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7 Quadrilaterals and Other Polygons 7 Quadrilaterals and Other Polygons 7.1 Angles of Polygons 7.2 Properties of Parallelograms 7.3 Proving That a Quadrilateral Is a Parallelogram 7.4 Properties of Special Parallelograms 7.5 Properties of Trapezoids and Kites SEE the Big Idea DiamondDiamond (p.((p. 406)406) WiWindow d ((p. 39395)5) AmAmusementusement Park Ride Ride (p(p. 377) ArArrowrow (p(p.. 37373)3) GazeboGazebo (p.(p. 36365)5) hhs_geo_pe_07co.indds_geo_pe_07co.indd 356356 11/19/15/19/15 111:411:41 AAMM MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Using Structure to Solve a Multi-Step Equation Example 1 Solve 3(2 + x) = −9 by interpreting the expression 2 + x as a single quantity. 3(2 + x) = −9 Write the equation. 3(2 + x) −9 — = — Divide each side by 3. 3 3 2 + x = −3 Simplify. −2 −2 Subtract 2 from each side. x = −5 Simplify. Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 4(7 − x) = 16 2. 7(1 − x) + 2 = −19 3. 3(x − 5) + 8(x − 5) = 22 Identifying Parallel and Perpendicular Lines Example 2 Determine which of the lines are parallel and which are perpendicular. a b y c Find the slope of each line. (−4, 3) d 3 − (−3) (2, 2) Line a: m = — = −3 2 (3, 2) −4 − (−2) −1 − (−4) Line b: m = — = −3 −2 (1, −1) 4 x 1 − 2 (−4, 0) (4, −2) − − −2 = —2 ( 2) = − Line c: m − 4 3 4 (−2, −3) (2, −4) −4 2 − 0 1 Line d: m = — = — 2 − (−4) 3 Because lines a and b have the same slope, lines a and b are parallel. Because 1 — − = − 3 ( 3) 1, lines a and d are perpendicular and lines b and d are perpendicular. Determine which lines are parallel and which are perpendicular. 4. a y d 5. c y b 6. (−4, 4) b y c 4 4 a 4 (3, 4) (−2, 4) (0, 3) (−4, 2) (−2, 2) d 2 d 2 (−2, 2) (4, −2) (0, 1) (3, 1) c (−3, 0) (1, 0) (2, 1) (3, 1) − x x − x 4 − − 1 3 4 2 4 b ( 3, 3) −2 −2 (−3, −2) −2 (2, −3) (−3, −2) (0, −4) (0, −3) (−2, −2) a (3, −3) (4, −4) (4, −4) 7. ABSTRACT REASONING Explain why interpreting an expression as a single quantity does not contradict the order of operations. Dynamic Solutions available at BigIdeasMath.com 357 hhs_geo_pe_07co.indds_geo_pe_07co.indd 357357 11/19/15/19/15 111:411:41 AAMM MMathematicalathematical PPracticesractices Mathematically profi cient students use diagrams to show relationships. Mapping Relationships CCoreore CConceptoncept Classifi cations of Quadrilaterals quadrilaterals parallelograms trapezoids rhombusessquares rectangles kites Writing Statements about Quadrilaterals Use the Venn diagram above to write three true statements about different types of quadrilaterals. SOLUTION Here are three true statements that can be made about the relationships shown in the Venn diagram. • All rhombuses are parallelograms. • Some rhombuses are rectangles. • No trapezoids are parallelograms. MMonitoringonitoring PProgressrogress Use the Venn diagram above to decide whether each statement is true or false. Explain your reasoning. 1. Some trapezoids are kites. 2. No kites are parallelograms. 3. All parallelograms are rectangles. 4. Some quadrilaterals are squares. 5. Example 1 lists three true statements based on the Venn diagram above. Write six more true statements based on the Venn diagram. 6. A cyclic quadrilateral is a quadrilateral that can be circumscribed by a circle so that the circle touches each vertex. Redraw the Venn diagram so that it includes cyclic quadrilaterals. 358 Chapter 7 Quadrilaterals and Other Polygons hhs_geo_pe_07co.indds_geo_pe_07co.indd 358358 11/19/15/19/15 111:411:41 AAMM 7.1 Angles of Polygons EEssentialssential QQuestionuestion What is the sum of the measures of the interior angles of a polygon? The Sum of the Angle Measures of a Polygon Work with a partner. Use dynamic geometry software. a. Draw a quadrilateral and a pentagon. Find the sum of the measures of the interior angles of each polygon. Sample B G F A C H E I D b. Draw other polygons and fi nd the sums of the measures of their interior angles. Record your results in the table below. CONSTRUCTING VIABLE ARGUMENTS Number of sides, n 3 456789 S To be profi cient in math, Sum of angle measures, you need to reason c. Plot the data from your table in a coordinate plane. inductively about data. d. Write a function that fi ts the data. Explain what the function represents. Measure of One Angle in a Regular Polygon Work with a partner. a. Use the function you found in Exploration 1 to write a new function that gives the measure of one interior angle in a regular polygon with n sides. b. Use the function in part (a) to fi nd the measure of one interior angle of a regular pentagon. Use dynamic geometry software to check your result by constructing a regular pentagon and fi nding the measure of one of its interior angles. c. Copy your table from Exploration 1 and add a row for the measure of one interior angle in a regular polygon with n sides. Complete the table. Use dynamic geometry software to check your results. CCommunicateommunicate YourYour AnswerAnswer 3. What is the sum of the measures of the interior angles of a polygon? 4. Find the measure of one interior angle in a regular dodecagon (a polygon with 12 sides). Section 7.1 Angles of Polygons 359 hhs_geo_pe_0701.indds_geo_pe_0701.indd 359359 11/19/15/19/15 111:501:50 AAMM 7.1 Lesson WWhathat YYouou WWillill LLearnearn Use the interior angle measures of polygons. Core VocabularyVocabulary Use the exterior angle measures of polygons. diagonal, p. 360 Using Interior Angle Measures of Polygons equilateral polygon, p. 361 p. 361 In a polygon, two vertices that are endpoints of Polygon ABCDE equiangular polygon, the same side are called consecutive vertices. p. 361 C regular polygon, A diagonal of a polygon is a segment that Previous joins two nonconsecutive vertices. B D polygon As you can see, the diagonals from one vertex convex divide a polygon into triangles. Dividing a diagonals interior angles polygon with n sides into (n − 2) triangles A E exterior angles shows that the sum of the measures of the A and B are consecutive vertices.— — interior angles of a polygon is a multiple Vertex B has two diagonals, BD and BE. of 180°. TTheoremheorem Theorem 7.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles n n − ° 2 REMEMBER of a convex -gon is ( 2) ⋅ 180 . 3 1 m∠ + m∠ + . + m∠n = n − ° A polygon is convex when 1 2 ( 2) ⋅ 180 4 no line that contains a side 6 5 of the polygon contains Proof Ex. 42 (for pentagons), p. 365 n = 6 a point in the interior of the polygon. Finding the Sum of Angle Measures in a Polygon Find the sum of the measures of the interior angles of the fi gure. SOLUTION The fi gure is a convex octagon. It has 8 sides. Use the Polygon Interior Angles Theorem. (n − 2) ⋅ 180° = (8 − 2) ⋅ 180° Substitute 8 for n. = 6 ⋅ 180° Subtract. = 1080° Multiply. The sum of the measures of the interior angles of the fi gure is 1080°. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 1. The coin shown is in the shape of an 11-gon. Find the sum of the measures of the interior angles. 360 Chapter 7 Quadrilaterals and Other Polygons hhs_geo_pe_0701.indds_geo_pe_0701.indd 360360 11/19/15/19/15 111:501:50 AAMM Finding the Number of Sides of a Polygon The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides. SOLUTION Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to fi nd the number of sides. (n − 2) ⋅ 180° = 900° Polygon Interior Angles Theorem n − 2 = 5 Divide each side by 180°. n = 7 Add 2 to each side. The polygon has 7 sides. It is a heptagon. CCorollaryorollary Corollary 7.1 Corollary to the Polygon Interior Angles Theorem The sum of the measures of the interior angles of a quadrilateral is 360°. Proof Ex. 43, p. 366 Finding an Unknown Interior Angle Measure x 108° 121° Find the value of in the diagram. SOLUTION x° 59° The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation. x° + 108° + 121° + 59° = 360° Corollary to the Polygon Interior Angles Theorem x + 288 = 360 Combine like terms. x = 72 Subtract 288 from each side. The value of x is 72. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 2. The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides. 3. The measures of the interior angles of a quadrilateral are x°, 3x°, 5x°, and 7x°. Find the measures of all the interior angles. In an equilateral polygon, In an equiangular A regular polygon is all sides are congruent. polygon, all angles in the a convex polygon that interior of the polygon are is both equilateral and congruent. equiangular. Section 7.1 Angles of Polygons 361 hhs_geo_pe_0701.indds_geo_pe_0701.indd 361361 11/19/15/19/15 111:501:50 AAMM Finding Angle Measures in Polygons A home plate for a baseball fi eld is shown. AB a. Is the polygon regular? Explain your reasoning.
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