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582726022 Name ___________________________ Chapter 5: Discovering and Proving Polygon Properties Lesson 5.1 Polygon Sum Conjecture & Lesson 5.2 Exterior Angles of a Polygon Warm up: Definition: Exterior angle is Measure the interior angles of QUAD to the nearest degree and put the measures into the diagram. Draw one exterior angle at each vertex of QUAD. Measure each exterior angle to the nearest degree and put the measures into the diagram. How could you have calculated the exterior angles if all you had was the interior angles? Are any of the angles equal? What is the sum of the interior angles? What is the sum of the exterior angles? Now repeat the above investigation for the triangle TRI at the right. Compare the different angle sums with the angle sums for the quadrilateral. Do you see a pattern? Are any of the angles equal? What is the sum of the interior angles? What is the sum of the exterior angles? R D Q I T U A S. Stirling Page 1 of 10 582726022 Name ___________________________ Page 258-259 5.1 Investigation: Is there a Polygon Sum Formula? Steps 1-2: Review your work from page 1 and examine the diagrams below. Step 3-5: Complete the sum of the interior angles column and drawing diagonals on the next page. Hexagon JKLMNO Quadrilateral ABCD mBCD = 113 mABC = 77 mDAB = 114 mCDA = 56 mLMN = 105 mMNO = 140 mNOJ = 96 mOJK = 112 mJKL = 159 mKLM = 108 mOJK+mJKL+mKLM+mLMN+mMNO+mNOJ = 720.00 mDAB+mABC+mBCD+mCDA = 360.00 B A L K J M C N D O Oc tagon PQRSTUVW mWPQ = 119 mPQR = 130 mQRS = 154 Pentagon EFGHI mGHI = 157 mIEF = 71 mEFG = 156 mHIE = 112 mFGH = 43 mIEF+mEFG+mFGH+mGHI+mHIE = 540.00 mRST = 132 mUVW = 131 mST U = 131 mVWP = 147 mTUV = 137 S R Q F T G E U P H I V W Page 262-263 5.2 Investigation: Is there an Exterior Angle Sum? Steps 1-5: Review your work from page 1 and examine the diagrams below. Complete the sum of the exterior angles column on the next page. F C E B I mHAB = 67 D mEBC = 103 C H mFCD = 84 A mGDA = 106 G H D mHAB+mEBC+mFCD+mGDA = 360.00 B J E C G G A D K F mFAB = 59 E M A m GAB+m HBC+m ICD+m JDE+m KEF+m MFA = 360 mGBC = 80 mHCD = 104 mIDE = 56 mJEA = 61 J mFAB+mGBC+mHCD+mIDE+mJEA = 360 S. Stirling m JDE = 33 m KEF = 54 m MFA = 72 I B F m GAB = 63 m HBC = 66 m ICD = 73 H Page 2 of 10 582726022 Name ___________________________ (Investigation 5.1 Step 5: Draw all possible diagonals from one vertex, which divides each polygon into triangles. Use these to develop a formula for the Polygon Sum Conjecture. Quadrilateral Pentagon Hexagon Diagonal forms 2 triangles, so Diagonals form __ triangles: Diagonals form __ triangles: ____ (180) = _______ ____ (180) = _______ ____ (180) = _______ Octagon Decagon Diagonals form __ triangles: Diagonals form __ triangles: ____ (180) = _______ ____ (180) = _______ Investigation 5.1 and 5.2 Summary Use the results from Lesson 5.1 and 5.2 to fill in the table at the left. (The last column of the table should be completed after 5.2 Investigation.) Number of sides of a polygon Sum of measures of interior angles Sum of measures of exterior angles (one at a vertex) 3 4 5 6 7 8 9 10 11 12 13 n 180 360 S. Stirling Polygon Sum Conjecture. The sum of the measures of the n angles of an ngon is Exterior Angle Sum Conjecture. The sum of the measures of a set of exterior angles of an n-gon is Page 3 of 10 582726022 Name ___________________________ Page 262-263 5.2 Investigation: Is there an Exterior Angle Sum? Steps 7-8: Use what you know about interior angle sums and exterior angle sums to calculate the measure of each interior and each exterior angle of any equiangular polygon. Try an example first: Find the measure of an interior and an exterior angle of an equiangular pentagon. Show your calculations below: One interior angle = One exterior angle = What is the relationship between one interior and one exterior angle? Equiangular Polygon Conjecture You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas: You can find the measure of each exterior angle of an equiangular n-gon by using the formula: One exterior angle = What is the relationship between one interior and one exterior angle? Use this relationship to find the measure of one interior angle. Use the formula to find the measure of one interior angle. Same results? S. Stirling Page 4 of 10 582726022 Lesson 5.3 Kite and Trapezoid Properties Definition of kite Name ___________________________ Label vocab in the drawing.: K E I Measure then compare the opposite angles of the kite. Which pair will be congruent? I T vertex angles (of a kite) The angles of congruent sides. mDIA = 90 nonvertex angles (of a kite) The two T angles between consecutive noncongruent The diagonals of the kite (c ompared to eachother): sides of a kite. DI = 1.57 cm IB = 1.57 cm Kites K between thekite: pairs Angle formed by the diagonals of the E IA = 1.68 cm CI = 4.57 cm Kites Kite Angles Conjecture (Dodiagonals proof onof Ch WS page 2.) Angle formed by the the5kite: The angles of the kite: mDIA = 90 The angles of a kite mDCB are = 38 mCBA = 118 The diagonals of the kite (c ompared mBAD to eachother): = 86 mADC = 118 Kites DI = 1.57 cm IB = 1.57 cm Additional measures: Label the diagram with the measures to help you write the conjectures. Angle formed by the diagonals of the kite: The diagonals of the kite and how they divide the angles of the kite: IA = 1.68 cm CI = 4.57 cm D mDIA = 90 The vertex angles: The angles of the kite: The diagonals of the kite (c ompared to eachother): mDCB = 38 DI = 1.57 cm IB = 1.57 cm mBAD = 86 IA = 1.68 cm CI = 4.57 cm The nonvertex angles: mDCI = 19 mCDI = 71 mCBA = 118 mICB = 19 mADC = 118 mIDA = 47 mBAI = 43 mIBA = 47 mIAD = 43 mIBC The diagonals of the kite and how they divide the angles of the kite: = 71 The angles of the kite: The vertex angles: The nonvertex angles: mDCB = 38 mBAD = 86 mCBA = 118 mDCI = 19 mADC = 118 mCDI = 71 mICB = 19 mIDA = 47 A I The diagonals of the kite and how angles of the kite: mBAI = 43they divide the mIBA = 47 The vertex angles: B mIAD = 43 The nonvertex angles:mIBC = 71 mDCI = 19 mCDI = 71 mICB = 19 mIDA = 47 C Kite Angle Conjecture mBAI = 43 Bisector mIBA = 47 mIAD = 43 anglesmIBC 71 are The vertex of a =kite Kite Diagonals Conjecture The diagonals of a kite are (Do proof on Ch 5 WS page 2.) by a (Do proof on Ch 5 WS page 3.) Kite Diagonal Bisector Conjecture (Do proof on Ch 5 WS page 3.) The diagonal connecting the vertex angles of a kite is the of the other diagonal. S. Stirling Page 5 of 10 582726022 Name ___________________________ Definition of trapezoid Label vocab in the drawing: Definition of isosceles trapezoid C B Measure the angles of the trapezoids below. Label the diagram with the measures to help you write the conjectures. B D mDC A = 59 D mCAB = 121 mBDC = 139 mABD = 41 A bases (of a trapezoid) The two parallel sides. base angles (of a trapezoid) A pair of angles with a base of the trapezoid as a common side. legs are the two nonparallel sides. C mABC = 143 mDAB = 37 mBCD = 143 mCDA = 37 B A D A R C Trapezoid Consecutive Angles Conjecture The consecutive angles between the bases of a trapezoid are A P T Isosceles Trapezoid [Base Angles] Conjecture The base angles of an isosceles trapezoid are I S C Measure diagonals of the trapezoids below. Isosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are S. Stirling (Do proof on Ch 5 WS page 4.) Page 6 of 10 O 582726022 Name ___________________________ Lesson 5.4 Properties of Midsegments Page 275-276 Investigation 1: Triangle Midsegment Properties Midsegment (of a triangle) is the line segment connecting the midpoints of the two sides. Steps 1 – 3: C A Three Midsegments Conjecture The three midsegments of a triangle divide it into B Steps 4 – 5: (Review Corresponding Angles Conjecture for parallel lines. The F shape!) Also label the drawing below. R Triangle Midsegment Conjecture A midsegment of a triangle is third side and the length of to the I T Page 276-277 Investigation 2: Trapezoid Midsegment Properties. Midsegment (of a trapezoid) is the line segment connecting the midpoints of the two nonparallel sides. P Steps 1 – 8: If you do not have tracing paper, you may measure instead. Label angles with measures!! A N M R T Trapezoid Midsegment Conjecture The midsegment of a trapezoid is to the bases and is equal in length to Side lengths: S. Stirling Page 7 of 10 582726022 Lesson 5.5 Properties of Parallelograms Name ___________________________ Page 281-282 Investigation: Four Parallelogram Properties Definition of a Parallelogram: A quadrilateral with two pairs of opposite sides parallel. Steps 1 – 4: Angles! Parallelogram Opposite Angles Conjecture (Do proof on Ch 5 WS page 5.) The opposite angles of a parallelogram L M J K Parallelogram Consecutive Angles Conjecture The consecutive angles of a parallelogram So to find all angles of a parallelogram: Steps 5 – 7: Side and diagonal lengths! P A Parallelogram Opposite Sides Conjecture (Do proof on Ch 5 WS page 5.) The opposite sides of a parallelogram Parallelogram Diagonals Conjecture (Do proof on Ch 5 WS page 6.) The diagonals of a parallelogram L S. Stirling R Page 8 of 10 582726022 5.6 Properties of Special Parallelograms Name ___________________________ Definition of a Rhombus: A quadrilateral with all sides congruent. Page 291 Investigation 1: What Can You Draw with the Double-Edged Straightedge? Steps 1 – 3: Complete in the space below. Use your ruler! Double-Edged Straightedge Conjecture If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a Page 292 Investigation 2: Do Rhombus Diagonals Have Special Properties? Steps 1 – 3: H R O M Rhombus Diagonals Conjecture The diagonals of a rhombus are (Do proof on Ch 5 WS page 7.) Rhombus Diagonals Angles Conjecture The of a rhombus S. Stirling , and they (Do proof on Ch 5 WS page 6.) the angles of the rhombus. Page 9 of 10 582726022 Name ___________________________ Definition of a Rectangle: A quadrilateral with all angles congruent. Page 293-4 Investigation 3: Do Rectangle Diagonals Have Special Properties? Steps 1 – 2: R E Rectangle Diagonals Conjecture The diagonals of a rectangle are and T C Definition of a Square: A quadrilateral with all angles and sides congruent Square Diagonals Conjecture The diagonals of a square are A U S Q S. Stirling Page 10 of 10