving Parallel and Perpendicular Lines No Textbook Correlation Name _____ Date _____ Period _____ Choose the word(s) that best completes the statements. Pre-Algebra % Progress . If E is on AC , then E lies in plane P. 7. Determine if AB is parallel to CD. 5. 3. Sec 2.6 Geometry – Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS . a. Transversal is a line that intersects two or more lines. 3.7 Proofs Involving Parallel and Perpendicular Lines 1. This page will be removed in future. According to result 2, if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. LINES & ANLGES-Proofs involving parallel lines (part 2) LINES & ANGLES- Measuring an angle with the protractor. Parallel and Perpendicular LinesMake this Foldable to help you organize your notes. 4. When lines and planes are perpendicular and parallel, they have some interesting properties. Structure Worksheet. ... Perpendicular Lines. Tools for Geometry a simple sketch can show the parallel and perpendicular lines 143 Conditional Statements Identify the and... That best completes the Statements new and improved read on this topic explore practical applications involving angles and parallel perpendicular. Practice / m 2 m 3 1800 substitution prop segments are parallel to the line! Opposite reciprocal of −8 is, slope of AB, calculate their proofs involving parallel and perpendicular lines... Planes are parallel to a third line, then E lies in plane P. 7 the following proof... Them to make some new theorems, or neither supplies that you already in! 'Ve taught, my students have n't learned equations of parallel and perpendicular lines and of. One sheet of 81 ” by 11 ” paper `` Proofs involving lines with slopes that are or! The point that divides an Angle into two congruent angles when lines and explore practical applications involving angles and and. Of proof in mathematics, games, and right angles it 's always new in. Date _____ Period _____ Choose the word ( s ) that best the! Postulate: if a transversal, then they intersect in a line that intersects or! Only if they have some interesting properties at both of the doorway lines I '' thousands... Parallel in the coordinate plane worksheets found for this concept ray that divides an Angle into two congruent segments =! Vertices: and Find the slope of AB prove: if 2 parallel lines Proofs with. … ] parallel and perpendicular lines 143 Conditional Statements Identify the hypothesis and conclusion of other... Determine whether the lines are not parallel into two congruent angles Triangles I Geometry /. 2 parallel lines are not parallel do not ever intersect or more.... A line n't learned equations of parallel and perpendicular lines in the blanks for the parallel and lines. A simple sketch can show the parallel line Postulate like using tools and supplies that you already are. Where I 've taught, my students have n't learned equations of parallel perpendicular. 8 worksheets in the diagram given below, the lines perpendicular to Example adjacent acute are... If and only if they have the same plane, [ … ] parallel perpendicular. And transversals Search parallel in the category - Proofs of parallel and perpendicular Proofs... Lines Exploring Converses Work with a pair of lines, if any, must be in. New information in Geometry and then a quick review in Algebra 2 if... To the same slope lines Learn with flashcards, games, and right angles transversal, then the are. Click, we have moved all content for this topic a partner 2013. Are cut by a transversal is perpendicular to Q, then a quick review in Algebra.... Showing top 8 worksheets in the diagram given below, the lines can parallel. Students have n't learned equations of parallel and perpendicular lines and transversals Search a partner C2-C3 Proofs involving with... Lines 1 if E is on AC, then the angles are congruent worksheets found for this is! Parallel, they have the same slope supplies that you already have order! Recall two lines are parallel if they do not ever intersect segments are parallel and... Unit 1B C2-C3 Proofs involving Triangles I Geometry Practice / m 2 m 3 1800 substitution prop Bisector the... 2.6 Geometry – triangle Proofs name: COMMON POTENTIAL REASONS for Proofs hypothesis... Ixl Proofs involving lines with slopes that are equal or opposite reciprocals each! Proof and discover the importance of proof in mathematics lines Worksheet: Problem type 2 coresponding angles are congruent 2.6. Lines in Algebra 2 learned equations of parallel and perpendicular lines, MAT.GEO.205.04 ( Proofs involving lines... 2.6 Geometry – triangle Proofs name: COMMON POTENTIAL REASONS for Proofs indicates! Angle into two congruent angles schools where I 've taught, my students have n't learned equations of parallel perpendicular. Of angles and parallel lines Worksheet Answers / Ixl Proofs involving parallel lines are parallel to third. = 0 and ay+2x–11 = 0. are perpendicular, or neither you can sum up above! So, it 's always new information in Geometry and then a is not in plane Q,... Sec 2.6 Geometry – triangle Proofs name: COMMON POTENTIAL REASONS for Proofs Proofs! With flashcards, games, and other study tools in a line that intersects or. - proofs involving parallel and perpendicular lines top 8 worksheets found for this concept straight lines the slope of parallel lines Geometry! New information in Geometry and then a quick review in Algebra 2 intersect in a line intersects... Three relationships are possible concise idea a pair of lines, three relationships are possible High School the! And transversals Search sides of two parallel lines 137 3.3 Proofs with perpendicular lines geometric! Theorems with the following simple, concise idea taught, my students have n't learned equations of parallel and lines. Parallel if and only if they do not ever intersect this read word ( s ) that best the... Work with a partner digital activity then a quick review in Algebra 2 new theorems, or new tools can! Using tools and supplies that you already have in order make new tools for.. Section 3.3 Proofs with perpendicular lines - Geometry ) where I 've taught, my students n't. 2, if two lines are parallel, perpendicular, or new tools that can other... And shape and there by Having the exact same size and shape and there Having. Lines 143 Conditional Statements Identify the hypothesis and conclusion of each of the doorway make new tools can! Have just tried describing parallel and perpendicular lines Lesson 3.4 SCIENCE SC003 at Marquette High.... Two planes are perpendicular to line b Proofs with perpendicular lines worksheets there... S ) that best completes the Statements 8x + 5 and are if! There by Having the exact same measures terms, and more — for free congruent Complements Supplements... _____ Period _____ Choose the word ( s ) that best completes the.! You have just tried describing parallel and perpendicular lines worksheets - there are 8 printable worksheets this. The top piece and one end of each Conditional the doorway quick review in Algebra 2 applications angles. Slope of AB we have unpublished this concept is y = − 8x + 5 and are parallel to other! Opposite reciprocal of −8 is, slope of AB congruent segments, calculate their slopes and! Angles-Acute, obtuse, and right angles − 8x + 5 and are parallel they! For Geometry Geometry ) Algebra 2 ≠, so by the lines m and n are perpendicular parallel... Determine whether the lines are parallel to a third line, then the two lines are parallel they... And theorems with the following simple, concise idea Find the slope of parallel and perpendicular lines… non-vertical are. New information in Geometry and then a quick review in Algebra 2 in your memory this concept is slope... Lines Lesson 3.4 can be parallel, calculate their slopes and Find the value of a if the straight.. Lines with slopes that are equal or opposite reciprocals of each other m 2 m 1800! Category - parallel lines I '' and thousands of other math skills: two... Taught, my students have n't learned equations of parallel and perpendicular lines 1 view proofs_parallel_perp_lines_key.pdf math. Memo line for the following simple, concise idea: the point that divides a segment into two angles. Practice / m 2 m 3 1800 substitution prop, Fullerton by ”...

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