# Geometry Companion Book, Volume 2

Geometry with Professor Burger

A companion book for Thinkwell's Geometry online video course. Volume #2 covers Chapters 6 ‒ 12.

No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without prior written permission of Thinkwell Corporation unless such copying is expressly permitted by federal copyright law. Address inquiries to: Permissions, Thinkwell Corp., 505 E. Huntland Drive, Suite 150, Austin, TX 78752.

Before we get too far into the fun, I wanted to personally introduce myself: I'm your virtual instructor, Professor Burger. Welcome to Thinkwell's wonderful world of Geometry!

Together, you and I will explore the ideas that are foundational to Geometry. I hope you ’ ll let me help you in MAKING MEANING of the math ideas we ’ ll see together — there ’ s no need to memorize if you focus on deep understanding. I invite (urge) you to take the time to truly think through the math we ’ ll explore together and to mindfully practice the skills behind the ideas every day. If you do, you ’ ll not only succeed in Geometry, but you ’ ll also be on solid ground for all the other math in your future! About This Book This is a companion book to Thinkwell's Geometry online video course. Use this book as a complement to the online materials. I always say that to learn math you must DO MATH! I encourage you to put pencil to paper and • take notes in this book • highlight key concepts and earmark ideas you want to remember • doodle, sketch, and visualize the math ideas presented in each topic. In a nutshell, make this book your own and keep it by your side as you study the concepts in this course.

This book is divided into chapters. Within each chapter are a series of Geometry topics. Every topic online contains my Video Lessons along with an electronic version of the Notes, Practice questions and Worksheet questions (although I wish we ’ d call them “ Funsheets ” , but that ’ s another story). How To Use This Book Use this book alongside the online course and TAKE NOTES here while watching my video lessons. Your own notes are a key to your own success — I promise. Since this book summarizes the concepts, vocabulary, and key examples presented in the Video Lessons, it is a great tool to help you navigate the videos — but this companion book is not intended as a shortcut to replace the Video Lessons. To get the most out of this learning experience, I urge you to watch (and think through) all of the online Video Lessons. Maybe even watch some videos twice! The online Geometry course offers lots of opportunities to practice the skills you ’ ll need for success in Geometry. Each topic ’ s Worksheet Practice and Interactive Practice is a collection of questions connected to the content presented in the Video Lessons. I've included those questions here in this book, so you can explore them offline and spend time really thinking through each question. I always say that the best way to learn math is to DO MATH. So, take advantage of all the opportunities to practice what you've learned! Finally, at the end of each section, I've included a Review Worksheet to give you even more opportunities to review and practice the concepts you learned in the Video Lessons. Put pencil to paper (or pen, if you dare) to answer each question. I recommend you complete these Reviews before taking the course Quizzes and Tests online.

LET'S GO! I look forward to our Geometry journey together! Remember to make meaning and focus on deep understanding … and also remember that YOU CAN DO IT! Have fun! If you have any questions, please reach out to my friends at Thinkwell. Email them at support@thinkwell.com. Also, I ’ m on Twitter @ebb663, if you want to say, “ hello ” .

I wish you all the best in your Geometry success!

— Prof. B.

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Tips for Success������������������������������������������������������������������������������������������������������������������������� 1 Suggested Pacing Guide��������������������������������������������������������������������������������������������������������������3 Chapter 6: Polygons and Quadrilaterals��������������������������������������������������������������������������������������5 6.1 Polygons and Parallelograms������������������������������������������������������������������������������������7 6.1.1 Properties and Attributes of Polygons �������������������������������������������������������� 9 6.1.2 Properties of Parallelograms ��������������������������������������������������������������������16 6.1.3 Conditions for Parallelograms ������������������������������������������������������������������22 6.1 Review Worksheet ��������������������������������������������������������������������������������������29 6.2 Other Special Quadrilaterals ������������������������������������������������������������������������������������33 6.2.1 Properties of Special Parallelograms ��������������������������������������������������������35 6.2.2 Conditions for Special Parallelograms �����������������������������������������������������41 6.2.3 Properties of Kites and Trapezoids ���������������������������������������������������������� 46 6.2 Review Worksheet ��������������������������������������������������������������������������������������55 Chapter 7: Similarity������������������������������������������������������������������������������������������������������������������59 7.1 Similarity Relationships��������������������������������������������������������������������������������������������61 7.1.1 Ratio and Proportion ��������������������������������������������������������������������������������63 7.1.2 Ratios in Similar Polygons ������������������������������������������������������������������������69 7.1.3 Triangle Similarity: AA, SSS, and SAS ����������������������������������������������������� 73 7.1 Review Worksheet ��������������������������������������������������������������������������������������79 Chapter 8: Right Triangles and Trigonometry����������������������������������������������������������������������������83 8.1 Trigonometric Ratios������������������������������������������������������������������������������������������������85 8.1.1 Similarity in Right Triangles ���������������������������������������������������������������������� 87 8.1.2 Trigonometric Ratios ��������������������������������������������������������������������������������92 8.1.3 Solving Right Triangles ����������������������������������������������������������������������������98 8.1 Review Worksheet ������������������������������������������������������������������������������������104 Chapter 9: Extending Perimeter, Circumference, and Area������������������������������������������������������ 109 9.1 Developing Geometric Formulas ����������������������������������������������������������������������������111 9.1.1 Developing Formulas for Triangles and Quadrilaterals ���������������������������113 9.1.2 Developing Formulas for Circles and Regular Polygons ������������������������122 9.1 Review Worksheet ������������������������������������������������������������������������������������127 Chapter 10: Spatial Reasoning������������������������������������������������������������������������������������������������131 10.1 Surface Area and Volume ������������������������������������������������������������������������������������ 133 10.1.1 Surface Area of Prisms and Cylinders �������������������������������������������������� 135 10.1.2 Surface Area of Pyramids and Cones �������������������������������������������������� 143 10.1.3 Volume of Prisms and Cylinders ����������������������������������������������������������151 10.1 Review Worksheet ����������������������������������������������������������������������������������158

Chapter 11: Circles������������������������������������������������������������������������������������������������������������������������165 11.1 Lines and Arcs in Circles��������������������������������������������������������������������������������������������167 11.1.1 Lines That Intersect Circles ������������������������������������������������������������������������169 11.1.2 Arcs and Chords �����������������������������������������������������������������������������������������175 11.1 Review Worksheet ����������������������������������������������������������������������������������������181 11.2 Angles and Segments in Circles ��������������������������������������������������������������������������������185 11.2.1 Inscribed Angles �����������������������������������������������������������������������������������������187 11.2.2 Angle Relationships in Circles ��������������������������������������������������������������������194 11.2.3 Segment Relationships in Circles ���������������������������������������������������������������201 11.2 Review Worksheet ����������������������������������������������������������������������������������������206 Chapter 12: Transformational Geometry�����������������������������������������������������������������������������������������211 12.1 Congruence Transformations�������������������������������������������������������������������������������������213 12.1.1 Reflections �������������������������������������������������������������������������������������������������215 12.1.2 Translations ������������������������������������������������������������������������������������������������221 12.1.3 Rotations ����������������������������������������������������������������������������������������������������226 12.1 Review Worksheet ����������������������������������������������������������������������������������������232 Blank Graph Paper����������������������������������������������������������������������������������������������������������������������� 237 Formulas & Symbols����������������������������������������������������������������������������������������������������������������������243

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Check out the suggested pacing guide in this book or download the online Lesson Plan and create a study schedule for yourself. Your schedule will be your plan for Geometry success!

Be an active learner. Before you begin studying, collect the tools you'll need: a pencil, scratch paper, highlighters, or graph paper are great things to have on-hand.

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As you watch the Video Lessons online, work out the examples along with Prof. Burger on the Lesson Notes here (or on your own paper). Highlight important points in the Lesson Notes, and earmark topics you want to go back to review before a Quiz or Test.

Practice as you go. After each Video example, complete the Worksheet questions for that example. Once you've watched all the video lessons and answered all the Worksheet questions, check your understanding by completing the Practice question set. Go online to check your answers and to see answer feedback with step- by-step explanations. Review to remember.

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Before a Quiz or Test, complete the Review Worksheet and re-do any exercises you need extra practice to master.

Reach out if you need help! Have questions? Need help? Reach out to us at support@thinkwell.com. We're here to help!

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Suggested Pacing guide

This pacing guide follows a 36-week plan to sequentially progress through Thinkwell's online Geometry course. Since the course is self-paced, feel free to go as quickly or as slowly through the material as you need to – this guide is just a suggestion. The list below corresponds with Thinkwell's Geometry online sequence.

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6.1

Polygons and Parallelograms

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6.1.1 Properties and Attributes of Polygons Key Objectives • Classify polygons based on their sides and angles. • Find and use the measures of interior and exterior angles of polygons. Key Terms • Each segment that forms a polygon is a side of a polygon . • The common endpoint of two sides is a vertex of the polygon .

• A segment that connects any two nonconsecutive vertices is a diagonal . • A regular polygon is one that is both equilateral and equiangular. • A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. • If no diagonal contains points in the exterior, then the polygon is convex . Theorems, Postulates, Corollaries, and Properties • Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is ( n − 2)180°. • Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. Example 1 Identifying Polygons A polygon is a closed figure on a plane formed by three or more line segments that intersect only at their endpoints.

Three figures are determined to be polygons or not polygons. The polygons are named according to the number of sides they have. The first and third figures are closed figures with only one interior and one exterior. The line segments defining each figure intersect only at their endpoints. These two figures are polygons. The second figure contains line segments that intersect away from their endpoints. It is not a polygon. The first figure has four sides; it is called a quadrilateral. The third figure has nine sides; it is called a nonagon.

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6.1.1 Properties and Attributes of Polygons (continued) Example 2 Classifying Polygons In a regular polygon, every side has the same length and every angle has the same measure. In a convex polygon, the diagonals do not include points exterior to the polygon. In a concave polygon, diagonals do include exterior points. Regular polygons are always convex.

Three figures are identified as regular polygons or irregular polygons, and as concave or convex. The hatch marks indicate congruent side lengths. In the first and third polygons all sides are the same length. In the first figure all the angles are congruent, so this is a regular polygon. In the third figure, the angles are not all congruent, so this is an irregular polygon. The first figure is convex, because its diagonals do not include points on the exterior. Figure three is concave because some diagonals include exterior points. The second figure is an irregular polygon, because one of the sides is a different length from the other two. It is a convex polygon because the diagonals do not include exterior points.

Example 3 Finding Interior Angle Measures and Sums in Polygons

According to the Polygon Angle Sum Theorem, the sum of the interior angle measures of a convex polygon with n sides is ( n − 2)180°. To see why the theorem is true, draw diagonals from one vertex in a polygon. This forms n − 2 triangles in a polygon with n sides. For example, in the septagon, there are 5 triangles formed. Each triangle contains 180° of angles. The total measure of the angles within the polygon is the sum of the angles contained in all the triangles, or 5 times 180°.

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6.1.1 Properties and Attributes of Polygons (continued)

The sum of the interior angles of a hexagon is determined in this example. The hexagon has 6 sides. According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Substitute 6 for n and simplify. The sum of the interior angles of a hexagon is 720°.

The measure of the interior angles of a regular octagon is determined in this example. The octagon has 8 sides. According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Substitute 8 for n and simplify. The sum of the interior angles of the octagon is 1080°. Since the octagon is regular, all the interior angles are congruent. The size of each angle is 1080°/8 = 135°. The measure of the interior angles of a quadrilateral is determined in this example. The measures of the angles of the quadrilateral are given as multiples of an unknown, x . According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Equate this value with the sum of the expressions we are given for the angle measures. Substitute 4 for n and solve for x . The value of x is 36°. This is the measure of angles P and R . The other two angles measure 4 x , or 144°.

Example 4 Finding Exterior Angle Measures in Polygons

According to the Polygon Exterior Angle Sum Theorem, the sum of the exterior angle measures of a convex polygon is 360°. Only one exterior angle per vertex is counted. The angle is made by extending the line segment that forms a side of the polygon.

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6.1.1 Properties and Attributes of Polygons (continued)

The measure of the exterior angles of a regular hexagon is determined in this example. According to the Polygon Exterior Angle Sum Theorem, the sum of the interior angles is 360°. The hexagon has 6 sides. Since the hexagon is regular, all the interior angles are congruent. The size of each angle is 360°/6 = 60°. The measures of unknown exterior angles of a polygon are determined in this example. The measures of the angles of the quadrilateral are given as multiples of an unknown x . According to the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles is 360°. Equate the sum of the expressions given for the angle measures to 360°. Solve for x . The value of x is 36°. This is the measure of the exterior angles at S and R . The measures of the other two angles can be determined by substitution.

Example 5 Problem-Solving Application

The measure of an exterior angle of a regular pentagon is determined in this application example. The building shape is given as a regular pentagon. According to the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles is 360°. Since the pentagon is regular, all the exterior angles are congruent. The size of each angle is 360°/5 = 72°.

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6.1.1 Properties and Attributes of Polygons - Worksheet

Example 1: Tell whether each outlined shape is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. 4.

Example 2: Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 5. 6. 7.

Example 3: 8. Find the measure of each interior angle of pentagon ABCDE.

C

5 z °

B

4 z °

D

3 z °

5 z °

3 z °

E

A

9. Find the measure of each interior angle of a regular dodecagon.

10. Find the sum of the interior angle measures of a convex 20-gon.

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6.1.1 Properties and Attributes of Polygons - Worksheet (continued)

Example 4:

11. Find the value of y in polygon JKLM .

K

4 y °

2 y °

J

L

4 y °

6 y °

M

12. Find the measure of each exterior angle of a regular pentagon.

Example 5:

Use the photograph of the traffic sign for Exercises 13 and 14.

13. Name the polygon by the number of its sides.

R

Q

S

P

T

14. In the polygon, ∠ P, ∠ R, and ∠ T are right angles, and ∠ Q ≅ ∠ S. What are m ∠ Q and m ∠ S ?

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6.1.1 Properties and Attributes of Polygons - Practice

1.  If the figure is a polygon, name it by its number of sides.

2.  Classify the polygon as regular or irregular, and concave or convex.

3.  Classify the polygon as regular or irregular, and concave or convex.

4.  Find the sum of the interior angle measures of a convex octagon.

5.  Find the measure of each exterior angle of a regular 18-gon.

6.  Find the measure of each exterior angle of a regular pentagon.

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6.1.2 Properties of Parallelograms Key Objectives • Prove and apply properties of parallelograms. • Use properties of parallelograms to solve problems. Key Terms

• A quadrilateral with two pairs of parallel sides is a parallelogram . Theorems, Postulates, Corollaries, and Properties

• Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. • Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. • Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. • Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. Example 1 Problem-Solving Application

The sides opposite each other in a parallelogram are congruent.

Consecutive angles, those on either end of a single side, in a parallelogram are supplementary, meaning their measures sum to 180°.

The length of a side and measure of an angle in a parallelogram are determined in this example. AD , m ∠ CDA , and AE are given. To find BC , recognize that BC ≅ AD by the Properties of Parallelograms (opposite sides are congruent). Then, by the definition of congruent line segments, BC = AD . Substitute the known length for AD to find BC = 60 in. To find m ∠ DAB , recognize that ∠ CDA is consecutive with ∠ DAB . Therefore, the sum of their measures equals 180°, according to the Properties of Parallelograms. Substitute the known value of 81° for m ∠ CDA . The measure of ∠ DAB is found to be 99°.

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6.1.2 Properties of Parallelograms (continued) Example 2 Using Properties of Parallelograms to Find Measures

The measures of a side and an angle in a parallelogram are determined in this example. AB and DC , m ∠ B , and m ∠ A are given as algebraic expressions. AB and DC are equal according to the Properties of Parallelograms. Set the expressions for the lengths of the two sides equal to each other and solve for x .The solution yields x = 7, and substituting this value for x in the expression for AB yields a length of 19 units. ∠ B and ∠ A are consecutive angles in a parallelogram, so according to the Properties of Parallelograms their measures sum to 180°. Substitute the expressions for m ∠ B and m ∠ A into the equation for their sum and solve for the unknown, y . The solution yields y = 11, and substituting this value for y in the expression for the measure of ∠ B yields 70°.

Example 3 Parallelograms in the Coordinate Plane

The coordinates are determined here for the unknown vertex of a parallelogram on the coordinate plane. The coordinates of three of the vertices for the parallelogram are given. The fourth vertex, D , will form the endpoint of a line segment, CD , that is parallel to line segment AB . The slope of the two sides will be equal, since this is a parallelogram. The length of the two sides will be equal according to the Properties of Parallelograms. Use the slope of AB , a rise of 3 units and a run of 1 unit, to find point D relative to point C . For the x coordinate, 2 + 1 = 3, and for the y coordinate, − 4 + 3 = − 1. The coordinates for D are (3, − 1).

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6.1.2 Properties of Parallelograms (continued) Example 4 Using Properties of a Parallelogram in a Proof

The angles opposite each other in a parallelogram are congruent.

The two diagonals of a parallelogram bisect each other.

A proof that ∠ B is congruent to ∠ 4 is developed in this example. It is given that there are three parallelograms ABCD , DEFG , and FIHJ , and that AE , CG , JG , and EI are line segments. According to the Properties of Parallelograms, ∠ B and ∠ 1 are congruent, and ∠ 2 and ∠ 3 are congruent, because they are opposite angles. According to the Vertical Angles Theorem ∠ 1 and ∠ 2 are congruent, and ∠ 3 and ∠ 4 are congruent. Therefore, because of the Transitive Property of Congruence, ∠ B ≅ ∠ 4.

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6.1.2 Properties of Parallelograms - Worksheet

Example 1: The handrail is made from congruent parallelograms. In ▱ ABCD, AB = 17.5, DE = 18, and m ∠ BCD = 110º. Find each measure .

B

C

E

A

1. BD

2. CD

D

3. BE

4. m ∠ ABC

6. m ∠ DAB

Example 2: JKLM is a parallelogram. Find each measure . 7. JK

8. LM

K

L

(2 z – 3)°

7 x

3 x + 14

(5 z – 6)°

J

M

9. m ∠ L

10. m ∠ M

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6.1.2 Properties of Parallelograms - Worksheet (continued)

Example 3: 11. Three vertices of ▱ DFGH are D ( − 9, 4), F ( − 1, 5), and G (2, 0). Find the coordinates of vertex H .

Example 4: 12. Write a two-column proof.

Q

 Given : PSTV is a parallelogram. PQ ≅ RQ Prove : ∠ STV ≅ ∠ R

S

T

P

R

V

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6.1.2 Properties of Parallelograms - Practice

1.  In parallelogram KLMN , NM = 8. Find KL .

2. PQRS is a parallelogram. QT = 20. Find QS .

3.  In parallelogram KLMN , NL = 15. Find NP .

4.  TPRS is a parallelogram. Find m ∠ R .

5.  Three vertices of parallelogram ABCD are A ( − 1, 5), B (2, 1) and D ( − 2, − 3). Find the coordinates of vertex C .

6. Given: KLMN is a parallelogram. Prove: m ∠ N + m ∠ M = 180°

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6.1.3 Conditions for Parallelograms Key Objectives • Prove that a given quadrilateral is a parallelogram.

Theorems, Postulates, Corollaries, and Properties • Theorem If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. • Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. • Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Example 1 Verifying Figures are Parallelograms

This is a condition of a quadrilateral being a parallelogram. If both pairs of opposite sides of the quadrilateral are congruent, then it is a parallelogram. Note the symbols for congruent sides in the figure. This is a condition of a quadrilateral being a parallelogram. If both pairs of the opposite angles of a quadrilateral are congruent, then that quadrilateral is a parallelogram. Note the symbols for congruent angles in the figure.

This is a condition of a quadrilateral being a parallelogram. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then that quadrilateral is a parallelogram.

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6.1.3 Conditions for Parallelograms (continued)

In this example, the quadrilateral ABCD is shown to be a parallelogram when the unknowns x and y have certain values. The lengths of the sides of the quadrilateral are given as algebraic expressions in x and y . Substitute the given values for x and y into the expressions for the lengths of the sides, solve for the lengths, and compare opposite sides. Since BC = DA = 15, and AB = CD = 11, then by the quadrilateral with congruent opposite sides condition for parallelograms, the figure is a parallelogram. The quadrilateral EFGH is shown here to be a parallelogram when the unknowns z and w have certain values. The measures of three of the angles of the quadrilateral are given as algebraic expressions in z and w . Substitute the given values for z and w into the expressions for the angle measures, and solve for the measures. This proof is more direct than the one shown. ∠ F and ∠ G are consecutive angles. Since m ∠ F + m ∠ G = 98° + 82° = 180°, ∠ F and ∠ G are supplementary. Therefore, EFGH is a parallelogram by the Properties of Parallelograms (Quad. with ∠ s supp. to cons. ∠ s → ▱ ). The same proof can be obtained using ∠ F and ∠ E .

Example 2 Applying Conditions for Parallelograms

This is a condition of a quadrilateral being a parallelogram. If a quadrilateral has one pair of opposite congruent and parallel sides, then it is a parallelogram. Note the symbols for congruent sides and those for parallel sides in the figure.

23

6.1.3 Conditions for Parallelograms (continued)

Quadrilateral figures are determined to be parallelograms or not.

It is given in the first figure that one of the pairs of opposite sides of the quadrilateral is both congruent and parallel. Therefore, this is a parallelogram by a condition for parallelograms. It is given in the second figure that one pair of opposite sides is congruent and one pair of consecutive angles is congruent. This is not enough information to determine that the quadrilateral is a parallelogram. Remember not to make any assumptions based on what the figure looks like. For example, if the congruent angles are not right angles, then the congruent sides are not parallel.

Example 3 Proving Parallelograms in the Coordinate Plane

A quadrilateral in the coordinate plane is proven here to be a parallelogram. The coordinates of the

vertices of the parallelogram are given. Calculate the slopes of the sides of the quadrilateral. For example the slope of AB = ( y B − y A )/( x B − x A ) = (1 − 3)/(1 − ( − 3)) = − 2/4 = − 1/2.

Calculating all slopes shows that AB || DC and AD || BC . Since both pairs of opposite sides are parallel, by the conditions for parallelograms, the quadrilateral ABCD is a parallelogram. A quadrilateral in the coordinate plane is proven here to be a parallelogram. The coordinates of the vertices of the parallelogram are given. Calculate the slopes of two of the sides of the quadrilateral. For example, the slope of FG = ( y G − y F )/( x G − x F ) = (5 − 5)/(1 − ( − 3)) = 0/4 = 0. The slope of JH is also 0. Therefore, FG || JH . Calculate the lengths FG and JH . For example, the length FG is (1 − ( − 3)) = 4. This is also the length JH . Since the opposite sides are congruent and parallel, by a condition for parallelograms, the quadrilateral FGHJ is a parallelogram.

24

6.1.3 Conditions for Parallelograms (continued) Example 4 Problem-Solving Application

This is a condition of a quadrilateral being a parallelogram. If a quadrilateral’s two diagonals bisect each other, then the quadrilateral is a parallelogram. Notice the symbols for congruent line segments in the figure. A quadrilateral is determined to be a parallelogram in this application example. It is given that AE and EC are equal in length and BE and ED are equal in length. The givens indicate that the diagonals of the quadrilateral bisect each other. Therefore, this is a parallelogram according to the conditions for parallelograms.

25

6.1.3 Conditions for Parallelograms - Worksheet

Example 1: 1. Show that EFGH is a parallelogram for s = 5 and t = 6.

2. Show that KLPQ is a parallelogram for m = 14 and n = 12.5.

F

L

P

(6 n – 1)°

(5 m + 36)°

G

J

E

(4 m + 50)°

H

K

Q

Example 3: Show that the quadrilateral with the given vertices is a parallelogram. 6. W ( − 5 , − 2) , X ( − 3 , 3), Y (3, 5), Z (1, 0)

7. R ( − 1 , − 5) , S ( − 2 , − 1), T (4, − 1), U (5, − 5)

26

6.1.3 Conditions for Parallelograms - Worksheet (continued)

Example 4: 8. A parallel rule can be used to plot a course on a navigation chart. The tool is made of two rulers connected at hinges to two congruent crossbars AD and BC. You place the edge of one ruler on your desired course and then move the second ruler over the compass rose on the chart to read the bearing for your course. If AD || BC, why is AB always parallel to CD ?

D

C

A

B

1200

27

6.1.3 Conditions for Parallelograms - Practice

1.  Show that PQRS is a parallelogram for x = 7 and p = 4.

2.   Show that EFGH is a parallelogram for y = 7 and z = 9.

3. Determine if KLMN must be a

4. Determine if IJKL must be a parallelogram. Justify your answer. Given: ∠ I and ∠ L are supplementary, ∠ J and ∠ K are supplementary, and ∠ I and ∠ J are congruent.

5. Show that the quadrilateral with vertices W ( − 2, 1), X (1, 5), Y (1, 2), and Z ( − 2, − 2) is a parallelogram.

6. Show that the quadrilateral with vertices L (0, 4), M (3, 5), N (2, 1), and O ( − 1, 0) is a parallelogram.

28

6.1 Review Worksheet

6.1.1 Properties and Attributes of Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

1.

2.

3.

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex .

4.

5.

6.

7. Find the measure of each interior angle of quadrilateral RSTV .

R

2 n °

S

6 n °

5 n °

2 n °

V

8. Find the measure of each interior angle of a regular 18-gon.

T

9. Find the sum of the interior angle measures of a convex heptagon.

29

6.1 Review Worksheet (continued)

10. Find the measure of each exterior angle of a regular nonagon.

11. A pentagon has exterior angle measures of 5 a°, 4 a °, 10 a °, 3 a °, and 8 a°. Find the value of a.

The folds on the lid of the gift box form a regular hexagon. Find each measure. 12. m ∠ JKM

* * *

* *

* *

*

*

*

*

L

*

*

*

* *

*

*

* *

M

*

* *

*

* *

*

*

*

*

*

*

*

*

*

*

* * *

*

*

*

* K

*

*

*

*

*

*

*

*

*

* *

*

*

*

*

*

*

*

*

*

13. m ∠ MKL

*

*

*

* * * J

*

*

*

*

*

*

*

*

6.1.2 Properties of Parallelograms Cranes can be used to load cargo onto ships. In ▱ JKLM , JL = 165.8, JK = 110, and m ∠ JML = 50° . Find the measure of each part of the crane . 14. JN 15. LM

M

N

L

J

16. LN

17. m ∠ JKL

K

18. m ∠ KLM

19. m ∠ MJK

30

6.1 Review Worksheet (continued)

WXYZ is a parallelogram. Find each measure . 20. WV 21. YW

X

Y

V

Z

W

22. XZ

23. ZV

24 .  Three vertices of ▱ PRTV are P ( − 4 , − 4) , R ( − 10, 0), and V (5, − l). Find the coordinates of vertex T.

25. Write a two-column proof.

B

C

Given : ABCD and AFGH are parallelograms. Prove : ∠ C ≅ ∠ G

G

F

A

D

H

6.1.3 Conditions for Parallelograms 26. Show that BCGH is a parallelogram for x = 3.2 and y = 7.

27. Show that TUVW is a parallelogram for a = 19.5 and b = 22.

6 y – 14

10 a – 6

U

V

C

G

(2 b + 41)°

8 x – 9

3 x + 7

(7 b – 59)°

T

W

B

H

8 a + 33

3 y + 7

31

6.1 Review Worksheet (continued)

28.

29.

30.

Show that the quadrilateral with the given vertices is a parallelogram .

31. J ( − l, 0), K ( − 3, 7), L (2, 6), M (4, − l)

32. P ( − 8, − 4), Q ( − 5, 1), R (1, − 5), S ( − 2, − 10)

33. The toolbox has cantilever trays that pull away from the box so that you can reach the items beneath them. Two congruent brackets connect each tray to the box. Given that AD = BC, how do the brackets AB and CD keep the tray horizontal?

B

C

A

D

32

6.2

33

6.2.1 Properties of Special Parallelograms Key Objectives • Prove and apply properties of rectangles, rhombuses, and squares. • Use properties of rectangles, rhombuses, and squares to solve problems. Key Terms • A rectangle is a quadrilateral with four right angles. • A rhombus is a quadrilateral with four congruent sides. • A square is a quadrilateral with four right angles and four congruent sides.

Theorems, Postulates, Corollaries, and Properties • Theorem If a quadrilateral is a rectangle, then it is a parallelogram. • Theorem If a parallelogram is a rectangle, then its diagonals are congruent. • Theorem If a quadrilateral is a rhombus, then it is a parallelogram. • Theorem If a parallelogram is a rhombus, then its diagonals are perpendicular. • Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Example 1 Craft Application This is a property of a rectangle. If a quadrilateral is a rectangle, then it is a parallelogram.

This is a property of a rectangle. If a parallelogram is a rectangle, then its diagonals are congruent.

A property of rectangles is used to find the length of the sides of a quilt patch in this application example. It is given that the patch is a rectangle. The length of one side, EH , and the length of a diagonal, EG , are given. By the properties of rectangles, opposite sides are congruent, so FG ≅ EH . EH is given as 12 in., so by the definition of congruent line segments, FG is also 12 in.

35

6.2.1 Properties of Special Parallelograms (continued)

A property of rectangles is used to find the length of half the diagonal of a quilt patch in this application example. It is given that the patch is a rectangle. The length of one side, EH , and the length of a diagonal, EG , are given. Here is a more direct proof than the one shown; try comparing them. By the properties of rectangles, the diagonals FH and EG are congruent. EG is given to be 20 in., so FH is 20 in. by the definition of congruent line segments. The diagonals of a rectangle bisect each other, so FJ = FH /2. Substitute 20 in. for FH to yield 10 in. for FJ .

Example 2 Using Properties of Rhombuses to Find Measures

This is a property of a rhombus. If a parallelogram is a rhombus, then its diagonals are perpendicular.

This is a property of a rhombus. If a parallelogram is a rhombus, then each of its diagonals bisects a pair of opposite angles.

This is a property of a rhombus. If a quadrilateral is a rhombus, then it is a parallelogram.

36

6.2.1 Properties of Special Parallelograms (continued)

The properties of a rhombus are used here to determine the length of a side of a rhombus. The figure is given to be a rhombus with the lengths of two sides given as algebraic expressions. To find DC , begin by solving for x . By the definition of a rhombus, all sides are congruent, so AD = AB . Substitute the given algebraic expressions for AD and AB and solve for x . The solution yields x = 3. Again, by the definition of a rhombus all sides are congruent, so to find DC substitute the value for x into either of the expressions for the length of the two given sides. DC = AD = 4 x = 4(3) = 12. The properties of a rhombus are used here to determine the measure of an angle of a rhombus. The figure is given to be a rhombus with the measures of two angles given as algebraic expressions with the unknown y . To find m ∠ ADB begin by solving for y . By the properties of a rhombus the diagonals are perpendicular, so m ∠ BEC = (11 y + 2)° = 90°. Solve for y to yield y = 8. Substitute the value of y to find m ∠ DAC . m ∠ DAC = (7 y + 4)° = (7(8) + 4)° = 60° By the properties of a rhombus each diagonal bisects opposite angles, so m ∠ CAB = m ∠ DAC = 60°. Substitute to find m ∠ DAB . m ∠ DAB = m ∠ DAC + m ∠ CAB = 60° + 60° = 120° By the Same Side Interior Angles Theorem, m ∠ ADC + m ∠ DAB = m ∠ ADC + 120° = 180°. Therefore, m ∠ ADC = 60°. According to the properties of a rhombus each diagonal bisects opposite angles, so m ∠ ADB = (1/2)m ∠ ADC = (1/2)60° = 30°.

37

6.2.1 Properties of Special Parallelograms (continued) Example 3 Verifying Properties of Squares

The diagonals of a square in the coordinate plane are shown to be congruent and to be perpendicular bisectors of each other. To show the diagonals are congruent, calculate their lengths using the Distance Formula. The lengths are both found to be 5 2. By the definition of congruent line segments, the diagonals are congruent. To show the diagonals bisect each other, calculate their midpoints using the Midpoint Formula. Both midpoints are the same point on the coordinate plane. This means the two diagonals intersect at their midpoints, or bisect each other. To show the diagonals are perpendicular, calculate their slopes. The slope of AC is − 1/7 and the slope of BD is 7. The product of the slopes is equal to − 1, indicating that the lines are perpendicular.

Example 4 Using Properties of Special Parallelograms in Proofs

A triangle is proven to be isosceles using properties of rectangles and parallelograms. ABCD is given to be a rectangle with E the midpoint of AD . By the definition of a rectangle, ∠ A and ∠ D are right angles and are congruent by the Right Angle Congruence Theorem. ABCD is a parallelogram since a rectangle is a parallelogram. AB ≅ DC according to the Properties of Parallelograms. AE ≅ ED by the definition of a midpoint. Therefore, △ ABE ≅ △ DCE by the SAS Congruence Postulate. BE ≅ CE by the Corresponding Parts of Congruent Triangles are Congruent Theorem. Thus by the definition of an isosceles triangle, △ BCE is isosceles.

38

6.2.1 Properties of Special Parallelograms - Worksheet

Example 1: The braces of the bridge support lie along the diagonals of rectangle PQRS. RS = 160 ft and QS = 380 ft. Find each length. 1. TQ 2. PQ R S

T

3. ST

4. PR

Q

P

Example 2:

(4 y – 1)°

4 x + 15

ABCD is a rhombus. Find each measure.

B

C

5. AB

6. m ∠ ABC

F

12 y °

7 x + 2

A

D

Example 3:

7. The vertices of square JKLM are J ( − 3, − 5), K ( − 4, 1), L (2, 2), and M (3, − 4). Show that the diagonals of square JKLM are congruent perpendicular bisectors of each other.

Example 4:

E

C

8. Given : RECT is a rectangle. RX ≅ TY Prove : △ REY ≅ △ TCX

T

R

X

Y

39

6.2.1 Properties of Special Parallelograms - Practice

1.  An artist needs to slice a rectangular cake EFGH into four triangles. Find EG , if FH = 22 ft.

2.  Molten metal is poured into a mold with diagonal spacers. In rectangle PQRS, PR = 46 in. Find TQ .

3. PFJA is a rhombus. Find JA .

4. RSTU is a rhombus. Find m ∠ STU .

5.  The vertices of square CDEF are C (1, 1), D (3, 1), E (3, − 1) and F (1, − 1). Prove that its diagonals are congruent perpendicular bisectors of each other.

6.  Given STUV is a rectangle. Prove that △ WVS ≅ △ WUT.

40

6.2.2 Conditions for Special Parallelograms Key Objectives • Prove that a given quadrilateral is a rectangle, rhombus, or square. Theorems, Postulates, Corollaries, and Properties • Theorem If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

• Theorem If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. • Theorem If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. • Theorem If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. • Theorem If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. Example 1 Carpentry Application This is a condition for a rectangle. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

This is a condition for a rhombus. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

This is a condition for a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

A quadrilateral ABCD is proved to be a rectangle in this application example. It is given that opposite sides of the quadrilateral are congruent, as are the diagonals. By the conditions for parallelograms, ABCD is a parallelogram because its opposite sides are congruent. By the conditions for rectangles, ABCD is a rectangle because the diagonals are congruent. In carpentry, the way to ensure a frame is “square,” meaning it has four right angles, is to make sure the lengths of the two diagonals are the same.

41

6.2.2 Conditions for Special Parallelograms (continued) Example 2 Applying Conditions for Special Parallelograms

This is a condition for a rectangle. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

This is a condition for a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

It is determined that a quadrilateral satisfies the conditions to be a square in this example. It is given that two pairs of opposite sides are congruent, that two sides are perpendicular, and that the diagonals are perpendicular. PQRS is a parallelogram by the conditions for parallelograms, because the opposite sides are congruent. PQRS is a rectangle by the conditions for rectangles, because two of the sides form a right angle. PQRS is a rhombus by the conditions for a rhombus, because the diagonals are perpendicular. PQRS is a rectangle, so it has four right angles, and a rhombus, so it has four congruent sides. Therefore, by definition, PQRS is also a square. It is determined whether a quadrilateral satisfies the conditions to be a rhombus in this example.

It is given that the diagonals are perpendicular.

There is insufficient information to determine if the quadrilateral is a rhombus. For example, a kite has perpendicular diagonals like the given figure. But a kite is not a rhombus because it does not have four congruent sides.

42

6.2.2 Conditions for Special Parallelograms (continued) Example 3 Identifying Special Parallelograms in the Coordinate Plane

The properties of the diagonals of a parallelogram in the coordinate plane are used here to determine whether the parallelogram is a rectangle, a rhombus, or a square. The coordinates of the parallelogram are given. First, test whether the parallelogram is a rectangle by determining if the diagonals are the same length. Calculate the length of the diagonals using the Pythagorean Theorem. The diagonals are congruent, so the figure is a rectangle by the conditions for rectangles. Second, test whether the parallelogram is a rhombus by determining if the slopes of the diagonals are perpendicular. The slopes of the diagonals are 3/7 and − 7/3, and the product of the slopes is − 1, so the diagonals are perpendicular and the figure is a rhombus by the conditions for a rhombus. The parallelogram is also a square because it is both a rectangle (four right angles) and a rhombus (four congruent sides).

43

6.2.2 Conditions for Special Parallelograms - Worksheet

Example 1: 1. A city garden club is planting a square garden. They drive pegs into the ground at each corner and tie strings between each pair. The pegs are spaced so that WX ≅ XY ≅ YZ ≅ ZW. How can the garden club use the diagonal strings to verify that the garden is a square?

X

W

Y

V

Z

Example 2:

Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

A

B

2. Given: AC ≅ BD

Conclusion: ABCD is a rectangle.

D

C

3. Given: AB || CD, AB ≅ CD , AB ⊥ BC Conclusion: ABCD is a rectangle.

Example 3:

Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus , or square. Give all the names that apply.

4. P ( − 5, 2), Q (4, 5), R (6, − 1), S ( − 3, − 4)

5. W ( − 6, 0), X (1, 4), Y (2, − 4), Z ( − 5, − 8)

44

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