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Quadrilaterals

Quadrilaterals. Chapter 8. 8.1 – Find Angle Measures in Polygons. Two vertices that are endpoints of the same side are called consecutive vertices in polygons Diagonal Segment that joins two non-consecutive vertices Theorem 8.1 – Polygon Interior Angles Theorem

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Quadrilaterals

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  1. Quadrilaterals Chapter 8

  2. 8.1 – Find Angle Measures in Polygons • Two vertices that are endpoints of the same side are called consecutive vertices in polygons • Diagonal • Segment that joins two non-consecutive vertices • Theorem 8.1 – Polygon Interior Angles Theorem • The sum of the measures of the interior angles of a convexn-gon is (n – 2)*180owhere n is the number of sides • Corollary to Thrm 8.1 - Interior angles of a quadrilateral: • Sum of measures of interior angles of a quadrilateral is 360degrees

  3. Examples • Example 1 • Find the sum of the measures of the interior angles of a convex octagon • Example 2 • The sum of the measures of the interior angles of a convex polygon is 900 degrees. Classify the polygon by the number of sides • Example 3 • Find the value of x (on board) • GP #1-4

  4. Exterior Angles • Sum of exterior angle measures does not depend on number of sides of polygon • Theorem 8.2 – Polygon Exterior Angles Theorem • Sum of measures of exterior angles of a convex polygon, one angle at each vertex, is 360 degrees • Example 4 • What is the value of x ? (on board)

  5. Example 5 • If you have a trampoline in the shape of a regular dodecagon, find the following • Measure of each interior angle • Measure of each exterior angle • GP #5-6

  6. 8.2 – Use Properties of Parallelograms • Parallelogram • Quadrilateral with both pairs of opposite sides parallel • Theorem 8.3 • If a quadrilateral is a parallelogram, then its opposite sides are congruent • Theorem 8.4 • If a quadrilateral is a parallelogram, then its opposite angles are congruent • Example 1: find values of x and y (on board)

  7. Interior Angles • Consecutive interior angle theorem states that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary • This holds true for parallelograms as well • Theorem 8.5 • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary • Theorem 8.6 • If a quadrilateral is a parallelogram, then its diagonals bisect each other

  8. Example 3 • The diagonals of parallelogram LMNO at point P. • What are the coordinates of P? (on board) • GP #1-6

  9. 8.3 – Show a Quad. is a Parallelogram • Converses of theorems 8.3 & 8.4 are stated below • Can be used to show a quadrilateral with certain properties is a parallelogram • Theorem 8.7 • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram • Theorem 8.8 • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

  10. More Theorems! • Theorem 8.9 • If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram • Theorem 8.10 • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram • Example 3 • For what value of x is CDEF a parallelogram? (on board) • GP #2-5

  11. Ways to Prove a Quad. is a Parallelogram • Show both pairs of opposite sides are parallel (DEFINITION) • Show both pairs of opposite sides are congruent (THEOREM 8.7) • Show both pairs of opposite angles are congruent (THEOREM 8.8) • Show one pair of opposite sides are congruent and parallel (THEOREM 8.9) • Show the diagonals bisect each other (THEOREM 8.10)

  12. 8.4 – Properties of Rhombuses, Rectangles, & Squares • Three special types of quadrilaterals exist: • Rhombus • Parallelogram with four congruent sides • Rectangle • Parallelogram with four congruent angles • Square • Parallelogram with four congruent sides and four congruent angles

  13. Corollaries • Rhombus Corollary • A quadrilateral is a rhombus if and only if it has four congruent sides • Rectangle Corollary • A quadrilateral is a rectangle if and only if it has four congruent angles • Square Corollary • A quadrilateral is a square if and only if it is a rhombus and a rectangle

  14. Venn Diagram of Parallelograms

  15. Diagonals of Rhombuses & Rectangles • Theorem 8.11 • A parallelogram is a rhombus if and only if its diagonals are perpendicular • Theorem 8. 12 • A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles • Theorem 8.13 • A parallelogram is a rectangle if and only if its diagonals are congruent

  16. 8.5 – Use Properties of Trapezoids & Kites • Other types of special quadrilaterals exist • Trapezoid • Quadrilateral with exactly one pair of parallel sides • Parallel sides are called bases, non-parallel sides are called legs • Has two pairs of base angles • Example 1 • Show that ORST is a trapezoid (on board)

  17. Isosceles Trapezoids • Isosceles trapezoid • A trapezoid is isosceles when the legs are congruent • Theorem 8.14 • If a trapezoid is isosceles, then each pair of base angles is congruent • Theorem 8.15 • If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid • Theorem 8.16 • A trapezoid is isosceles if and only if its diagonals are congruent

  18. Midsegments • Midsegment of a trapezoid • Segment that connects the midpoints of its legs • Theorem 8.17 – Midsegment Theorem for Trapezoids • The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases (average of the bases) • Example 3 • In the diagram (on board), MN is the midsegment of trapezoid PQRS. Find length of MN

  19. Kites • Kite • Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent • Theorem 8.18 • If a quadrilateral is a kite, then its diagonals are perpendicular • Theorem 8.19 • If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent

  20. Example 4 • Find m <D in the kite (on board) • GP #5 & 6

  21. 8.6 – Identify Special Quadrilaterals

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