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Polygons and Angles: Exploring Quadrilaterals, Rectangles, Rhombi, and Squares

Discover the properties of different types of polygons, including quadrilaterals, rectangles, rhombi, and squares, and learn how to calculate interior and exterior angles. Explore the concepts of parallel lines and diagonals within these polygons.

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Polygons and Angles: Exploring Quadrilaterals, Rectangles, Rhombi, and Squares

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  1. Chapter 6 Quadrilaterals

  2. Types of Polygons • Triangle – 3 sides • Quadrilateral – 4 sides • Pentagon – 5 sides • Hexagon – 6 sides • Heptagon – 7 sides • Octagon – 8 sides • Nonagon – 9 sides • Decagon – 10 sides • Dodecagon – 12 sides • All other polygons = n-gon

  3. Interior Angle Sum Theorem The sum of the measures of the interior angles of a polygon is found by S=180(n-2) Ex: Hexagon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of a polygon is 360 no matter how many sides. Lesson 6.1 : Angles of Polygons

  4. Find the measure of an interior and an exterior angle for each polygon. 24-gon 3x-gon Find the measure of an exterior angle given the number of sides of a polygon 260 sides Lesson 6.1 : Angles of Polygons

  5. The measure of an interior angle of a polygon is given. Find the number of sides. 175 168.75 A pentagon has angles (4x+5), (5x-5), (6x+10), (4x+10), and 7x. Find x. Lesson 6.1: Angles of Polygons

  6. A. Find the value of x in the diagram.

  7. Lesson 6.2: Parallelograms Opposite angles in a parallelogram are congruent Opposite sides of a parallelogram are congruent Consecutive angles in a parallelogram are supplementary Properties of Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel If a parallelogram has 1 right angle, it has 4 right angles. The diagonals of a parallelogram split it into 2 congruent triangles The diagonals of a parallelogram bisect each other

  8. ? ? ____ ?

  9. A. ABCD is a parallelogram. Find AB. B. ABCD is a parallelogram. Find mC. C. ABCD is a parallelogram. Find mD.

  10. A. If WXYZ is a parallelogram, find the value of r, s and t.

  11. What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?

  12. Lesson 6.3 : Tests for Parallelograms • If… • Both pairs of opposite sides are parallel • Both pairs of opposite sides are congruent • Both pairs of opposite angles are congruent • The diagonals bisect each other • One pair of opposite sides is congruent and parallel • Then the quadrilateral is a parallelogram

  13. Determine whether the quadrilateral is a parallelogram. Justify your answer.

  14. Which method would prove the quadrilateral is a parallelogram?

  15. Determine whether the quadrilateral is a parallelogram.

  16. Determine whether the quadrilateral is a parallelogram.

  17. Find x and y so that the quadrilateral is a parallelogram.

  18. COORDINATE GEOMETRYGraph quadrilateral QRST with vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

  19. Given quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram. (The graph does not determine for you)

  20. 6.4-6.6 Foldable • Fold the construction paper in half both length and width wise • Unfold the paper and hold width wise • Fold the edges in to meet at the center crease • Cut the creases on the tabs to make 4 flaps

  21. Characteristics of a rectangle: Both sets of opp. Sides are congruent and parallel Both sets opp. angles are congruent Diagonals bisect each other Diagonals split it into 2 congruent triangles Consecutive angles are supplementary If one angle is a right angle then all 4 are right angles In a rectangle the diagonals are congruent. If diagonals of a parallelogram are congruent, then it is a rectangle. Lesson 6.4 : Rectangles

  22. Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

  23. Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

  24. Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.

  25. Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

  26. A quadrilateral with 4 congruent sides Characteristics of a rhombus: Both sets of opp. sides are congruent and parallel Both sets of opp. angles are congruent Diagonals bisect each other Diagonals split it into 2 congruent triangles Consecutive angles are supplementary If an angle is a right angle then all 4 angles are right angles In a rhombus: Diagonals are perpendicular Diagonals bisect the pairs of opposite angles Lesson 6.5 : Rhombi(special type of parallelogram)

  27. A quadrilateral with 4 congruent sides Characteristics of a square: Both sets of opp. sides are congruent and parallel Both sets of opp. angles are congruent Diagonals bisect each other Diagonals split it into 2 congruent triangles Consecutive angles are supplementary If an angle is a right angle then all 4 angles are right angles Diagonals bisect the pairs of opposite angles A square is a rhombus and a rectangle. 6.5: Squares (special type of parallelogram)

  28. A. The diagonals of rhombus WXYZ intersect at V.If mWZX = 39.5, find mZYX. B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

  29. A. ABCD is a rhombus. Find mCDB if mABC = 126. B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

  30. QRST is a square. Find n if mTQR = 8n + 8.

  31. QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11.

  32. Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

  33. A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of non-parallel sides (legs) Isosceles Trapezoid: A trapezoid with congruent legs and congruent base angles Diagonals of an isosceles trapezoid are congruent Median (of a trapezoid): The segment that connects the midpoints of the legs The median is parallel to the bases 6.6: Trapezoids A B AC = BD base D C leg leg Base angle Base angle base Median = ½ (base + base)

  34. A.Each side of the basket shown is an isosceles trapezoid. If mJML= 130, KN = 6.7 feet, and LN = 3.6 feet, find mMJK. B. Each side of the basket shown is an isosceles trapezoid. If mJML= 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.

  35. In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

  36. WXYZ is an isosceles trapezoid with medianFind XY if JK = 18 and WZ = 25.

  37. A. If WXYZ is a kite, find mXYZ.

  38. A. If WXYZ is a kite, find mXYZ.

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