Number of Number of Sides Interior Angles Triangle 3 3 Quadrilateral 4 4 Pentagon 5 5 Hexagon 6 6 Heptagon 7 7 Octagon 8 8 Nonagon 9 9 Decagon 10 10

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1 A closed figure made up of line segments is called a polygon. Most two-dimensional geometric figures are polygons, and they are named according to the number of sides they have: Name Number of Number of Sides Interior Angles Triangle 3 3 Quadrilateral 4 4 Pentagon 5 5 Hexagon 6 6 Heptagon 7 7 Octagon 8 8 Nonagon 9 9 Decagon One common figure that is missing from the table above is a circle, because a circle is not a polygon. A circle is a collection of points in a plane (in two-dimensional space) that are all the same distance away from a fixed point. This fixed point is the center of the circle, and a line segment connecting the center to any other point on the circle is called a radius of the circle: r A circle is not a polygon because it is not made up of line segments. Another common geometric figure that is not a polygon is an ellipse: Besides circles and ellipses, just about every other geometric figure you will ever work with will be a polygon. One special type of polygon (first defined in Lesson 36) is a regular polygon (which is a polygon whose sides are all congruent and whose interior angles are all congruent).

2 Triangles A triangle is a three-sided polygon. The sum of the measures of the interior angles of a triangle is 180º. Types of triangles include: An equilateral triangle (or equiangular triangle) is a triangle with all three sides having the same length. The interior angles of an equilateral triangle are also congruent and each measure 60º. An equilateral triangle is also a regular triangle. An isosceles triangle is a triangle with two sides that are of equal length. A scalene triangle is a triangle with three sides that each have a different length. An acute triangle is a triangle whose angles are all acute angles. (An acute angle has a measure less than 90º.)

3 An obtuse triangle is a triangle with an obtuse angle. (An obtuse angle has a measure greater than 90º.) A right triangle is a triangle with a right angle. (A right angle has a measure of exactly 90º.) Quadrilaterals A quadrilateral is a four-sided polygon. The sum of the measures of the interior angles of a quadrilateral is 360º. Types of quadrilaterals include: A parallelogram is a quadrilateral with two pairs of parallel sides. A rhombus is a quadrilateral with four sides of equal length.

4 A rectangle is a quadrilateral with four interior angles of equal measure, or in other words, all four angles are right angles. A square is a regular quadrilateral, or in other words, it is a quadrilateral with four sides of equal length and four interior angles of equal measure (which is 90º). A trapezoid is a quadrilateral with exactly one pair of parallel sides. The two parallel sides are called the bases of the trapezoid. example 1 A partial drawing of a quadrilateral is shown below. A B C If no other sides or angles are congruent, which best describes the figure? A. square B. rectangle C. parallelogram D. trapezoid

5 Investigating each answer: Answer A: To be a square, the other sides and angles must be congruent, so the answer cannot be A. Answer B: To be a rectangle, opposite sides must be congruent, and all angles must be congruent (right angles), so the answer cannot be B. Answer C: To be a parallelogram, opposite sides must be parallel, but if two adjacent sides are given as being congruent and perpendicular to each other, the only possible parallelogram would be a square, which has already been ruled out, so the answer cannot be C. Answer D: Answers A, B, and C have already been eliminated, so the answer must be D, even if you don t know what a trapezoid is. However, a trapezoid is a quadrilateral (four-sided polygon) with exactly one pair of parallel sides. The given figure could easily be drawn as a trapezoid and meet the criteria listed in the problem (no other sides or angles being congruent) The answer is D. example 2 Ricardo drew a rectangle. Which statement must be true? A. Ricardo s figure is a parallelogram. B. Ricardo s figure is a regular polygon. C. Ricardo s figure is a square. D. Ricardo s figure is a rhombus. To answer this question, the definition of rectangle must be examined. A rectangle is a polygon with four right interior angles. This implies that it is also a parallelogram. However, this does not mean that all four sides must be congruent (although opposite sides are congruent). Now let s look at the four statements given: Answer A) A rectangle is, in fact, a parallelogram. But, to be sure this is the answer, the other answers must be checked. Answer B) A rectangle is not a regular polygon, unless it is a square. Remember that all of the sides and interior angles of a regular polygon are congruent, which is true for a square, but isn t always true for all rectangles. Answer C) A rectangle can be a square, but does not have to be a square.

6 Answer D) A rectangle is a rhombus, only when the rhombus is a square. This is because a rhombus is a quadrilateral with four congruent sides, but the interior angles of a rhombus do not have to be right angles, although they can be. The answer is A. Pentagons A pentagon is a five-sided polygon. The sum of the measures of the interior angles of a pentagon is 540º. Hexagons A hexagon is a six-sided polygon. The sum of the measures of the interior angles of a hexagon is 720º. Octagons An octagon is an eight-sided polygon. The sum of the measures of the interior angles of a octagon is 1080º. N-gons An n-gon is an n-sided polygon. The sum of the measures of the interior angles of an n-gon is (n 2) 180º.

7 Name Match the appropriate label below with each quadrilateral. Some figures will require more than one label. A) B) C) D) E) parallelogram rhombus rectangle square trapezoid 1) 2) 3) 4)

8 Match the appropriate label below with each triangle. Some figures will require more than one label. A) B) C) D) E) F) equilateral triangle isosceles triangle scalene triangle acute triangle obtuse triangle right triangle 5) 6) 7) 8) 9)