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- 1. CHAPTER III: POLYGONS3.1 INTRODUCING POLYGONS.Objectives To identify and name polygons and their parts To identify and draw convex and concave polygons To identify regular polygonsA polygon is the union of three or more coplanar segments such that: • each endpoint is shared by exactly two segments; • segments intersect only at their endpoints; • intersecting segments are noncollinear. Number of side Name Number of side Name 3 Triángulo 8 Octágono 4 Cuadrilátero 9 Nonagono 5 Pentágono 10 Decágono 6 Hexágono 11 Endecágono 7 Heptágono 12 Dodecágono 1
- 2. A diagonal of a polygon is a segment joining two nonconsecutive vertices.A polygon is convex if the segment XY joining any two interior points of the polygon isin the interior of the polygon. If a polygon is not convex, then it concave. Convex polygons Concave PolygonsA regular polygon is a convex polygon that is both equilateral and equiangular. 2
- 3. EXERCISESComplete the statement for polygon PQRST. S RS and _____________ are adjacent sides. T R TP and ______________ are nonadjacent sides < S and ______________ are consecutive angles < P and ______________ are nonconsecutive angles P Q Name the polygon. State whether it is convex or concave.Complete the table.Number of side of polygon Number of diagonals from Number of diagonals one vertex 3 4 5 6 7 50 n 3
- 4. 3.2 INTERIOR ANGLES OF POLYGONSObjectives To find the sum of the angle measures of a convex polygon To find angle measures and the number of sides of polygons To find angle measures and the number of sides of regular polygons. Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2) 180. Corollary The sum of the measures of the interior angles of a convex quadrilateral is 360°. The measure o fan angle of a regular polygon with n sides is (n-2)180 n Polygon Number of sides Number of Sum of Angle Triangles Measures Quadrilateral 4 2 2 . 180 = 360 Pentagon 5 3 3 . 180 = 540 Hexagon 6 4 4 . 180 = 720 Heptagon 7 5 5 . 180 = 900 Octagon 8 6 6 . 180 = 1,080EXERCISES.Find the sum of the measures of the angles of the convex polygon. 1. a pentagon 2. a hexagon 3. a decagon 4. a 30-gon 5. a 62-gon 6. a 100-gon 4
- 5. The sum of the measures of the angles of a convex polygon is given. Find the numberof sides of the polygon. 1. 900 2. 1260 3. 1980 4. 3600 5. 4500 6. 7560Find the measure of an angle of the regular polygon. 1. a square 2. a pentagon 3. a decagon 4. a 20-gon 5. a 30-gon 6. a 100-gonEach of a regular polygon has the given measure. How many sides does the polygonhave? 1. 60 2. 135 3. 108Find the measure of each angle of quadrilateral ABCD. Draw the figure. 5
- 6. • m < A = 10x, m < B = 6x + 10, m < C = 12x – 10, m < D = 8x • m < A = 8x + 5, m < B = 10x + 5, m < C = 10x – 8, m < D = 13x – 11.Solve the problems. 1. the sum of the measures of four angle of a pentagon is 498. what is the measure of the unknown angle? 2. the sum of the measures of nine angles of a decagon is 1320. What is the measure of the missing angle? 3. Three angle of a hexagon are congruent. The other three angles are also congruent. Each of the first three angles has a measure twice that one of the second three angles. What is the measure of each angle of the hexagon? 4. In a pentagon, the measure of one angle is twice that of a second angle. The remaining angles are congruent, each having a measure of three times that of the second angle. What is the measure of each angle of the pentagon? 6
- 7. 3.3 EXTERIOR ANGLES OF POLYGONSObjective To identify the exterior angles of a polygon. To find the measure of exterior angles of polygons To explain why the exterior angles of regular polygon are congruent. 7
- 8. The sumo f the measures of the exterior angles, one at each vertex, of any convex polygon is 360°. The measure of an exterior angle of a regular polygon with n sides is 360. nEXERCISES.Find the measure of each exterior angle of the given regular polygon. 1. a pentagon 2. an octagon 3. a decagon 4. a 15-gonDraw a triangle with its exterior angles. 3.4 QUADRILATERALS AND PARALLELOGRAMSObjectives To identify parts an investigate properties of quadrilaterals To prove an apply theorems about parallelograms 8
- 9. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.( EFGH means parallelogram EFGH). A diagonal of a parallelogram forms two congruent triangles. The diagonals of parallelogram bisect each other. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Consecutive angles of a parallelogram are supplementary.EXERCISESComplete the statement for the parallelogram TUVW. WV is opposite ___________ W T < U and ________ are supplementary. < T = ________ TU and ___________ are adjacent. < W is opposite __________ < T and ________ are consecutive angles. V UEFGH is a parallelogram. Find the unknown measure. 9
- 10. 1. EF = 17, GH = _________ H G2. m < EFG = 67, m < GHE = ___________3. m < HEF = 119, m < GHE = __________4. m < HGF = 125, m < EHG = __________ E F5. EF = 12x, GH = 10x + 12, GH = _______6. EF = 5x – 7, GH = 3x + 1, EF = ________7. EH = 2x + 2, FG = 3x -5, FG = _________8. EO = 3x + 2, GO = 5x – 8, EO = ________9. FO = 4x + 13, HO = 5x + 1, FH = _______10. m < EFG = 6x + 6, m < FGH = 3x + 3, M < FGH = _____________11. m < EFG = 12x -24, m < GHE = 9x + 12,m < EFG = _____________ 10

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