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All Things Algebra®

Geometry Unit 2: Logic & Proof

This unit includes 73 pages of guided notes, homework assignments, four quizzes, a study guide, and a unit test that cover the topics listed in the description below.

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• Additional Information
• What Educators Are Saying

This unit contains the following topics:

• Inductive Reasoning and Conjectures • Compound Statements and Truth Tables • Conditional Statements • Related Conditionals (Inverse, Converse, Contrapositive) • Biconditional Statements • Venn Diagrams with Logic Statements • Deductive Reasoning (Law of Detachment and Law of Syllogism) • Properties of Equality (Addition, Subtraction, Multiplication, Division, Distributive, Substitution, Reflexive, Symmetric, Transitive • Algebraic Proofs • Properties of Congruence (Reflexive, Symmetric, Transitive) • Segment Proofs • Angle Proofs

This unit does not contain activities.

This is the guided notes, homework assignments, quizzes, study guide, and unit test only.  For suggested activities to go with this unit, check out the ATA Activity Alignment Guides .

This resource is included in the following bundle(s):

Geometry Curriculum Geometry Curriculum (with Activities)

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This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another.  No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses.  A t ransferable license is not available for this resource.

Copyright Terms:

No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

What standards is this curriculum aligned to?

What format are the files in, will i have access to materials if they are updated, are answer keys included, are videos included.

LOVE these units! I intend to buy them all! Makes it so easy to reinforce what they're leaning in class. The notes and answer keys help ME so much!

-RACHELLE P.

Math 2 students are expected to be able to solve proofs without ever being exposed to them. This unit is the perfect bridge to teach them HOW to solve proofs before getting into geometry.

I teach online geometry and love using the filled in notes and videos as an additional resource for students to access. It is an amazing resource for my online students to have and the notes are very well put together! This has saved me hours of time making additional resources, well worth the money!

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Geometry Unit 4: Congruent Triangles

Geometry Unit 3: Parallel & Perpendicular Lines

Geometry Unit 1: Geometry Basics

• Kindergarten
• Greater Than Less Than
• Measurement
• Multiplication
• Place Value
• Subtraction
• Punctuation
• 1st Grade Reading
• 2nd Grade Reading
• 3rd Grade Reading
• Cursive Writing

Unit 2 Logic And Proofs Homework 8 Segment Proofs

Unit 2 Logic And Proofs Homework 8 Segment Proofs - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Geometry chapter 2 reasoning and proof, Chapter 2, Geometry beginning proofs packet 1, Algebraic proofs, Unit 1 tools of geometry reasoning and proof, Geometry proofs work with answers, Geometry proofs work with answers, Name geometry unit 2 note packet triangle proofs.

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1. GEOMETRY CHAPTER 2 Reasoning and Proof

2. chapter 2, 3. geometry beginning proofs packet 1, 4. 2.6 algebraic proofs, 5. unit 1: tools of geometry / reasoning and proof, 6. geometry proofs worksheets with answers, 7. geometry proofs worksheets with answers, 8. name geometry unit 2 note packet triangle proofs.

Free Printable Math Worksheets for Geometry

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• Review of equations
• Simplifying square roots
• Adding and subtracting square roots
• Multiplying square roots
• Dividing square roots
• Line segments and their measures inches
• Line segments and their measures cm
• Segment Addition Postulate
• Angles and their measures
• Classifying angles
• Naming angles
• The Angle Addition Postulate
• Angle pair relationships
• Understanding geometric diagrams and notation
• Parallel lines and transversals
• Proving lines parallel
• Points in the coordinate plane
• The Midpoint Formula
• The Distance Formula
• Parallel lines in the coordinate plane
• Classifying triangles
• Triangle angle sum
• The Exterior Angle Theorem
• Triangles and congruence
• SSS and SAS congruence
• ASA and AAS congruence
• SSS, SAS, ASA, and AAS congruences combined
• Right triangle congruence
• Isosceles and equilateral triangles
• Midsegment of a triangle
• Angle bisectors
• The Triangle Inequality Theorem
• Inequalities in one triangle
• Classifying quadrilaterals
• Angles in quadrilaterals
• Properties of parallelograms
• Properties of trapezoids
• Properties of rhombuses
• Properties of kites
• Areas of triangles and quadrilaterals
• Introduction to polygons
• Polygons and angles
• Areas of regular polygons
• Solving proportions
• Similar polygons
• Using similar polygons
• Similar triangles
• Similar right triangles
• Proportional parts in triangles and parallel lines
• The Pythagorean Theorem and its Converse
• Multi-step Pythagorean Theorem problems
• Special right triangles
• Multi-step special right triangle problems
• Trig. ratios
• Inverse trig. ratios
• Solving right triangles
• Multi-step trig. problems
• Rhombuses and kites with right triangles
• Trigonometry and area
• Identifying solid figures
• Volume of prisms and cylinders
• Surface area of prisms and cylinders
• Volume of pyramids and cones
• Surface area of pyramids and cones
• More on nets of solids
• Similar solids
• Arcs and central angles
• Arcs and chords
• Circumference and area
• Inscribed angles
• Tangents to circles
• Secant angles
• Secant-tangent and tangent-tangent angles
• Segment measures
• Equations of circles
• Translations
• Reflections
• All transformations combined
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations
• Line segments
• Perpendicular segments
• Medians of triangles
• Altitudes of triangles

The Perpendicular Bisector Theorem

8.1: Which One Doesn’t Belong: Intersecting Lines (5 minutes)

CCSS Standards

Building On

Building Towards

Routines and Materials

Instructional Routines

• Which One Doesn’t Belong?

This activity prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. In particular, students will be focused on the characteristics of perpendicular lines.

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn‘t belong.

Student Facing

Which one doesn’t belong?

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as perpendicular. Also, press students on unsubstantiated claims.

8.2: Lots of Lines (15 minutes)

• HSG-CO.D.12
• MLR6: Three Reads
• Notice and Wonder

The proof that if a point \(C\) is the same distance from \(A\) as it is from \(B\) , then \(C\) must be on the perpendicular bisector of \(AB\) is challenging. Students may struggle to make sense of what it means to prove that a point must be on a line. This proof, like the proof of the Isosceles Triangle Theorem, requires thinking carefully about how to define an auxiliary line.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Arrange students in groups of 2. Use the Notice and Wonder instructional routine with this image to launch the activity.

Display the image for all to see. Ask student to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

\(\overline{AP} \cong \overline{BP}\)

Things students may notice:

• It’s a triangle.
• Segment \(PA\) is congruent to segment \(PB\) .
• Angle \(A\) is congruent to angle \(B\) (by the Isosceles Triangle Theorem).
• There’s a dotted line.

Things students may wonder:

• Is the dotted line the line of symmetry?
• Is the dotted line the angle bisector of angle \(APB\) ?
• Is the dotted line the perpendicular bisector of segment \(AB\) ?

The purpose of this launch is to elicit the idea that special lines through \(P\) when segments \(PA\) and \(PB\) are congruent have many overlapping properties, which will be useful when students draw auxiliary lines later in this activity. While students may notice and wonder many things about these images, wondering about the line through \(P\) is an important focus of the discussion.

If students have access to GeoGebra Geometry from Math Tools, suggest that it might be a helpful tool in this activity.

Diego, Jada, and Noah were given the following task:

Prove that if a point \(C\) is the same distance from \(A\) as it is from \(B\) , then \(C\) must be on the perpendicular bisector of \(AB\) .

At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their teacher gave them the hint, “Another way to think about it is to draw a line that you know \(C\) is on, and prove that line has to be the perpendicular bisector.”

They each drew a line and thought about their pictures. Here are their rough drafts.

Diego’s approach: “I drew a line through \(C\) that was perpendicular to \(AB\) and through the midpoint of \(AB\) . That line is the perpendicular bisector of \(AB\) and \(C\) is on it, so that proves \(C\) is on the perpendicular bisector.”

Jada’s approach: “I thought the line through \(C\) would probably go through the midpoint of \(AB\) so I drew that and labeled the midpoint \(D\) . Triangle \(ACB\) is isosceles, so angles \(A\) and \(B\) are congruent, and \(AC\) and \(BC\) are congruent. And \(AD\) and \(DB\) are congruent because \(D\) is a midpoint. That made two congruent triangles by the Side-Angle-Side Triangle Congruence Theorem. So I know angle \(ADC\) and angle \(BDC\) are congruent, but I still don’t know if \(DC\) is the perpendicular bisector of \(AB\) .”

Noah’s approach: “In the Isosceles Triangle Theorem proof, Mai and Kiran drew an angle bisector in their isosceles triangle, so I’ll try that. I’ll draw the angle bisector of angle \(ACB\) . The point where the angle bisector hits \(AB\) will be \(D\) . So triangles \(ACD\) and \(BCD\) are congruent, which means \(AD\) and \(BD\) are congruent, so \(D\) is a midpoint and \(CD\) is the perpendicular bisector.”

• What do you notice that this student understands about the problem?
• What question would you ask them to help them move forward?
• Using the ideas you heard and the ways you think each student could make their explanation better, write your own explanation for why \(C\) must be on the perpendicular bisector of \(A\) and \(B\) .

Are you ready for more?

Elena has another approach: “I drew the line of reflection. If you reflect across  \(C\) , then  \(A\)  and  \(B\)  will switch places, meaning \(A'\) coincides with \(B\) , and \(B'\)  coincides with  \(A\) . \(C\)  will stay in its place, so the triangles will be congruent.”

1. What feedback would you give Elena?

2. Write your own explanation based on Elena‘s idea.

Anticipated Misconceptions

Remind students who struggle with their critique or edits to make use of the tips from the display, such as asking for 3 statements and 3 reasons, or looking for congruent triangles.

Focus discussion on what students proved and the implications. Display the image from the launch of this activity, and ask what must be true about this image, based on what they just proved. (The dotted line is the perpendicular bisector of \(AB\) . The dotted line is the line of reflection of \(AB\) .)

Display this image of the construction of a perpendicular bisector and ask students how what they just proved explains why this construction works. (Both circles have radius \(AB\) , so the intersection points of the circles are the same distance from \(A\) and \(B\) . Therefore, both intersection points are on the perpendicular bisector of \(AB\) .)

Ask students to add part 1 of the Perpendicular Bisector Theorem to their reference charts as you add it to the class reference chart:

If a point \(C\) is the same distance from \(A\) as it is from \(B\) , \(C\) must be on the perpendicular bisector of \(AB\) . (Theorem)

\(\overline{AC} \cong \overline{BC}\) , so  \(C\)  is on the line through midpoint  \(M\)  perpendicular to  \(\overline{AB}\)

8.3: Not Too Close, Not Too Far (15 minutes)

• MLR1: Stronger and Clearer Each Time
• Think Pair Share

In a previous activity, students had a chance to practice giving feedback on proofs. In this activity, they will put their proof-writing and feedback-giving skills to work as they draft proofs and then give feedback to a partner. Students have already drawn diagrams of this situation in an earlier lesson, so the focus of this activity is writing a proof that is clear and easy to understand (MP3). Students will make revisions to the proofs they write in the cool-down, so focus discussion on common concerns.

Arrange students in groups of 2.

Display one of the correct, complete diagrams from the cool-down of the Side-Angle-Side Triangle Congruence lesson. Ask students which statement this image illustrates and why. (If a point is on the perpendicular bisector of a line segment, then that point must be the same distance from each endpoint of the segment.)

• If \(P\) is a point on the perpendicular bisector of \(AB\) , prove that the distance from \(P\) to \(A\) is the same as the distance from \(P\) to \(B\) .
• What do you notice that your partner understands about the problem?

Students may think this is the same proof as the previous activity. Ask students what the difference is between this claim and that one. (This one is backwards.) Inform students this is the converse, and the converse of a true statement is not always true, so they will need to write a new proof for this claim. 

Students will make revisions to the proofs they write in the cool-down, so focus discussion on common concerns about the draft proofs. Invite a few students to share their progress. 

Lesson Synthesis

Display these statements for all to see:

• If a point \(C\) is the same distance from \(A\) as it is from \(B\) , prove that \(C\) must be on the perpendicular bisector of \(AB\) .

Ask students what is the same about the two statements and what is different. (They are the same thing—only the ifs and thens are switched.)

Tell students that when two statements are switched like this, we call them converses . Ask student if these statements are converses: “If you practice, then you’ll get better,” and “If you don’t practice, then you won’t get better.” (No; one is the negation of the other, but they don’t switch the order of what is given and what is concluded.)

Ask students for the converse of this statement: “If I won the lottery, then I am rich.” (If I am rich, then I won the lottery.)

Ask students if the converse of a true statement has to be true. (No; you could get rich by receiving an inheritance or having an insanely popular YouTube channel.)

“Today, we proved these converse statements about perpendicular bisectors are both true, which means that a point being the same distance from two endpoints of a segment and a point being on the perpendicular bisector of that segment is the exact same thing.” A statement and its converse will be true if they are from a definition, such as: “If a quadrilateral has four equal sides and four equal angles, then it is a square.” and “If a quadrilateral is a square, then it has four equal sides and four equal angles.”

Display one of the correct, complete images from the cool-down of the Side-Angle-Side Triangle Congruence lesson. Ask students what they can conclude about this image, and which statement from the Perpendicular Bisector Theorem they are using to draw the conclusion. (Point \(P\) is the same distance from points \(A\) and \(B\) . Part 2, the converse.)

Ask students to add part 2 of the Perpendicular Bisector Theorem (the converse of part 1) to their reference charts as you add it to the class reference chart:

If \(C\) is a point on the perpendicular bisector of \(AB\) , the distance from \(C\) to \(A\) is the same as the distance from \(C\) to \(B\) . (Theorem)

\(\overline{AB} \perp \overline{CM}, \overline{AM} \cong \overline{BM},\)  so  \(\overline{AC} \cong \overline{BC}\)

8.4: Cool-down - Reflect and Revise (5 minutes)

Student lesson summary.

The perpendicular bisector of a line segment is exactly those points that are the same distance from both endpoints of the line segment. This idea can be broken down into 2 statements:

• If a point is on the perpendicular bisector of a segment, then it must be the same distance from both endpoints of the line segment.
• If a point is the same distance from both endpoints of a line segment, then it must be on the perpendicular bisector of the segment.

These statements are converses of one another. Two statements are converses if the “if” part and the “then” part are swapped. The converse of a true statement isn‘t always true, but in this case, both statements are true parts of the Perpendicular Bisector Theorem.

A line of reflection is the perpendicular bisector of segments connecting points in the original figure with corresponding points in the image. Therefore, these 3 lines are all the same:

• The perpendicular bisector of a segment.
• The set of points equidistant from the 2 endpoints of a segment.
• The line of reflection that takes the 2 endpoints of the segment onto each other, and the segment onto itself.

Description: <p>A figure: A large quadrilateral with a small triangle inside. The two upward facing sides of the quadrilateral each have two tick marks, the downward facing sides each have three tick marks. Two sides of the downward facing small triangle each have one tick mark. A perpendicular bisector is drawn through the figure, intersecting at three points. A horizontal line segment is drawn in the center of the figure.<br>  </p>

It is useful to know that the perpendicular bisector of a line segment is also all the points which are the same distance from both endpoints of the line segment, because then:

• If 2 points are both equidistant from the endpoints of a segment, then the line through those points must be the perpendicular bisector of the segment (because 2 points define a unique line).
• If 2 points are both equidistant from the endpoints of a segment, then the line through those must be the line of reflection that takes the segment to itself and swaps the endpoints.
• If a point is on the line of reflection, then it is the same distance from that point to a point in the original figure and to its corresponding point in the image.
• If a point is on the perpendicular bisector of a segment, then it is the same distance from that point to both endpoints of the segment.

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Geometry Unit 2 Homework 8

Displaying top 8 worksheets found for - Geometry Unit 2 Homework 8 .

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1. Geometry Unit 2 (teacher) -

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Polygons and Quadrilaterals (Geometry Curriculum - Unit 8) | All Things Algebra®

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Description

This Polygons and Quadrilaterals Unit Bundle contains guided notes, homework assignments, two quizzes, a study guide and a unit test that cover the following topics:

• Angles of Polygons

• Parallelograms

• Tests for Parallelograms in the Coordinate Plane:

Using distance and slope formulas to prove parallelograms

• Parallelogram Proofs

• Rectangles

• Rhombi and Squares

• Quadrilaterals in the Coordinate Plane:

Is it a parallelogram, rectangle, rhombus, or square?

• Non-Isosceles and Isosceles Trapezoids

• Midsegment of a Trapezoid

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Geometry Second Semester Notes Bundle

Geometry Curriculum

Geometry Curriculum (with Activities)

More Geometry Units:

Unit 1 – Geometry Basics

Unit 2 – Logic and Proof

Unit 3 – Parallel and Perpendicular Lines

Unit 4 – Congruent Triangles

Unit 5 – Relationships in Triangles

Unit 6 – Similar Triangles

Unit 7 – Right Triangles and Trigonometry

Unit 9 – Transformations

Unit 10 – Circles

Unit 11 – Volume and Surface Area Unit 12 – Probability

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

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Unit 2 – Tools of Geometry

Starting Concepts in Geometry

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Angles and Their Measures

Angle Pairs

Geometric Terminology

Parallel Lines

More Work with Parallel Lines

Geometry with Coordinates

Congruent Figures

Congruent Triangles

Unit Review

Unit 2 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 2 Assessment Form A

EDITABLE ASSESSMENT

Unit 2 Assessment – Form B

Unit 2 Exit Tickets

Unit 2 Mid-Unit Quiz – Form A

Unit 2 Mid-Unit Quiz – Form B

U02.AO.01 – Angle Pair Practice

EDITABLE RESOURCE

U02.AO.02 – Practice with Parallel Lines

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