Describing Polygons Q A polygon is a figure that is: -formed by 3 or more segments called sides, such that no 2 sides with a common endpoint are collinear - each side intersects exactly 2 other sides, one at each endpoint. Each endpoint of the side is called a vertex. Vertex P R Side T S Vertex
Identifying Convex and Concave A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon is that is not convex is called nonconvex or concave.
Definitions: A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is both equilateral and equiangular.
Theorem 6.1: Polygon Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is (n-2)180. If you draw a diagonal in a polygon, you create triangles. Using the Triangle Sum Theorem you can conclude that the sum of the measures of the interior angles of a quadrilateral is 2(180)=360°.
Corollary to Polygon Angle-Sum Theorem The measure of each of the interior angles of a regular polygon is (Where n is the number of sides.) Theorem 6.2 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360 degrees.
PARALLELOGRAM: • A quadrilateral with both pairs of opposite sides parallel.
Theorem 6.3: • If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Theorem 6.4: • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
Theorem 6.5: • If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Theorem 6.6: • If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Special Parallelograms A square is a parallelogram with 4 right angles and 4 congruent sides A rhombus is a parallelogram with 4 congruent sides. A rectangle is a parallelogram with 4 right angles Rectangle Square Rhombus
Diagonals of Special Parallelograms Thm. 6.13: If a parallelogram is a rhombus, then its diagonals are perpendicular. Thm. 6.14: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. • Thm. 6.15: If a parallelogram is a rectangle, then its diagonals are congruent.
Properties of Trapezoids • A trapezoid is a quadrilateral with exactly one pair of parallel sides. Trapezoid Terminology: The parallel sides are called BASES. The nonparallel sides are called LEGS. There are two pairs of base angles, the two touching the top base, and the two touching the bottom base.
ISOSCELES TRAPEZOID If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. Thm. 6.19: If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. Thm. 6.20: If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.
Midsegment of a Trapezoid The midsegment of a trapezoid connects the midpoints of its legs. Thm. 6.21: If a quadrilateral is a trapezoid, then… • The midsegment is parallel to both bases and • The length of the midsegment is half the sum of the lengths of the bases.
Kites A kite is a quadrilateral that has 2 pairs of consecutive congruent sides, but opposite sides are not congruent. Thm. 6.22: If a quadrilateral is a kite, then its diagonals are perpendicular. Interesting fact: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
Relationships Among Quadrilaterals • Fill in Chart
Formulas and the Coordinate Plane Formula Distance Formula Midpoint Formula Slope Formula • When to Use it…. • To determine whether… • Sides are congruent • Diagonals are congruent • To determine … • The coordinates of the midpoint of a side • Whether diagonals bisect each other • To determine whether… • Opposite sides are parallel • Diagonals are perpendicular • Sides are perpendicular
Reminders on Ratios: • It is a comparison of two quantities by division • Notation: or , read a to b. • Measured in same units. • Denominator can not be zero. • Usually expressed in simplified form: 6:8 simplified to 3:4
An equation that equals two ratios. Proportion Extremes Means
Similar Polygons Two polygons are SIMILAR if and only if: 1-their corresponding angles are congruent, 2- the measures of their corresponding sides are proportional.
Similar Polygons • Symbol to indicate similarity: ~ • ABCD ~ GHIJ (ratio of the lengths of two corresponding sides)
Angle-Angle Similarity Postulate Postulate 7.1 Angle-Angle Similarity (AA~) Postulate: If two angles of one triangle are congruent to 2 angles of another triangle, then the two triangles are similar. Hint: From earlier information we know that if we have two congruent angles then we also know the third angles are congruent. Thus AA is the same as AAA. This is the most common proof of two triangles to be similar.
THEOREMS X M P N Z Y XY MN ZX PM If XM and= THEOREM 7.1 Side-Angle-Side (SAS~) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. then XYZ ~ MNP.
THEOREMS P A AB PQ BC QR CA RP Q R If = = B C THEOREM 7.2 Side-Side-Side (SSS~) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. then ABC ~ PQR.
Theorem 7.3 • The altitude to the hypotenuse of a right triangle divides the triangle into 2 triangles that are similar to the original triangle and to each other. ΔCBD ~ ΔABC ΔACD ~ ΔABC ΔCBD ~ ΔACD
The GEOMETRIC MEAN between two positive numbers a and b is the positive number x where • What is the geometric mean of 5 and 12? • What is the geometric mean of 6 and 16?
Theorem 7.4: Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. C E A B D
Theorem 7.5 Triangle-Angle Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.
The Pythagorean Theorem *Remember the Pythagorean Theorem is only true for RIGHT triangles!! *The hypotenuse (c) is always the longest side and opposite the right angle! *The legs (a and b) are the two sides that form the right angle.
Theorem 8.3: If , then the triangle is obtuse. Theorem 8.4: If , then the triangle is acute. Classifying Triangles:
Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures. These formulas let us arrive at the answer very quickly. Theorem 8-5 - 45º-45º-90º
Theorem 8.6 Hypotenuse = 2 x shorter leg Longer Leg = x shorter leg
Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.
Writing Trigonometric Ratios *What are the ratios for angle G?
Angle of Elevation/Depression: The angle formed by a horizontal line and the line of sight to an object either above or below the horizontal line.
Vectors: • Vector: any quantity with both magnitude (size) and direction • The magnitude corresponds to the distance from the initial point to the terminal point of the vector • The direction corresponds to which way the arrow is pointed
Describing a Vector • You can indicate a vector by using an ordered pair. For example, <-2, 4> is a vector with its initial point at the origin and its terminal point at (-2, 4). • We use brackets to represent a vector (called Component Form)
Direction of a Vector: • You can use a compass arrangement on the coordinate grid to describe a vector’s direction. This vector is 30 south of east. This vector is 40 east of north.
Magnitude of a Vector • The magnitude of a vector is its length. You can use the distance formula to determine the length, or magnitude, of a vector.
Adding Two Vectors • The sum of two vectors is called the Resultant