Polygon Interior Angles

Transcription

1 Polygons can be named by the number of sides. A regular polygon has All other polygons are irregular. A concave polygon has All other polygons are convex, with all vertices facing outwards. Name each polygon by the number of sides. # of sides n Name Tell whether the following polygons are regular or irregular, and whether they are convex or concave The sum of the interior angle measures of a convex polygon with n sides is: Polygon Interior Angles The measure of each angle of a regular convex polygon with n sides is: Find the sum of the interior angle measures of each convex polygon. 8. pentagon 9. Octagon 10. nonagon

2 Find the measure of each interior angle of each regular polygon. 11. pentagon 12. heptagon gon Name the convex polygon whose interior angle measures have the given sum Name the convex polygon whose interior angles each have the given measure Polygon Exterior Angles Sum The sum of the exterior angle measures of a convex polygon is: Find the measure of each exterior angle of each regular polygon. 18. pentagon 19. quadrilateral 20. octagon Name the regular polygon that has the given exterior angle. Then find the measure of each interior angle Find the value of each variable. Then find the measure of each angle

3 Properties of Parallelograms is a right angle, so K, L, and M are right angles. J AB DC and AD BC Find each measure. 1. AB = m A= 3. m D = 4. ML = 5. LP = 6. m LPM 7. m MLN = 8. QN = = H 9. GH = 10. EF = 11. m F = 12. m E = G

4 13. Determine the coordinates of the intersection of the diagonals of FGHJ with vertices F( 2, 4), G(3, 5), H(2, 3), and J( 3, 4). The perimeter of PQRS is 84. Find the length of each side under the given conditions. 14. PQ = QR 15. RS = SP 7 Complete each statement about KMPR. Justify each answer. 16. PRK _ 17. PR 18. MT 19. MK 20. m MKR m PRK _ Find them values of x, y, and z in each parallelogram

5 You can use the following conditions to determine whether a quadrilateral is a parallelogram. Conditions for Parallelograms Determine whether each quadrilateral must be a parallelogram. Justify your answer

6 Show that the quadrilateral with the given vertices is a parallelogram by using the given method. 8. J( 2, 2), K( 3, 3), L(1, 5), M(2, 0) 9. N(5, 1), P(2, 7), Q(6, 9), R(9, 3) Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Find the values of a and b that would make the quadrilateral a parallelogram EFGH is a quadrilateral. Which pieces of information would allow you to conclude that EFGH is a parallelogram? Hint draw a picture for each! a) m E = 125, m F = 55, m G = 125 b) EF GH and FG EH c) EF FG and GH EH d) E G, F H

7 Rectangle Rectangles have all the properties of parallelograms, plus two more: Properties of Rectangles BD AC ABCD is a rectangle. Find each length. 1. BD = 2. CD = 3. AC = 4. AE = 5. Find the measure of each numbered angle in the rectangle. m 1 m 2 m 3 m 4 m 5 6. JKLM is a rectangle. Find x when m KJL 2x and m JLK 7x 5.

8 7. Quadrilateral PQRS has vertices P( 5, 3), Q(1, 1), R( 1, 4), and S( 7, 0). Determine whether PQRS is a rectangle by using the distance formula. 8. Quadrilateral JKLM has vertices J( 10, 2), K( 8, 6), L(5, 3), and M(2, 5). Determine whether JKLM is a rectangle by using the slope formula.

9 Rhombus Rhombi have all the properties of parallelograms, plus two more: Properties of a rhombus KLMN is a rhombus. Find each measure. 1. KL = 2. m MNK = Find the measures of the numbered angles in each rhombus Square A square is both a and a.

10 Sometimes, always, or never true? 5. A rectangle is a parallelogram. 6. A rhombus is a square. 7. A parallelogram is a rhombus. 8. A rhombus is a rectangle. 9. A square is a rhombus. 10. A rectangle is a quadrilateral. 11. A square is a rectangle. 12. A rectangle is a square. Tell whether each quadrilateral is a parallelogram, rectangle, rhombus, or square. Give all names that apply Given VWXY with vertices V(3, 0), W(6, 4), X(11, 4), and Y(8, 0). Use the diagonals to determine whether it is a rectangle, rhombus, or square. Give all names that apply. 18. Given ABCD with vertices A( 5, 3), B(1, 1), C( 1, 4), and D( 7, 0). Use the diagonals to determine whether it is a rectangle, rhombus, or square. Give all names that apply.

11 Trapezoid Isosceles trapezoid Isosceles Trapezoid Theorems If a trapezoid is isosceles, then each pair of base angles is congruent. If a trapezoid has one pair of congruent base angles, then it is isosceles. A trapezoid is isosceles if and only if its diagonals are congruent. Each parallel side is a base. 1. m J 2. If FH = 9, then AG = 3. m T 4. AC = 2z+ 9, and BD = 4z 3. Find the value of z. Trapezoid Midsegment Theorem Trapezoid midsegment The midsegment is parallel to each base. The length of the midsegment is the average length of the bases (one half the sum of the bases).

12 5. KL = 6. PQ = 7. EF = E F Kite The diagonals are perpendicular. Properties of Kites Exactly one pair of opposite angles is congruent. In kite ABCD, m BCD 98, m ADE 47, BC = 21, and AD = m BAD 14. m BCE 10. m CDE 15. m ABC 11. m AED 16. m ABE 12. m DAE 17. AB = 13. m DCE 18. CD = Find the measure of each numbered angle Parallelograms have pairs of parallel sides. Trapezoids have pairs of parallel sides. Kites have pairs of parallel sides.