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C H A P T E R Quadrilaterals © 2010 Carnegie Learning, Inc. 8 8 Carpenters build, install, and maintain wooden objects such as buildings and furniture. There are almost 1.5 million carpenters in the United States, making it the nation's largest building trades occupation. When building, carpenters often use math, along with specialized tools such as levels and squares. You will use math to calculate whether a bookcase was properly built. 8.1 Squares and Rectangles 8.4 Properties of Squares and Rectangles | p. 417 8.2 Parallelograms and Rhombi Sum of the Interior Angle Measures of a Polygon | p. 451 8.5 Properties of Parallelograms and Rhombi | p. 429 8.3 Exterior and Interior Angle Measurement Interactions Sum of the Exterior Angle Measures of a Polygon | p. 459 Kites and Trapezoids Properties of Kites and Trapezoids | p. 437 Decomposing Polygons 8.6 Quadrilateral Family Categorizing Quadrilaterals | p. 467 Chapter 8 | Quadrilaterals 413 Introductory Problem for Chapter 8 Ms. Smith was teaching her class how to make Venn diagrams. She used the following example. S 2 7 L FH 6 8 5 4 3 T 1 Rectangle S represents all students in the high school. Circle L represents all students on the lacrosse team. Circle FH represents all students on the field hockey team. Circle T represents all students on the track team. The numbers on the Venn diagram represent 8 different regions. 8 Region 2: Region 3: Region 4: Region 5: Region 6: Region 7: Region 8: 414 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 1. Describe which sport(s) the students in each designated region play. Region 1: 2. Create a Venn diagram that describes the relationship between all of the terms listed. Number each region and name the polygons found in each region. Use RH for rhombus and RE for rectangle. Trapezoids Kites Rhombi Rectangles Parallelograms Squares Quadrilaterals © 2010 Carnegie Learning, Inc. 8 3. Write a description for each numbered region. Be prepared to share your solutions and methods. Chapter 8 | Introductory Problem for Chapter 8 415 © 2010 Carnegie Learning, Inc. 8 416 Chapter 8 | Quadrilaterals 8.1 Squares and Rectangles Properties of Squares and Rectangles OBJECTIVES KEY TERM In this lesson you will: l l l l l Prove the Perpendicular/Parallel Line Theorem. Construct a square and a rectangle. Determine the properties of a square and rectangle. Prove the properties of a square and a rectangle. PROBLEM 1 Perpendicular/Parallel Line Theorem Know the Properties or be Square! A quadrilateral is a four-sided polygon. A diagonal of a polygon is a line segment connecting two non-adjacent vertices. 8 A square is a quadrilateral with four right angles and all sides congruent. © 2010 Carnegie Learning, Inc. Quadrilaterals have different properties directly related to the measures of their interior angles and lengths of their sides. Perpendicular lines and right angles are useful when proving properties of certain quadrilaterals. 1. Devon is trying to think of quadrilaterals that have four right angles. Can you help him? Lesson 8.1 | Squares and Rectangles 417 2. Ramira is helping Jessica with her math homework. She tries to explain the theorem: “If two lines are perpendicular to the same line, then the two lines are parallel to each other.” Jessica doesn’t understand why this is true. Use the diagram shown and complete the proof to help Jessica understand this theorem. ᐉ3 1 2 ᐉ2 3 4 ᐉ1 5 6 7 8 Given: ᐉ1 ⊥ ᐉ3; ᐉ2 ⊥ ᐉ3 Prove: ᐉ1 ᐉ2 Statements Reasons 8 © 2010 Carnegie Learning, Inc. Perpendicular/Parallel Line Theorem: If two lines are perpendicular to the same line, then the two lines are parallel to each other. 418 Chapter 8 | Quadrilaterals 3. Draw a square with two diagonals. Label the vertices and the intersection of the diagonals. List all of the properties you know to be true. 8 ___ ___ ___ 4. Use AB to construct square ABCD with diagonals AC and BD intersecting at point E. B © 2010 Carnegie Learning, Inc. A Lesson 8.1 | Squares and Rectangles 419 5. Create a two-column proof of the statement DAB CBA. D C E A ___ ___ B Given: Square ABCD with diagonals AC and BD intersecting at point E Prove: DAB CBA Statements 8 Reasons ___ ___ Congratulations! You have just proven a property of a square. Property of a Square: Diagonals of a square are congruent. You can now use this property as a valid reason in future proofs. 420 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 6. Do you have enough information to conclude AC BD ? Explain. ___ ___ ___ ___ 7. Create a two-column proof of the statement DA CB and DC AB . D C A B Given: Square ___ ___ABCD___ ___ Prove: DA CB and DC AB Statements Reasons 8 © 2010 Carnegie Learning, Inc. 8. If a parallelogram is a quadrilateral with opposite sides parallel, do you have enough information to conclude square ABCD is a parallelogram? Explain. Congratulations! You have just proven another property of a square! Property of a Square: Opposite sides of a square are parallel. You can now use this property as a valid reason in future proofs. Lesson 8.1 | Squares and Rectangles 421 ___ ___ ___ ___ 9. Create a two-column proof. Use DEC and BEA to prove DE BE and CE AE . D C E A ___ ___ B Given: Square with diagonals AC and BD intersecting at point E ___ ___ABCD ___ ___ Prove: DE BE and CE AE Statements Reasons 8 Congratulations! You have just proven another property of a square! Property of a Square: The diagonals of a square bisect each other. You can now use this property as a valid reason in future proofs. 422 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 10. Do you have enough information to conclude the diagonals of a square bisect each other? Explain. 11. Write a paragraph proof to conclude the diagonals of a square bisect the vertex angles. Use the square in Question 9 and the Property of a Square: The diagonals of a square bisect each other. Congratulations! You have just proven another property of a square! Property of a Square: Opposite sides of a square are parallel. You can now use this property as a valid reason in future proofs. 12. Write a paragraph proof to conclude the diagonals of a square are perpendicular to each other. Use the square in Question 9. © 2010 Carnegie Learning, Inc. 8 Congratulations! You have just proven another property of a square! Property of a Square: The diagonals of a square are perpendicular to each other. You can now use this property as a valid reason in future proofs. 13. Revisit Question 3 to make sure you have listed all of the properties of a square. Lesson 8.1 | Squares and Rectangles 423 PROBLEM 2 The Rectangle A rectangle is a quadrilateral with opposite sides congruent and all angles congruent. 1. Draw a rectangle with two diagonals. Label the vertices and the intersection of the two diagonals. List all of the properties you know to be true. (Do not draw a square.) 8 ___ ___ R 424 Chapter 8 | Quadrilaterals E © 2010 Carnegie Learning, Inc. ___ 2. Use RE to construct rectangle RECT with diagonals RC and ET intersecting at point A. Do not construct a square. 3. Create a two-column proof of the statement RCT ETC. R E A T ___ C ___ Given: Rectangle RECT with diagonals RC and ET intersecting at point A Prove: RCT ETC Statements Reasons 8 © 2010 Carnegie Learning, Inc. ___ ___ 4. Do you have enough information to conclude RT EC ? Explain. Lesson 8.1 | Squares and Rectangles 425 5. In a paragraph proof, describe how you could prove the second pair of opposite sides of the rectangle are congruent. 6. Do you have enough information to conclude rectangle RECT is a parallelogram? Explain your reasoning. 7. Do you have enough information to conclude the diagonals of a rectangle are congruent? 8. Do you have enough information to conclude the diagonals of a rectangle bisect each other? Explain. 8 © 2010 Carnegie Learning, Inc. 9. Revisit Question 1 of Problem 2 and make sure you have listed all of the properties of a rectangle. 426 Chapter 8 | Quadrilaterals PROBLEM 3 Application Problems 1. Ofelia is making a square mat for a picture frame. How can she make sure the mat is a square using only a ruler? 2. Gretchen is putting together a bookcase. It came with diagonal support bars that are to be screwed into the top and bottom on the back of the bookcase. Unfortunately, the instructions were lost and Gretchen does not have the directions or a measuring tool. She has a screwdriver, a marker, and a piece of string. How can Gretchen attach the supports to make certain the bookcase will be a rectangle and the shelves are parallel to the ground? © 2010 Carnegie Learning, Inc. 8 Lesson 8.1 | Squares and Rectangles 427 3. Matsuo knows this birdhouse has a rectangular base, but he wonders if it has a square base. a. What does Matsuo already know to conclude the birdhouse has a rectangular base? b. What does Matsuo need to know to conclude the birdhouse has a square base? 8 © 2010 Carnegie Learning, Inc. Be prepared to share your solutions and methods. 428 Chapter 8 | Quadrilaterals 8.2 Parallelograms and Rhombi Properties of Parallelograms and Rhombi OBJECTIVES KEY TERM In this lesson you will: l Construct a parallelogram. Construct a rhombus. l Prove the properties of a parallelogram. l Prove the properties of a rhombus. l Parallelogram/Congruent-Parallel Side Theorem l PROBLEM 1 The Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 8 © 2010 Carnegie Learning, Inc. 1. Draw a parallelogram with two diagonals. Label the vertices and the intersection of the diagonals. List all of the properties you know to be true. Do not draw a square or a rectangle. Lesson 8.2 | Parallelograms and Rhombi 429 ___ ___ ___ 2. Use PA to construct parallelogram PARG with diagonals PR and AG intersecting at point M. A P 3. To prove opposite sides of a parallelogram are congruent, which triangles would you prove congruent? P A 8 M R © 2010 Carnegie Learning, Inc. G 430 Chapter 8 | Quadrilaterals 4. Use PGR and RAP in the parallelogram from Question 3 to prove opposite sides of a parallelogram Create a two-column proof of the ___ ___ are ___congruent. ___ statement PG AR and GR PA . ___ ___ Given: Parallelogram ___ ___ PARG ___ with ___ diagonals PR and AG intersecting at point M Prove: PG AR and GR PA Statements Reasons Congratulations! You have just proven a property of a parallelogram! Property of a Parallelogram: Opposite sides of a parallelogram are congruent. You can now use this property as a valid reason in future proofs. 5. Do you have enough information to conclude PGR RAP ? Explain. © 2010 Carnegie Learning, Inc. 8 6. What additional angles would you need to show congruent to prove opposite angles of a parallelogram are congruent? What two triangles do you need to prove congruent? Lesson 8.2 | Parallelograms and Rhombi 431 7. Use APG and GRA in the diagram from Question 3 to prove opposite angles of a parallelogram are congruent. Create a two-column proof of the statement GPA ARG. ___ ___ Given: Parallelogram PARG with diagonals PR and AG intersecting at point M Prove: GPA ARG (You have already proven PGR RAP in Question 5.) Statements Reasons Congratulations! You have just proven another property of a parallelogram! Property of a Parallelogram: Opposite angles of a parallelogram are congruent. You can now use this property as a valid reason in future proofs. 8 Congratulations! You have just proven another property of a parallelogram! Property of a Parallelogram: The diagonals of a parallelogram bisect each other. You can now use this property as a valid reason in future proofs. 9. Ray told his math teacher that he thinks a quadrilateral is a parallelogram if only one pair of opposite sides is known to be both congruent and parallel. Is Ray correct? Write a paragraph proof to verify Ray’s conjecture. Use the diagram from Question 3. 432 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 8. Write a paragraph proof to conclude the diagonals of a parallelogram bisect each other. Use the parallelogram in Question 3. Parallelogram/Congruent-Parallel Side Theorem: If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. 10. Revisit Question 1 to make sure you have listed all of the properties of a parallelogram. PROBLEM 2 The Rhombus A rhombus is a quadrilateral with all sides congruent. 1. Draw a rhombus with two diagonals. Label the vertices and the intersection of the two diagonals. List all of the properties you know to be true. (Do not draw a square.) © 2010 Carnegie Learning, Inc. 8 Lesson 8.2 | Parallelograms and Rhombi 433 ___ ___ ___ 2. Use RH to construct rhombus RHOM with diagonals RO and HM intersecting at point B. Do not construct a square. H R 3. Write a paragraph proof to conclude rhombus RHOM is a parallelogram. R H B 8 O Congratulations! You have just proven a property of a rhombus! Property of a Rhombus: A rhombus is a parallelogram. You can now use this property as a valid reason in future proofs. 4. Since a rhombus is a parallelogram, what properties hold true for all rhombi? 5. Write a paragraph proof to conclude the diagonals of a rhombus are perpendicular. Use the rhombus in Question 3. 434 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. M 6. Write a paragraph proof to conclude the diagonals of a rhombus bisect the vertex angles. Use the rhombus in Question 3. 7. Revisit Question 1 and make sure you have listed all of the properties of a rhombus. PROBLEM 3 Application Problems 1. Jim tells you he is thinking of a quadrilateral that is either a square or a rhombus, but not both. He wants you to guess which quadrilateral he is thinking of and allows you to ask one question about the quadrilateral. Which question should you ask? 8 © 2010 Carnegie Learning, Inc. 2. Mrs. Baker told her geometry students to bring in a picture of a parallelogram for extra credit. Albert brought in a picture of the flag shown. The teacher handed Albert a ruler and told him to prove it was a parallelogram. What are two ways Albert could prove the picture is a parallelogram? Lesson 8.2 | Parallelograms and Rhombi 435 3. Mrs. Baker held up two different lengths of rope shown and a piece of chalk. She asked her students if they could use this rope and chalk to construct a rhombus on the blackboard. Rena raised her hand and said she could construct a rhombus with the materials. Mrs. Baker handed Rena the chalk and rope. What did Rena do? Be prepared to share your solutions and methods. 436 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 8 8.3 Kites and Trapezoids Properties of Kites and Trapezoids OBJECTIVES KEY TERMS In this lesson you will: l l l l l l Construct a kite. Construct a trapezoid. Prove the properties of a kite. Prove the properties of a trapezoid. Prove a biconditional statement. PROBLEM 1 l l base angles of a trapezoid isosceles trapezoid biconditional statement Let's Go Fly a Kite! A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite sides that are not congruent. 8 © 2010 Carnegie Learning, Inc. 1. Draw a kite with two diagonals. Label the vertices and the intersection of the two diagonals. List all of the properties you know to be true. Lesson 8.3 | Kites and Trapezoids 437 __ ___ 2. Construct kite KITE with diagonals IE and KT intersecting at point S. 3. To prove one pair of opposite angles of a kite is congruent, which triangles in the kite would you prove congruent? Explain your reasoning. I 8 K S T 4. Prove one pair of opposite angles ___of a kite __ congruent. Given: Kite KITE with diagonals KT and IE intersecting at point S. Prove: KIT KET Statements 438 Chapter 8 | Quadrilaterals Reasons © 2010 Carnegie Learning, Inc. E Congratulations! You have just proven a property of a kite! Property of a Kite: One pair of opposite angles is congruent. You are now able to use this property as a valid reason in future proofs. ___ 5. Do you have enough information to conclude KT bisects IKE and ITE? Explain your reasoning. __ ___ 6. What two triangles could you use to prove IS ES ? __ ___ 7. If IS ES , is that enough information to determine that one diagonal of a kite bisects the other diagonal? Explain. 8 © 2010 Carnegie Learning, Inc. 8. Write a paragraph proof to conclude the diagonals of a kite are perpendicular to each other. Congratulations! You have just proven another property of a kite! Property of a Kite: The diagonals of a kite are perpendicular to each other. You are now able to use this property as a reason in future proofs. 9. Revisit Question 1 to make sure you have listed all of the properties of a kite. Lesson 8.3 | Kites and Trapezoids 439 PROBLEM 2 The Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. The bases of a trapezoid are its parallel sides. The base angles of a trapezoid are either pair of angles that share a base as a common side. The legs of a trapezoid are its non-parallel sides. 1. Draw a trapezoid. Identify the vertices, bases, base angles, and legs. ___ 2. Use TR to construct trapezoid TRAP. T R © 2010 Carnegie Learning, Inc. 8 440 Chapter 8 | Quadrilaterals An isosceles trapezoid is a trapezoid with congruent non-parallel sides. One property of an isosceles trapezoid is that the base angles of an isosceles trapezoid are congruent. 3. Create a two-column proof of this property of an isosceles trapezoid. You will need to draw an auxiliary line parallel to one of the congruent legs to prove this property. You will also need to do the proof in two parts because there are two pairs of base angles. ___ ___ ___ ___ Given: Isosceles Trapezoid TRAP with TR PA , TP RA Prove: T R T Z P Statements R A Reasons © 2010 Carnegie Learning, Inc. 8 You must also prove A TPA. Write a paragraph proof to prove A TPA. Lesson 8.3 | Kites and Trapezoids 441 4. Kala insists that if a trapezoid has only one pair of congruent base angles, then the trapezoid must be isosceles. She thinks proving two pairs of base angles are congruent is not necessary. Prove the given statement using a two column proof to show that Kala is correct. ___ ___ Given: Isosceles Trapezoid TRAP with TR PA , T R ___ ___ Prove: TP RA T Z P R A Statements Reasons An if and only if statement is called a biconditional statement because it consists of two separate conditional statements rewritten as one statement. It is a combination of both a conditional statement and the converse of that conditional statement. A biconditional statement is true only when the conditional statement and the converse of the statement are both true. Consider the following property of an isosceles trapezoid: The diagonals of an isosceles trapezoid are congruent. The property clearly states that if a trapezoid is isosceles, then the diagonals are congruent. Is the converse of this statement true? If so, then this property can be written as a biconditional statement. Rewording the property as a biconditional statement becomes: “A trapezoid is isosceles if and only if its diagonals are congruent.” To prove this biconditional statement is true, rewrite it as two conditional statements and prove each statement. Statement 1: If a trapezoid is an isosceles trapezoid, then the diagonals of the trapezoid are congruent. (Original statement) Statement 2: If the diagonals of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. (Converse of original statement) 442 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 8 5. Use the trapezoid shown to prove each statement. T R A P ___ ___ ___ ___ Given: Isosceles Trapezoid ___ ___TRAP with TP RA , TR PA , and diagonals TA and PR . ___ ___ Prove: TA PR Statements Reasons 8 ___ ___ ___ ___ © 2010 Carnegie Learning, Inc. Given: Trapezoid TRAP with TP RA , and diagonals TA PR Prove: Trapezoid TRAP is isosceles ___ To prove the converse, auxiliary lines must be drawn such that RA is extended to ___intersect ___ a perpendicular line passing through point T perpendicular to RA (TE ) and intersect a second perpendicular line passing through point P ___ ___ perpendicular to RA (PZ ). T E R A P Z Notice that quadrilateral TEZP is a rectangle. Lesson 8.3 | Kites and Trapezoids 443 6. Write a paragraph proof to prove the converse is true. The property of an isosceles trapezoid can now be written as a biconditional statement because the conditional statement and its converse have both been proven to be true. Property of an Isosceles Trapezoid: A trapezoid is isosceles if and only if its diagonals are congruent. PROBLEM 3 Construction Segment AD is the perimeter of an isosceles trapezoid. Follow the steps to construct the isosceles trapezoid. 8 A ___ Choose a segment on segment AD for the shorter base (___ AB ). Choose a segment on segment AD for the longer base ( BC ). Segment ___CD represents the sum of the length of the two legs. Bisect CD to determine the length of each congruent leg. Label the midpoint E. © 2010 Carnegie Learning, Inc. 1. 2. 3. 4. D 444 Chapter 8 | Quadrilaterals 5. Copy segment AB onto segment ___BC (creating segment BF ) to determine the difference between the bases (FC ). ___ 6. Bisect FC to determine point G (FG ⫽ CG). © 2010 Carnegie Learning, Inc. 8 Lesson 8.3 | Kites and Trapezoids 445 ___ 7. Take FG (half the ___ difference of the base ___ lengths) and copy it onto the left end of the long base BC (creating distance ____ BH ). Notice that it is already marked off on the right end of the long base (GC ). 8. Construct the perpendicular through point H. Note the distance between the two most left perpendiculars is the length of the short base. © 2010 Carnegie Learning, Inc. 8 446 Chapter 8 | Quadrilaterals 9. Place the compass point on B and mark off the distance of one leg (CE) on the left most perpendicular, name the new point I. This forms one leg of the isosceles trapezoid. 10. Place the compass point on C and mark off the length of one leg (CE) on the other perpendicular, name the new point J. This is one leg of the isosceles trapezoid. Note that IJ ⫽ AB. BIJC is an isosceles trapezoid! © 2010 Carnegie Learning, Inc. 8 Lesson 8.3 | Kites and Trapezoids 447 PROBLEM 4 Application Problems 1. Solve for the perimeter of the kite. © 2010 Carnegie Learning, Inc. 8 448 Chapter 8 | Quadrilaterals 2. Could quadrilaterals 1, 2, and 3 on this kite be squares? Explain. 2 3 1 5 4 © 2010 Carnegie Learning, Inc. 8 Lesson 8.3 | Kites and Trapezoids 449 3. Trevor used a ruler to measure the height of each trapezoid and the length of each leg. He tells Carmen the three trapezoids must be congruent because they are all the same height and have congruent legs. What does Carmen need to do to convince Trevor that he is incorrect? 8 © 2010 Carnegie Learning, Inc. Be prepared to share your solutions and methods. 450 Chapter 8 | Quadrilaterals 8.4 Decomposing Polygons Sum of the Interior Angle Measures of a Polygon KEY TERM OBJECTIVES In this lesson you will: l Write a formula for the sum of the interior angles of any polygon. l Calculate the sum of the interior angles of any polygon, given the number of sides. l Calculate the number of sides of a polygon, given the sum of the interior angles. l Write a formula for the measure of each interior angle of any regular polygon. l Calculate the measure of an interior angle of a regular polygon, given the number of sides. l Calculate the number of sides of a regular polygon, given the sum of the interior angles. © 2010 Carnegie Learning, Inc. PROBLEM 1 l interior angle of a polygon 8 Who’s Correct? In geometry, it is necessary to know the sum of the interior angles of various polygons to determine other information. An interior angle of a polygon is an angle which faces the inside of a polygon and is formed by consecutive sides of the polygon. Is there a quick method for calculating the sum of the measures of different polygons? Let’s find out! Ms. Lambert asked her class to determine the sum of the interior angle measures of a quadrilateral. She that two of her students, Carson and Juno, were already engaged in a heated disagreement. Lesson 8.4 | Decomposing Polygons 451 Carson drew a quadrilateral and added one diagonal as shown. He concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 360º. 1. Describe Carson’s reasoning. Juno drew a quadrilateral and added two diagonals as shown. She concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 720º. 8 3. Who is correct? Explain. 452 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 2. Describe Juno’s reasoning. PROBLEM 2 The Sum of the Interior Angle Measures of a Polygon As always, you must start with what you know to be true. The Triangle Sum Theorem states that the sum of the three interior angles of any triangle is equal to 180º. You can use this information to calculate the sum of the interior angles of other polygons. 1. Calculate the sum of the interior angle measures of a quadrilateral by completing each step. Step 1: Draw a quadrilateral. Step 2: Draw all possible diagonals using only one vertex of the quadrilateral. Remember, a diagonal is a line segment connecting non-adjacent vertices. 8 © 2010 Carnegie Learning, Inc. Step 3: How many triangles are formed when the diagonal(s) divide the quadrilateral? Step 4: If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)? Lesson 8.4 | Decomposing Polygons 453 2. Calculate the sum of the interior angle measures of a pentagon by completing each step. Step 1: Draw a pentagon. Step 2: Draw all possible diagonals using only one vertex of the pentagon. Step 3: How many triangles are formed when the diagonal(s) divide the pentagon? 8 3. Calculate the sum of the interior angle measures of a hexagon by completing each step. Step 1: Draw a hexagon. 454 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. Step 4: If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)? Step 2: Draw all possible diagonals using one vertex of the hexagon. Step 3: How many triangles are formed when the diagonal(s) divide the hexagon? Step 4: If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)? 4. Complete the table shown. Number of sides of the polygon 3 4 5 6 Number of diagonals drawn 8 Number of triangles formed © 2010 Carnegie Learning, Inc. Sum of the measures of the interior angles 5. What pattern do you notice between the number of possible diagonals drawn from one vertex of the polygon, and the number of triangles formed by those diagonals? 6. Compare the number of sides of the polygon to the number of possible diagonals drawn from one vertex. What do you notice? 7. Compare the number of sides of the polygon to the number of triangles formed by drawing all possible diagonals from one vertex. What do you notice? Lesson 8.4 | Decomposing Polygons 455 8. What pattern do you notice about the sum of the interior angle measures of a polygon as the number of sides of each polygon increases by 1? 9. Predict the number of possible diagonals drawn from one vertex and the number of triangles formed for a seven-sided polygon using the table you completed. 10. Predict the sum of all the interior angle measures of a seven-sided polygon using the table your completed. 11. Continue the pattern to complete the table. Number of sides of the polygon 7 8 9 16 Number of diagonals drawn Number of triangles formed Sum of the measures of the interior angles 12. When you calculated the number of triangles formed in the 16-sided polygon, did you need to know how many triangles were formed in a 15-sided polygon first? Explain your reasoning. 8 14. What is the sum of all the interior angle measures of a 100-sided polygon? Explain your reasoning. 15. If a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex? Explain. 16. What is the sum of all the interior angle measures of an n-sided polygon? Explain your reasoning. 456 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 13. If a polygon has 100 sides, how many triangles are formed by drawing all possible diagonals from one vertex? Explain. 17. Use the formula to calculate the sum of all the interior angle measures of a polygon with 32 sides. 18. If the sum of all the interior angle measures of a polygon is 9540º, how many sides does the polygon have? Explain your reasoning. PROBLEM 3 Sum of the Interior Angle Measures of a Regular Polygon 1. Use the formula developed in Problem 2, Question 16 to calculate the sum of the all the interior angle measures of a decagon. 2. Calculate each interior angle measure of a decagon if each interior angle is congruent. How did you calculate your answer? 8 3. Complete the table. Number of sides of regular polygon 3 4 5 6 7 8 © 2010 Carnegie Learning, Inc. Sum of measures of interior angles Measure of each interior angle 4. If a regular polygon has n sides, write a formula to calculate the measure of each interior angle. 5. Use the formula to calculate each interior angle measure of a regular 100-sided polygon. Lesson 8.4 | Decomposing Polygons 457 6. If each interior angle measure of a regular polygon is equal to 150º, determine the number of sides. How did you calculate your answer? 7. Apply what you have learned about the interior angle measures of a regular polygon on the star shown. PENTA is a regular pentagon. Solve for x. P E x A N T Be prepared to share your methods and solutions. 458 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 8 8.5 Exterior and Interior Angle Measurement Interactions Sum of the Exterior Angle Measures of a Polygon OBJECTIVES KEY TERM In this lesson you will: l exterior angle of a polygon Write a formula for the sum of the exterior angles of any polygon. l Calculate the sum of the exterior angles of any polygon, given the number of sides. l Write a formula for the measure of each exterior angle of any regular polygon. l Calculate the measure of an exterior angle of a regular polygon, given the number of sides. l Calculate the number of sides of a regular polygon, given the measure of each exterior angle. l 8 © 2010 Carnegie Learning, Inc. PROBLEM 1 Is There a Formula? In the previous lesson, you wrote a formula for the sum of all the interior angle measures of a polygon. In this lesson, you will write a formula for the sum of all the exterior angle measures of a polygon. Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 459 Each interior angle of a polygon can be paired with an exterior angle. An exterior angle of a polygon is formed adjacent to each interior angle by extending one side of each vertex of the polygon as shown in the triangle. Each exterior angle and the adjacent interior angle form a linear pair. Exterior Angle Exterior Angle Exterior Angle 1. Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a triangle. © 2010 Carnegie Learning, Inc. 8 460 Chapter 8 | Quadrilaterals 2. Calculate the sum of the exterior angle measures of a quadrilateral by completing each step. Step 1: Draw a quadrilateral and extend each side to locate an exterior angle at each vertex. Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a quadrilateral. © 2010 Carnegie Learning, Inc. 8 3. Calculate the sum of the exterior angle measures of a pentagon by completing each step. Step 1: Draw a pentagon and extend each side to locate an exterior angle at each vertex. Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 461 Step 2: Use the formula for the sum of the interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a pentagon. 4. Calculate the sum of the exterior angle measure of a hexagon by completing each step. Step 1: Without drawing a hexagon, how many linear pairs are formed by each interior and adjacent exterior angle? How do you know? 8 Step 3: Use the formula for the sum of the interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the measures of the exterior angles of a hexagon. 462 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. Step 2: What is the relationship between the number of sides of a polygon and the number of linear pairs formed by each interior angle and its adjacent exterior angle? 5. Complete the table. Number of sides of the polygon 3 4 5 6 7 15 Number of linear pairs formed Sum of measures of linear pairs Sum of measures of interior angles Sum of measures of exterior angles 6. When you calculated the sum of the exterior angle measures in the 15-sided polygon, did you need to know anything about the number of linear pairs, the sum of the linear pair measures, or the sum of the interior angle measures of the 15-sided polygon? Explain. © 2010 Carnegie Learning, Inc. 7. If a polygon has 100 sides, calculate the sum of the exterior angle measures. Explain how you calculated your answer. 8 8. What is the sum of the exterior angle measures of an n-sided polygon? 9. If the sum of the exterior angle measures of a polygon is 360⬚, how many sides does the polygon have? Explain how you got this answer. Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 463 10. Explain why the sum of the exterior angle measures of any polygon is always equal to 360⬚. PROBLEM 2 Regular Polygons 1. Calculate the measure of each exterior angle of an equilateral triangle. Explain your reasoning. 2. Calculate the measure of each exterior angle of a square. Explain your reasoning. 8 4. Calculate the measure of each exterior angle of a regular hexagon. Explain your reasoning. 5. Complete the table shown to look for a pattern. Number of sides of a regular polygon Sum of measures of exterior angles Measure of each interior angle Measure of each exterior angle 464 Chapter 8 | Quadrilaterals 3 4 5 6 7 15 © 2010 Carnegie Learning, Inc. 3. Calculate the measure of each exterior angle of a regular pentagon. Explain your reasoning. 6. When you calculated the measure of each exterior angle in the 15-sided regular polygon, did you need to know anything about the measure of each interior angle? Explain. 7. If a regular polygon has 100 sides, calculate the measure of each exterior angle. Explain how you calculated your answer. 8. What is the measure of each exterior angle of an n-sided regular polygon? 9. If the measure of each exterior angle of a regular polygon is 18⬚, how many sides does the polygon have? Explain how you calculated your answer. © 2010 Carnegie Learning, Inc. 8 Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 465 PROBLEM 3 Old Sibling Rivalry Two sisters, Molly and Lily, were arguing about who was better at using a compass and a straightedge. 1. Molly challenged Lily to construct a regular hexagon. Undaunted by the challenge, Lily took the compass and went to work. What did Lily do? 2. Lily then challenged Molly to construct a square. Molly grabbed her compass with gusto and began the construction. What did Molly do? 3. Both sisters were now glaring at each other and their mother, a math teacher, walked into the room. Determined to end this dispute, she gave her daughters a challenge. She told them the only way to settle the argument was to see who could be the first to come up with a construction for a regular pentagon. Give it a try! Hint: There are only six steps. The first two steps are to draw a starter line and to construct a perpendicular line. © 2010 Carnegie Learning, Inc. 8 Be prepared to share your solutions and methods. 466 Chapter 8 | Quadrilaterals 8.6 Quadrilateral Family Categorizing Quadrilaterals OBJECTIVES In this lesson you will: List the properties of quadrilaterals. Categorize quadrilaterals based upon their properties. l Construct quadrilaterals given a diagonal. l l PROBLEM 1 Characteristics of Quadrilaterals Complete the table by placing a checkmark in the appropriate row and column to associate each figure with its properties. © 2010 Carnegie Learning, Inc. Square Rectangle Rhombus Kite Parallelogram Trapezoid Quadrilateral 8 No parallel sides Exactly one pair of parallel sides Two pairs of parallel sides One pair of sides are both congruent and parallel Two pairs of opposite sides are congruent Exactly one pair of opposite angles are congruent Two pairs of opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other All sides are congruent Diagonals are perpendicular to each other Diagonals bisect the vertex angles All angles are congruent Diagonals are congruent Lesson 8.6 | Quadrilateral Family 467 PROBLEM 2 The introduction problem to this chapter involved creating a Venn diagram associating various quadrilaterals. The properties of each quadrilateral dictate how to correctly position and relate each quadrilateral in a Venn diagram. To conclude this unit, we will revisit the Venn diagram problem. 1. Create a Venn diagram describing the relationships between all of the quadrilaterals listed. Number each region and name the figure located in each region. Use RH for rhombus and RE for rectangle. 2. Write a description for each region. Trapezoids Kites Rhombi Rectangles Parallelograms Squares Quadrilaterals © 2010 Carnegie Learning, Inc. 8 468 Chapter 8 | Quadrilaterals PROBLEM 3 True or False Determine whether each statement is true or false. If it is false, explain why. 1. A square is also a rectangle. 2. A rectangle is also a square. 3. The base angles of a trapezoid are congruent. 4. A parallelogram is also a trapezoid. 5. A square is a rectangle with all sides congruent. 8 © 2010 Carnegie Learning, Inc. 6. The diagonals of a trapezoid are congruent. 7. A kite is also a parallelogram. 8. The diagonals of a rhombus bisect each other. Lesson 8.6 | Quadrilateral Family 469 PROBLEM 4 Can You Read Joe’s Mind? Joe is thinking of a specific polygon. He has listed six hints. As you read each hint, use deductive reasoning to try and guess Joe’s polygon. By the last hint you should be able to read Joe’s mind. 1. The polygon has four sides. 2. The polygon has at least one pair of parallel sides. 3. The diagonals of the polygon bisect each other. 4. The polygon has opposite sides congruent. 8 6. The polygon does not have four congruent angles. 470 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 5. The diagonals of the polygon are perpendicular to each other. PROBLEM 5 Using Diagonals Knowing certain properties of each quadrilateral makes it possible to construct the quadrilateral given only a single diagonal. ____ 1. Describe how you could construct parallelogram WXYZ given only diagonal WY . ___ 2. Describe how you could construct rhombus RHOM given only diagonal RO . ___ 3. Describe how you could construct kite KITE given only diagonal KT . 8 © 2010 Carnegie Learning, Inc. Be prepared to share your solutions and methods. Lesson 8.6 | Quadrilateral Family 471 Chapter 8 Checklist KEY TERMS l l base angles of a trapezoid (8.3) isosceles trapezoid (8.3) l l biconditional statement (8.3) interior angle of a polygon (8.4) l exterior angle of a polygon (8.5) l trapezoid (8.3) isosceles trapezoid (8.3) THEOREMS l Perpendicular/Parallel Line Theorem (8.1) l Parallelogram/CongruentParallel Side Theorem (8.2) l rhombus (8.2) kite (8.3) CONSTRUCTIONS l l l 8 square (8.1) rectangle (8.1) parallelogram (8.2) 8.1 l l Using the Perpendicular/Parallel Line Theorem The Perpendicular/Parallel Line Theorem states: “If two lines are perpendicular to the same line, then the two lines are parallel to each other.” p m n Because line p is perpendicular to line m and line p is perpendicular to line n, lines m and n are parallel. 472 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. Example: 8.1 Determining Properties of Squares A square is a quadrilateral with four right angles and all sides congruent. You can use the Perpendicular/Parallel Line Theorem and congruent triangles to determine the following properties of squares. • The diagonals of a square are congruent. • Opposite sides of a square are parallel. • The diagonals of a square bisect each other. • The diagonals of a square bisect the vertex angles. • The diagonals of a square are perpendicular to each other. Example: For square PQRS, the following statements are true: ___ ___ • PR QS ___ ___ Q ___ R ___ • PQ || RS and PS || QR ___ ___ ___ ___ • PT RT and QT ST • PQS RQS, QRP SRP, RSQ PSQ, and SPR QPR ___ T P S ___ • PR QS 8.1 8 Determining Properties of Rectangles © 2010 Carnegie Learning, Inc. A rectangle is a quadrilateral with opposite sides congruent and with four right angles. You can use the Perpendicular/Parallel Line Theorem and congruent triangles to determine the following properties of rectangles. • Opposite sides of a rectangle are congruent. • Opposite sides of a rectangle are parallel. • The diagonals of a rectangle are congruent. • The diagonals of a rectangle bisect each other. Example: For rectangle FGHJ, the following statements are true. ___ ___ ___ ___ • FG HJ and FJ GH ___ ___ ___ G H ___ • FG || HJ and FJ || GH ___ ___ ___ ___ • FH GJ ___ ___ K • GK JK and FK HK F Chapter 8 J | Checklist 473 8.2 Determining Properties of Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel. You can use congruent triangles to determine the following properties of parallelograms. • Opposite sides of a parallelogram are congruent. • Opposite angles of a parallelogram are congruent. • The diagonals of a parallelogram bisect each other. Example: For parallelogram WXYZ, the following statements are true. ____ ___ ____ ___ • WX YZ and WZ XY Y X V • WXY WZY and XYZ XWZ ____ ___ ___ ___ • WV YV and XV ZV 8.2 W Z Using the Parallelogram/Congruent-Parallel Side Theorem The Parallelogram/Congruent-Parallel Side Theorem states: “If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.” 8 Example: ___ ___ ___ C ___ In quadrilateral ABCD, AB CD and AB || CD. So, quadrilateral ABCD is a parallelogram, and thus has all of the properties of a parallelogram. 5 cm B D A 474 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 5 cm 8.2 Determining Properties of Rhombi A rhombus is a quadrilateral with all sides congruent. You can use congruent triangles to determine the following properties of rhombi. • Opposite angles of a rhombus are congruent. • Opposite sides of a rhombus are parallel. • The diagonals of a rhombus are perpendicular to each other. • The diagonals of a rhombus bisect each other. • The diagonals of a rhombus bisect the vertex angles. Example: For rhombus ABCD, the following statements are true: • ABC CDA and BCD DAB ___ ___ ___ B C ___ • AB || CD and BC || DA ___ ___ ___ ___ • AC BD ___ X ___ • AX CX and BX DX • BAC DAC, ABD CBD, BCA DCA, and CDB ADB 8.3 A D 8 Determining Properties of Kites © 2010 Carnegie Learning, Inc. A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite sides that are not congruent. You can use congruent triangles to determine the following properties of kites. • One pair of opposite angles of a kite is congruent. • The diagonals of a kite are perpendicular to each other. • The diagonal that connects the opposite vertex angles that are not congruent bisects the diagonal that connects the opposite vertex angles that are congruent. • The diagonal that connects the opposite vertex angles that are not congruent bisects the vertex angles. Example: For kite KLMN, the following statements are true: M • LMN LKN ____ ___ ___ ____ • KM LN L N P • KP MP • KLN MLN and KNL MNL K Chapter 8 | Checklist 475 8.3 Determining Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a trapezoid with congruent non-parallel sides. You can use congruent triangles to determine the following properties of isosceles trapezoids: • The base angles of a trapezoid are congruent. • The diagonals of a trapezoid are congruent. Example: For isosceles trapezoid PQRS, the following statements are true: • QPS RSP ___ Q R ___ • PR QS T P 8.4 S Determining the Sum of the Interior Angle Measures of Polygons You can calculate the sum of the interior angle measures of a polygon by using the formula 180°(n ⫺ 2), where n is the number of sides of the polygon. You can calculate the measure of each interior angle of a regular polygon by dividing the formula by n, the number of sides of the regular polygon. 8 Examples: The sum of the interior angle measures of a pentagon is 180°(5 ⫺ 2) ⫽ 540°. 720° ⫽ 120°. Each interior angle of a regular hexagon measures _____ 6 The sum of the interior angle measures of a decagon is 180°(10 ⫺ 2) ⫽ 1440°. 1440° ⫽ 144°. Each interior angle of a regular decagon measures ______ 10 The sum of the interior angle measures of an 18-gon is 180°(18 ⫺ 2) ⫽ 2880°. 2880° ⫽ 160°. Each interior angle of a regular 18-gon measures ______ 18 476 Chapter 8 | Quadrilaterals © 2010 Carnegie Learning, Inc. 540° ⫽ 108°. Each interior angle of a regular pentagon measures _____ 5 The sum of the interior angle measures of a hexagon is 180°(6 ⫺ 2) ⫽ 720°. 8.5 Determining the Sum of the Exterior Angle Measures of Polygons You can use the formula for the sum of the interior angle measures of a polygon and the Linear Pair Postulate to determine that the sum of the exterior angle measures of any polygon is 360°. Examples: You can use the formula for the sum of the interior angle measures of a polygon to determine that the interior angle measures of the hexagon is 720°. Then, you can use the Linear Pair Postulate to determine that the sum of the angle measures formed by six linear pairs is 6(180°) ⫽ 1080°. Next, subtract the sum of the interior angle measures from the sum of the linear pair measures to get the sum of the exterior angle measures: 1080° ⫺ 720° ⫽ 360°. 8 © 2010 Carnegie Learning, Inc. You can use the formula for the sum of the interior angle measures of a polygon to determine that the interior angle measures of the nonagon is 1260°. Then, you can use the Linear Pair Postulate to determine that the sum of the angle measures formed by nine linear pairs is 9(180°) ⫽ 1620°. Next, subtract the sum of the interior angle measures from the sum of the linear pair measures to get the sum of the exterior angle measures: 1620° ⫺ 1260° ⫽ 360°. Chapter 8 | Checklist 477 8.6 Identifying Characteristics of Quadrilaterals The table shows the characteristics of special types of quadrilaterals. Rhombus Rectangle Square • • • • One pair of sides are both congruent and parallel • • • • Two pairs of opposite sides are congruent • • • • Exactly one pair of parallel sides Kite Parallelogram Trapezoid Two pairs of parallel sides No parallel sides • • Exactly one pair of opposite angles are congruent • Two pairs of opposite angles are congruent • • • • Consecutive angles are supplementary • • • • Diagonals bisect each other • • • • All sides are congruent 8 Diagonals are perpendicular to each other Diagonals bisect the vertex angles • • • • • • • • • Diagonals are congruent • • © 2010 Carnegie Learning, Inc. All angles are congruent 478 Chapter 8 | Quadrilaterals