MATH Integrated Math I  - Unit 20: Polygons and Quadrilaterals; Sequences
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Polygons

1) What name is given to the point of intersection where two segments meet in a polygon?

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2) What name is given to a line segment that joins two nonconsecutive vertices of a polygon?

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3) What is the difference between the interior and the exterior of a polygon?

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To find the sum of the angles in a polygon, draw diagonals from the same vertex to divide the polygon into as many non-overlapping triangles as possible.  In the example below, a hexagon is divided into four triangles.  Since the sum of the angles in one triangle is 180 degrees, then the sum of the interior angles of the hexagon can be found by multiplying 180 times 4.  Thus, the sum of the interior angles of any hexagon is 720 degrees.

4) Now, check another polygon to find the sum of the interior angles. 1) Draw a pentagon.  (2) Draw a diagonal from one vertex to all of the other vertices. Do not draw any overlapping lines. (3) Count the number of triangles formed.  (4) Recall that in each triangle, the sum of the angles equals 180 degrees.  Multiply the number of triangles times 180 degrees to find the sum of the angles in the pentagon.  What is the sum of the angles?

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5) Use the technique of drawing nonoverlapping diagonals from one vertex to all of the other vertices to determine the sum of the measures of the interior angles of an octagon.  What is the sum of the angles?

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Activity:  Print out (right-click, print) or copy the table below.  Fill in the results for the pentagon, hexagon, and octagon from above.  Draw the other figures and complete the same activity.  Draw nonoverlapping diagonals from one vertex to the other vertices to create all possible triangles, and then compute the sum of the interior angles by multiplying by 180 degrees.

6) What is the sum of the interior angles of a (a) heptagon, (b) nonagon, (c) decagon, and (d) dodecagon?

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7) Look for a pattern in the table between the number of sides in the polygon and the number of triangles determined by connecting one vertex to all the other vertices. If “n” represents the number of sides of a polygon, which formula could be used to determine the total numbers of degrees of the interior angles in any polygon?

8) What is the sum of the interior angles of a 20-sided polygon? 

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9) What is the difference between a polygon and a regular polygon?

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10) The formula to find the measure of one interior angle of a REGULAR polygon is shown below. In words, explain how to determine the measure of one angle of any REGULAR polygon. (Note:  In the given formula, “n” represents the number of sides of the regular polygon.)

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11) What is the measure of one angle of a REGULAR decagon (10 sides)?

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12) When a ray is extended out from a side of a polygon, an exterior angle of a polygon is created.  Notice that, in the given regular pentagon, the interior angle and the exterior angle together make a straight angle.  (a) What is the measure of one interior angle of a REGULAR pentagon? (b) What is the measure of one exterior angle of a REGULAR pentagon?

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13) What is (a) the measure of an interior angle of the REGULAR hexagon and (b) the measure of an exterior angle of a regular hexagon?   

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14) The lot of a new house is shaped like a trapezoid, a special quadrilateral, with two right angle corners. If the third corner forms a 75-degree angle, (a) what is the measure of the fourth angle and (b) what is the measure of the exterior angle adjacent to this angle?

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15) Darius and his father are constructing a set of bunk beds as shown in the diagram below. What is the measure of the angle represented by “x”? State the letter of the correct answer.

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Quadrilaterals

16) Which statement is INCORRECT?

17) True or False.  The figure below has the properties of a parallelogram, a rectangle, a rhombus, and a square.  (Note:  The blue hash marks denote congruent sides and the corner squares denote right angles.)

18) Print out or sketch the quadrilateral and draw one diagonal. (a) How many triangles are formed? (b) Since the sum of the angles in one triangle totals 180 degrees, the four angles in a quadrilateral must total how many degrees?

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19) The measure of three of the four angles a quadrilateral are shown in the given figure.  What is the measure of angle A?

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20) What are the measures of two consecutive angles in the given parallelogram where “x” represents the smaller angle and “3x” represents the larger angle? 

21) Which figure is NOT a parallelogram? State the letter of the correct answer.

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Sequences

22) Is the given sequence an ARITHMETIC sequence?  If so, what is the common difference and what are the next three terms. If not, just state “no”.

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23) The first term and the common difference for an ARITHMETIC sequence are given below. What are the first five terms for the arithmetic sequence?

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24) For the given GEOMETRIC sequence, (a) what is the common ratio of the sequence, and (b) what are the next three terms of the sequence? 

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25)

For the given GEOMETRIC sequence, (a) what is the common ratio of the sequence, and (b) what are the next four terms of the sequence?

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26) The first term and the common ratio for a geometric sequence are given below. What are the first five terms for the geometric sequence?

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27) The given sequence is what type of sequence?

28) What is the twenty-second (22nd) term for the ARITHMETIC sequence?

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29) Sharon earned 6000 bonus miles for opening a new credit card that has a frequent-flyer program. She earns 250 miles for every $100 she spends. When the total miles reaches 25,000 miles, Sharon will be able to reserve a free round-trip ticket to anywhere in the United States. How many $100-purchases must she make to get the free trip? (Hint: Once the number of $100-purchases is determined, check the work the following way: 6000 + (number of purchases)(250) = 25000).

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30) What is the seventh term of a geometric sequence that has a common ratio of –3 and the first term is –4?   

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31) A rubber ball is dropped from a height of 250 feet. After each bounce, the height of the ball is recorded in the table below. The heights in the table form a geometric sequence. (a) What is the common ratio of the sequence? (b) Estimate the height of the ball after the 7th bounce. Round the answer to the nearest foot.

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32) The sequence below is a set of numbers that are referred to as triangular numbers since they can be modeled as shown.  The sequence is neither arithmetic nor geometric, but there is a pattern to determine each term of the sequence.  Look for a pattern in the numbers and/or figures, and then state the next five terms in the sequence.

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33) There are many kinds of sequences other than arithmetic or geometric. The    Fibonacci sequence is just one special sequence where addition is used to determine the next term; but, it is not an arithmetic sequence. (a) What are the next three terms in the given sequence?   (b) Explain how to calculate any term in the Fibonacci sequence.

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The spiral seed patterns in sunflowers are connected to the Fibonacci Sequence.  Two numbers in the Fibonacci sequence will occur as the number of seeds in the two crisscrossing spiral seed patterns in any sunflower.   Can you find the two numbers of the Fibonacci Sequence show in the picture below?

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