Unit 8 – Geometry QUADRILATERALS

1 Unit 8 Geometry QUADRILATERALS . NAME _____. Period _____. 1. A little background . Polygon is the generic term for a closed figure with any number of sides. Depending on the number, the first part of the word Poly is replaced by a prefix. The prefix used is from Greek. The Greek term for 5 is Penta, so a 5-sided figure is called a Pentagon. We can draw figures with as many sides as we want, but most of us don't remember all that Greek, so when the number is over 12, or if we are talking about a general polygon, many mathematicians call the figure an n-gon. So a figure with 46 sides would be called a 46-gon.. Vocabulary Polygon - A closed plane (two-dimensional) figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex. Types of Polygons Regular - all angles are equal and all sides are the same length.

2 Regular polygons are both equiangular and equilateral. Irregular Any polygon with any angles NOT equal and any sides NOT the same length. Equiangular - all angles are equal. Equilateral - all sides are the same length. Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180 . Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180 . Polygon Parts Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by one side of a triangle and the extension of another side.

3 2. special Polygons special QUADRILATERALS - square, rhombus, parallelogram, rectangle, and the trapezoid. special Triangles - right, equilateral, isosceles, scalene, acute, obtuse. Polygon Names Generally accepted names Sides Name n N-gon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 10 Decagon 12 Dodecagon Vocabulary from 3. NUMBER. NUMBER. OF ONE ONE. NUMBER OF. NUMBER TRANGLE SUM OF INTERIO EXTERIO SUM OF. OF NAME DIAGONAL. OF S MEASURE R ANGLE R ANGLE EXTERIO. SIDES OF S. INTERIO FORMED S OF MEASUR MEASUR R ANGLES. OF THE POLYGO POSSIBLE. R FROM INTERIO E E MEASURE. POLYGO N FROM ONE. ANGLES ONE R ANGLES (REGULAR (REGULAR S. N VERTEX POLYGON) POLYGON). VERTEX. POINT. POINT. 3 3 0 1. 4. 5 5 2 3 540o 6. 7. 8. 9. 10. 11. 12 1800o n a) Compare the number of triangles to the number of sides. Do you see a pattern? b) How can you use the number of triangles formed by the diagonals to figure out the sum of all the interior angles of a polygon?

4 C) Write an expression for the sum of the interior angles of an n-gon, using n and the patterns you found from the table. 4. n=4. n=3 n=5. n=6. n=7 n=8. n=9. n = 10 n = 12. 5. Date: _____. Section 8 1: Angles of Polygons Notes Diagonal of a Polygon: A segment that _____ any two _____ vertices. Theorem : Interior Angle Sum Theorem: If a convex polygon has n sides and S is the sum of the _____ of its interior angles, then _____. Example #1: Find the sum of the measures of the interior angles of the regular pentagon below. Example #2: The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Example #3: Find the measure of each interior angle. 6. Theorem : Exterior Angle Sum Theorem: If a polygon is _____, then the sum of the measures of the _____ angles, one at each _____, is 360. Example #4: Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.

5 7. NAME _____ DATE _____ PERIOD _____. 8-1 Practice Angles of Polygons Find the sum of the measures of the interior angles of each convex polygon. 1. 11-gon 2. 14-gon 3. 17-gon The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon. 4. 144 5. 156 6. 160. Find the measure of each interior angle using the given information. 7. J K 8. quadrilateral RSTU with (2x 1 15)8 (3x 2 20)8. m/R 5 6x 2 4, m/S 5 2x 1 8. R S. (x 1 15)8 x8. N M. U T. Find the measures of an interior angle and an exterior angle for each regular polygon. Round to the nearest tenth if necessary. 9. 16-gon 10. 24-gon 11. 30-gon Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary. 12. 14 13. 22 14. 40. 15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems.

6 The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram. Find the sum of the measures of the interior angles of one such face. Glencoe/McGraw-Hil 8. l Glencoe Geometry Date: _____. Properties of Parallelograms Activity Step 1 Using the lines on a piece of graph paper as a guide, draw a pair of parallel lines that are at least 10 cm long and at least 6 cm apart. Using the parallel edges of your straightedge, make a parallelogram. Label your parallelogram MATH. Step 2 Look at the opposite angles. Measure the angles of parallelogram MATH. Compare a pair of opposite angles using your protractor. The opposite angles of a parallelogram are _____. Step 3 Two angles that share a common side in a polygon are consecutive angles. In parallelogram MATH, MAT and HTA are a pair of consecutive angles.

7 The consecutive angles of a parallelogram are also related. Find the sum of the measures of each pair of consecutive angles in parallelogram MATH. The consecutive angles of a parallelogram are _____. Step 4 Next look at the opposite sides of a parallelogram. With your ruler, compare the lengths of the opposite sides of the parallelogram you made. The opposite sides of a parallelogram are _____. Step 5 Finally, consider the diagonals of a parallelogram. Construct the diagonals MT and HA . Label the point where the two diagonals intersect point B. Step 6 Measure MB and TB. What can you conclude about point B? Is this conclusion also true for diagonal HA ? How do the diagonals relate? The diagonals of a parallelogram _____. 9. Date: _____. Section 8 2: Parallelograms Notes Key Concept (Parallelogram): A _____ is a quadrilateral with _____ pairs of opposite sides _____.

8 Ex: Symbols: Theorem : Opposite sides of a parallelogram are _____. Theorem : _____ angles in a parallelogram are congruent. Theorem : Consecutive angles in a parallelogram are _____. Theorem : If a parallelogram has one _____ angle, it has four right angles. Theorem : The _____ of a parallelogram bisect each other. 10. Example #1: RSTU is a parallelogram. Find m URT , m RST , and y. Theorem : Each diagonal of a _____ separates the parallelogram into _____ congruent triangles. 11. NAME _____ DATE _____ PERIOD _____. 8-2 Practice Parallelograms Complete each statement about ~LMNP. Justify your answer. L M. Q. 1. ww>. LQ ? P N. 2. /LMN > ? 3. nLMP > ? 4. /NPL is supplementary to ? . wM. 5. L w> ? ALGEBRA Use ~RSTU to find each measure or value. R 258. S. 308 B 23. 6. m/RST 5 7. m/STU 5 4b 2 1. U T. 8. m/TUR 5 9. b 5. COORDINATE Geometry Find the coordinates of the intersection of the diagonals of parallelogram PRYZ given each set of vertices.

9 10. P(2, 5), R(3, 3), Y(22, 23), Z(23, 21) 11. P(2, 3), R(1, 22), Y(25, 27), Z(24, 22). 12. PROOF Write a paragraph proof of the following. Given: ~PRST and ~PQVU P Q R. Prove: /V > /S U V. T S. 13. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to 1 2. design a herringbone pattern for a paving stone. He will use the paving 4 3. stone for a sidewalk. If m/1 is 130, find m/2, m/3, and m/4. Glencoe/McGraw-Hill 12 Glencoe Geometry Date: _____. Section 8 3: Tests for Parallelograms Notes Conditions for a Parallelogram: By definition, the opposite sides of a parallelogram are parallel. So, if a quadrilateral has each pair of opposite sides parallel it is a parallelogram. Key Concept (Proving Parallelograms): Theorem : If both pairs of _____ sides of a quadrilateral are _____, then the quadrilateral is a parallelogram. Ex: Theorem : If both pairs of opposite _____ of a quadrilateral are _____, then the quadrilateral is a parallelogram.

10 Ex: Theorem : If the _____ of a quadrilateral _____. each other, then the quadrilateral is a parallelogram. Ex: Theorem : If one pair of opposite sides of a quadrilateral is both _____ and _____, then the quadrilateral is a parallelogram. Ex: 13. Example #1: Find x and y so that each quadrilateral is a parallelogram and justify your reasoning. a.). b.). 14. Given: VZRQ and WQST Q. R. A. W. Prove: Z T T. V. Z. Statements Reasons 1. VZRQ 1. Given 2. Z Q 2. Opposite angles of a parallelogram are congruent 3. WQST 3. Given 4. Q T 4. Opposite angles of a parallelogram are congruent 5. Z T 5. Transitive 15. NAME _____ DATE _____ PERIOD _____. 8-3 Practice Tests for Parallelograms Determine whether each quadrilateral is a parallelogram. Justify your answer. 1. 2. 3. 1188 628 4. 628 1188. COORDINATE Geometry Determine whether a figure with the given vertices is a parallelogram.