Unit 4: Polygons 180 Complimentary
Opposite exterior
Practice Makes Perfect!
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Practice Makes Perfect! Def: Midsegment of a triangle - a segment that connects the midpoints of two sides of the triangle - every triangle has three. B M N A P C
Investigate Segments in Triangles Question: How are the midsegments of a triangle related to the sides of the triangle? A midsegment of a triangle connects the midpoints of two sides of a triangle. Step 1: Draw right triangle ABO, with vertices at point A(0, 8); B(6, 0) and O(0, 0) Step 2: Find the midpoint of segment OA, and segment OB, label then D and E, and connect them to form segment DE. Step 3: Construct the following Table and fill in the missing information. Once completed complete case 2. Then us it to answer the questions below. 1. What do you notice about the slopes of segments AB and DE? 2. What do you notice about the lengths of segments AB and DE? Midsegment Theorem A segment that connects the midpoints of two sides is parallel to the third side AND The segment is half the length of the third side
Example: - What would the third midsegment be? - If UW is 81in' what would VS be?
Example: Perpendicular Bisectors Def: A segment bisector intersects a segments at its midpoint. Def: perpendicular bisector a segment, ray, line, or plane, that is perpendicular to a segment at its midpoint. Def: A point is equidistant from two figures if the point is the same distance from each figure.
Perpendicular Bisector Theorem Points on the perpendicular bisector of a segment are equidistant from the segment's endpoints. Converse of the Perpendicular Bisector Theorem If points are equidistant from a segment's endpoints then the point lies on the perpendicular bisector Example: Example:
Def: Concurrent lines, segments, or rays - when three or more lines, rays or segments intersect in the same point. Def: Point of Concurrency - the point of intersection of the lines, rays, or segments Concurrency of Perpendicular Bisectors of a Triangle Theorem The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. Def: circumcenter the point of concurrency of the three perpendicular bisectors of a triangle
Practice Makes Perfect! Angle Bisectors of Triangles Remember an angle bisector is a ray that divides an angle into two congruent adjacent angles. Remember the distance from a point to a line is the length of the perpendicular segment from the point to the line.
Angle Bisector Theorem - If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Converse of the Angle Bisector Theorem If a point is in the interior of an angle and it is equidistant from the sides of the angle, then it lies on the bisector of the angle Example: Example:
Example: Concurrency of an Angle Bisector Theorem Def: incenter of the triangle - the point of concurrency of the three angle bisectors of a triangle
Example: Use Medians and Altitudes Def: median of a triangle: a segment from a vertex to the midpoint of the opposite side Def: centroid: the point of concurrency for the three medians of a triangle Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
Example: Example: Def: Altitude of a Triangle - the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. Concurrancey of
Def: orthocenter - the point at which the lines containing the three altitudes of a triangle intersect Fun Fact! In an Isosceles Triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment. Practice Makes Perfect!
Quadrilateral Sum Theorem The interior angles of a Quadrilateral add up to 360 o I. What is a polygon? Polygons! A polygon is a plane figure that is formed by three of more segments called sides, such that the following are true: 1. Each side intersects two other sides, once at each endpoint. 2. No two sides with a common endpoint are collinear. II. Some familiar terms Diagonal is a segment that joins two nonconsecutive vertices Equilateral its sides are congruent Equiangular all its angles are congruent Polygons that are equiangular and equilateral are called Regular polygons. Name of Polygon Figure Number of Sides Number of Triangles Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon Hexagon Heptagon Octagon Nonagon Decagon n-sided
Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n-gon is If the n-gon is regular, then the measure of each interior angles is Example 1: Find the sum of the measures of the interior angles of a convex octagon Example 2: The sum of the measures of the interior angles of a convex polygon in 900 o. Classify the polygon by the number of sides.
Example 3: Find the value of x in the diagram shown. Polygon Exterior Angles Theorem All polygons have exterior angles that add up to 360 o. If the n-gon is regular each exterior angle is equal to 360/n
Example 4: What is the value of x in the diagram? Practice Makes Perfect!
Use Properties of Parallelograms Def: a parallelogram is a quadrilateral whose of opposite sides are parallel. write parallelogram PQRS as A quadrilateral is a parallelogram if - it has two pairs of congruent sides or - it has opposite angles congruent
Rhombus, Rectangles and Squares Oh My! Three special types of parallelograms Square Corollary: a quadrilateral is a square iff (if and only if) it is a rhombus and a rectangle
A parallelogram is a rhombus iff - the diagonals are perpendicular Diagonals or - the diagonals bisect a pairs opposite angles. If a parallelogram is a rectangle then it's diagonals are congruent. Example: Sketch rectangle ABCD. List everything that you know about it. Now sketch square PQRS. List everything you know about the square.
Practice Makes Perfect!
Properties of Trapezoids and Kites Trapezoids Trapezoid - a quadrilateral with exactly one pair of parallel sides. The base of a trapezoid is the parallel lines, a trapezoid has two pair of base angles Legs of a trapezoid are the non parallel sides Isosceles Trapezoids Isosceles Trapezoid - if the legs of the trapezoid are congruent Theorem: Each pair of base angles in an isosceles trapezoid is congruent Theorem: A trapezoid is isosceles iff its diagonals are congruent
Example 1: In Trapezoid ZOID, name the bases, legs and the base angles. D Z I O Midsegment of a Trapezoid The Midsegment of a trapezoid is a segment that joins the midpoints of the legs M I G H T Y Theorem : a.the midsegment of a trapezoid is parallel to the bases. b. The length of the midsegment equals one-half to sum of the lengths of the bases. IT = 1/2(GH + MY)
Example: Kites a Kite is a quadrilateral that has two pair of consecutive congruent sides, but opposite sides are not congruent. Theorem: a kite has diagonals that are perpendicular Theorem: a kite has exactly one pair of opposite angles congruent
Example: Classify Quadrilaterals 1, 2, 3, 4, 5, 6, 8, 11, 12 2, 3, 4, 5, 6, 11, 12 2, 3, 5, 6, 7, 10 3, 4, 6, 12 2, 3, 5, 6 3, 4, 6, 12 3, 6 squares
1, 2, 3, 4, 5, 6, 8, 11, 12 1, 3, 4, 6, 8, 12 2, 3, 4, 5, 6, 11, 12 7, 10, 3, 4, 6, 12 2, 3, 5, 6 Practice Makes Perfect!
Characteristics Parallelogram Rectangle Rhombus Square Trapezoid Kites Opposite sides are parallel Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary. Diagonals bisect each other Diagonals are congruent Diagonals are perpendicular Each diagonal bisects two angles