Secondary Math II Honors Unit 4 Notes Polygons Name: Per:
Day 1: Interior and Exterior Angles of a Polygon Unit 4 Notes / Secondary 2 Honors Vocabulary: Polygon: Regular Polygon: Example(s): Discover the formula for the sum of interior angles of a polygon. 1. Determine the sum of the interior angles of a pentagon by completing each step. a. Draw a pentagon. Then draw all possible diagonals using only one vertex of the polygon. b. How many triangles are formed? c. The sum of the interior angles of ONE triangle =. Since a pentagon has triangles, the sum of all interior angles in a pentagon is. 2. Determine the sum of the interior angles of a hexagon by completing each step. a. Draw a hexagon. Then draw all possible diagonals using only one vertex of the polygon. b. How many triangles are formed? c. The sum of the interior angles of ONE triangle =. Since a hexagon has triangles, the sum of all interior angles in a hexagon is. Polygon # of Sides Number of Triangles Sum of interior angles Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon n-gon
Find the sum of the measures of the interior angles of each of the following. 3. 14-gon 4. 25-gon 5. 32-gon Find the value of x in the figures below. 6. 7. The sum of the measures of the interior angles of a polygon is given. Determine the number of sides for each polygon. 8. 1620 9. 2700 10. 1080 For each regular polygon, calculate the measure of each of its interior angles. 11. 12. Vocabulary: Exterior angle of a polygon: Formed adjacent to each interior angle by extending one side of each vertex of the polygon. Interior Angle + Exterior Angle = Example(s): Given each interior angle, calculate the measure of the adjacent exterior angle. 13. interior angle is 100 14. interior angle measures 80
Discover the sum of exterior angles of any polygon: 15. Given the triangle to the right answer the following: a. How many linear pairs are formed? b. What is the sum of the linear pairs? c. What is the sum of all the interior angles? d. What is the sum of the exterior angles? 16. Draw a quadrilateral and extend each side to locate an exterior angle at each vertex (like the picture above). a. How many linear pairs are formed? b. What is the sum of the linear pairs? c. What is the sum of all the interior angles? d. What is the sum of the exterior angles? 17. Complete the table. Number of Sides of the Polygon 3 4 5 6 15 Number of Linear Pairs Formed Sum of Measures of Linear Pairs Sum of Measures of Interior Angles Sum of Measures of Exterior Angles Find the measure of each exterior angle of the regular polygon. 18. 19. 20. If a regular polygon has 100 sides, calculate the measure of each exterior angle.
Calculate the number of sides for each regular polygon given the following information. 21. The measure of each exterior angle is 30 22. The measure of each exterior angle is18. 23. The measure of each interior angle is 156. 24. The measure of each interior angle is 162. Summary: **To find the sum of interior angles use. **The sum of exterior angles is always. **If you have a regular polygon then divided by will give you the measure of each exterior angle. Day 2: Squares, Rectangles, Parallelograms, & Rhombi Vocabulary: Quadrilateral: sided polygon. Parallelogram: A quadrilateral with pairs of sides. Rectangle: A quadrilateral with opposite sides and and all angles are. Square: A quadrilateral with and all sides congruent. Rhombus: A quadrilateral with sides. Properties of a Parallelogram: Opposite sides are congruent Opposite sides are parallel Opposite angles are congruent Diagonals bisect each other Consecutive angles are supplementary + = 180 + = 180 + = 180 + = 180
Find the missing measures using the parallelogram below. 1) x 2) h 3) m B Find the missing measures using the parallelogram below. 4) x = 5) a = 6) f = 7) BD = Properties of a Rectangle: Opposite sides are congruent AND parallel All angles congruent (90 degrees) Consecutive angles are supplementary Diagonals bisect each other Given ABCD is a rectangle determine the following: 8) ADC = 9) DC= 10) BD = 11) DEA =
Properties of a Square: All sides are congruent All angles are congruent Opposite sides are parallel Consecutive angles are supplementary The diagonals are congruent The diagonals bisect each other The diagonals bisect the vertex angles The diagonals are perpendicular to each other Properties of a Rhombus: Opposite angles are congruent Opposite sides are parallel All sides are congruent The diagonals bisect the vertex angles The diagonals bisect each other The diagonals are perpendicular to each other
Use rhombus SPQR to determine the following: 12) SP = 13) If m QPS = 14) Complete a two-column proof to prove that AC BD in square ABCD ** Summary: Replace the question marks with the correct vocabulary term. Use rectangle STUV to answer the following: 15) If m 1 = 30, m 2 = 16) If m 6 = 57, m 4 = 17) If m 8 = 133, m 2 = 18) If m 5 = 16, m 3 =
19) ABCD is a rhombus. If the perimeter of ABCD = 68 in. and BD = 16in, find AC. 20) ABCD is a square. If m DBC = x - 4, find x. Day 3: Properties of Kites and Trapezoids Vocabulary: Kite: A quadrilateral with pairs of consecutive sides, but sides ARE NOT congruent. Trapezoid: A quadrilateral with exactly pair of sides. The parallel sides are called the. The nonparallel sides are called the. Isosceles Trapezoid: A trapezoid with congruent sides. Midsegment of a Trapezoid is parallel to each base and its length is 1 EF ( BC AD ) 2 Properties of a Kite: One pair of opposite angles of a kite is congruent The diagonals are perpendicular to each other The diagonal that connects the opposite vertex angles that are not congruent bisects the diagonal that connects the opposite vertex angles that are congruent. The diagonal that connects the opposite vertex angles that are not congruent bisects the vertex angles.
Example(s): 1. Given ABCD is a kite, determine the length of segments CD, AD, CB, AB, and DB, and AC. 2. Find the perimeter of the kite. Round answer to the nearest inch. Given trapezoid ABCD, find the other angles: 3. 4. 5. Maria told Sam that an isosceles trapezoid must also be a parallelogram because there is a pair of congruent sides in an isosceles trapezoid. Is Maria correct? Explain? Given trapezoid ABCD find x. 6. 7. 8. Write a two-column proof to prove HF JF in kite GHIJ. Statements Reasons 1. Kite GHIJ with diagonals HJ and GI intersecting at 1. point F. 2. 2. Definition of a kite 3. 3. 4. GHI GJI 4. 5. 5. CPCTC 6. GF GF 6. 7. 7. SAS 8. 8.
9. Complete the flow chart for quadrilaterals. 10. Place a check mark in the box that has the given characteristic. Characteristic Trapezoid Parallelogram Kite Rhombus Rectangle Square No parallel sides Exactly one pair of parallel sides Two pairs of parallel sides One pair of sides are both congruent and parallel Two pairs of opposite sides are congruent Exactly one pair of opposite angles are congruent Two pairs of opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other All sides are congruent Diagonals are perpendicular to each other Diagonals bisect the vertex angles All angles are congruent Diagonals are congruent
Determine whether each statement is true or false. If false, explain why. 11. A square is also a rectangle. 12. The base angles of a trapezoid are always congruent. 13. The diagonals of a rhombus bisect each other. 14. A parallelogram is also a trapezoid. 15. Classify the quadrilaterals by their properties. State its most specific name.