# Unit 3: Triangle Bisectors and Quadrilaterals

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1 Unit 3: Triangle Bisectors and Quadrilaterals

2 Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties of parallelograms, trapezoids, and kites

3 Lesson 5.1 Perpendicular and Angle Bisectors

4 Lesson 5.1 Objectives Define perpendicular bisector Utilize the Perpendicular Bisector Theorem and its converse Define and utilize the Angle Bisector Theorem

5 Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

6 Theorem 5.1: Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

7 suur PQ Example suur In the diagram below, PQ is the perpendicular bisector of CD. What segment lengths in the diagram are equal? suur Explain why T is on PQ. T C Q D P

8 Theorem 5.2: Perpendicular Bisector Converse If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

9 Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. The point of concurrency for this figure is called the circumcenter of the triangle. The circumcenter is the center of a circle that is drawn through the vertices of the triangle. Radius

10 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. Distance is found perpendicular to the segment.

11 Example H C G F Which segment is the perpendicular bisector of segment BC? Which segment is on the bisector of angle ACB? If the measure of angle ACD is 35, what is the measure of angle ACB? If the distance from D to F is 4, what is the distance from D to H? A E D B b

12 Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The point of concurrency for this figure is called the incenter of the triangle. The incenter is the center of a circle that is drawn inside the triangle.

13 Midsegment A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

14 Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. E C D CD = 1 / 2 AB A B

15 Given that CD, DF, and CF are all midsegments of the triangle. The problem is to find AC and AB. 5 C Example Problem E 12 5 D To first find AC, notice that DF is the side of a parallelogram (ACDF). So opposite sides are congruent. Thus AC = 5. The same idea can be used to find AF using CD. Since F is the midpoint of AB, then AB = 2AF. Thus AB = 2(12) = 24 A 12 F B Thanks to the Midsegment Theorem, DF is parallel to AC.

16 Lesson 5.5 Inequalities in One Triangle

17 Lesson 5.5 Objectives Use triangle measurements to determine which side or angle is the largest/smallest. Identify possible sides lengths for a triangle.

18 Theorem 5.10: Side Lengths of a Triangle Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Basically, the larger the side, the larger the angle opposite that side. 2 nd Largest Angle Smallest Side Largest Angle 2 nd Longest Side Smallest Angle

19 Theorem 5.11: Angle Measures of a Triangle Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Basically, the larger the angle, the larger the side opposite that angle. 2 nd Largest Angle Smallest Side Largest Angle 2 nd Longest Side Smallest Angle

20 Theorem 5.13: Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side Add each combination of two sides to make sure that they are longer than the third remaining side.

21 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A AB + BC >AC AC + BC > AB AB + AC > BC B C

22 Inequalities in One Triangle If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. C If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. B AB > AC > BC C > B > A A

23 Lesson 6.1 Polygons

24 Lesson 6.1 Objectives Identify a figure to be a polygon. Recognize the different types of polygons based on the number of sides. Identify the components of a polygon. Use the sum of the interior angles of a quadrilateral.

25 Definition of a Polygon A polygon is plane figure (two-dimensional) that 1. It meets is formed the by three following or more segments conditions. called sides. 2. The sides must be straight lines. 3. Each side intersects exactly two other sides, one at each endpoint. 4. The polygon is closed in all the way around with no gaps. 5. Each side must end when the next side begins. No tails. Polygons Not Polygons

26 Polygon Parts Each segment that is used to close a polygon in is called a side. Where each side ends is called a vertex. A vertex is simply a corner of the polygon. vertices sides

27 Types of Polygons Number of Sides n Type of Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

28 Concave v Convex 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 concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Concave polygons have dents in the sides, or you could say it caves in.

29 Regular Polygons 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. The best way to draw these is to label each sides and angle with the proper congruent marks.

30 Diagonals of a Polygon A diagonal of a polygon is a segment that joins two nonconsecutive vertices. A diagonal does not go to the point next to it. That would make it a side! Diagonals cut across the polygon to all vertices on the other side. There is always more than one diagonal.

31 Theorem 6.1: Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior angles of a quadrilateral is 360 o m 1 +m 2 + m 3 + m 4 = 360 o

32 Lesson 6.2 Properties of Parallelograms

33 Lesson 6.2 Objectives Define a parallelogram Identify properties of parallelograms Use properties of parallelograms to determine unknown quantities of the parallelogram

34 Definition of a Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

35 Theorem 6.2: Congruent Sides of a Parallelogram If a quadrilateral is a parallelogram, then its opposite sides are congruent.

36 Theorem 6.3: Opposite Angles of a Parallelogram If a quadrilateral is a parallelogram, then its opposite angles are congruent.

37 Theorem 6.4: Consecutive Angles of a Parallelogram If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Q R P m P + m S = 180 o S m Q + m R = 180 o m P + m Q = 180 o m R + m S = 180 o

38 Theorem 6.5: Diagonals of a Parallelogram If a quadrilateral is a parallelogram, then its diagonals bisect each other. Remember that means to cut into two congruent segments.

39 Lesson 6.3 Proving Quadrilaterals are Parallelograms

40 Lesson 6.3 Objectives Verify that a quadrilateral is a parallelogram. Utilize coordinate geometry with parallelograms

41 Theorem 6.6: Congruent Sides of a Parallelogram Converse If both pairs of opposite sides are congruent, then it is a parallelogram.

42 Theorem 6.7: Opposite Angles of a Parallelogram Converse If both pairs of opposite angles are congruent, then it is a parallelogram.

43 Theorem 6.8: Consecutive Angles of a Parallelogram Converse If an angle of a quadrilateral is supplementary to its consecutive angles, then it is a parallelogram. Q R P m P + m S = 180 o m P + m Q = 180 o S m Q + m R = 180o m R + m S = 180 o

44 Theorem 6.9: Diagonals of a Parallelogram Converse If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

45 Theorem 6.10: Opposite Sides of a Parallelogram If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

46 Lesson 6.4 Rhombuses, Rectangles, and Squares

47 Lesson 6.4 Objectives Identify characteristics of a rhombus. Identify characteristics of a rectangle. Identify characteristics of a square.

48 Rhombus A rhombus is a parallelogram with four congruent sides.

49 Theorem 6.11: Perpendicular Diagonals A parallelogram is a rhombus if and only if its diagonals are perpendicular.

50 Theorem 6.12: Opposite Angle Bisector A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

51 Rectangle A rectangle is a parallelogram with four congruent angles.

52 Theorem 6.13: Four Congruent Diagonals A parallelogram is a rectangle iff all four segments of the diagonals are congruent.

53 Square A square is a parallelogram with four congruent sides and four congruent angles.

54 Square Corollary A quadrilateral is a square iff it s a rhombus and a rectangle. So that means that all the properties of rhombuses and rectangles work for a square at the same time.

55 Lesson 6.5 Trapezoids and Kites

56 Lesson 6.5 Objectives Identify properties of a trapezoid. Recognize an isosceles trapezoid. Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid. Identify a kite.

57 Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the bases. The nonparallel sides are called legs. The angles formed by the bases are called the base angles.

58 Isosceles Trapezoid If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

59 Theorem 6.14: Bases Angles of a Trapezoid If a trapezoid is isosceles, then each pair of base angles is congruent. That means the top base angles are congruent. The bottom base angles are congruent. But they are not all congruent to each other!

60 Theorem 6.15: Base Angles of a Trapezoid Converse If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

61 Theorem 6.16: Congruent Diagonals of a Trapezoid A trapezoid is isosceles if and only if its diagonals are congruent. Notice this is the entire diagonal itself. Don t worry about it being bisected.

62 Midsegment The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.

63 Theorem 6.17: Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of Cthe lengths of the bases. D M MN = 1 / 2 (AB + CD) N A B

64 Kite A kite is a quadrilateral that has two pairs of consecutive sides that are congruent, but opposite sides are not congruent. It looks like the kite you got for your birthday when you were 5! There are no sides that are parallel.

65 Theorem 6.18: Diagonals of a Kite If a quadrilateral is a kite, then its diagonals are perpendicular.

66 Theorem 6.19: Opposite Angles of a Kite If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. The angles that are congruent are between the two different congruent sides. You could call those the shoulder angles. NOT

68 Lesson 6.6 Objectives Create a hierarchy of polygons Identify special quadrilaterals based on limited information

69 Polygon Hierarchy Polygons Triangles Quadrilaterals Pentagons Parallelogram Trapezoid Kite Rhombus Rectangle Isosceles Trapezoid Square

70 NEVER How to Read the Hierarchy Polygons Triangles Quadrilaterals Pentagons ALWAYS Parallelogram Rhombus Square Rectangle Trapezoid Kite Isosceles Trapezoid But a parallelogram is sometimes a rhombus and sometimes a square. SOMETIMES So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon. However, a parallelogram is never a trapezoid or a kite.

71 Using the Hierarchy Remember that a square must fit all the properties of its ancestors. That means the properties of a rhombus, rectangle, parallelogram, quadrilateral, and polygon must all be true! So when asked to identify a figure as specific as possible, test the properties working your way down the hierarchy. As soon as you find a figure that doesn t work any more you should be able to identify the specific name of that figure.

72 Lesson 6.7 Areas of Triangles and Quadrilaterals

73 Lesson 6.7 Objectives Find the area of any type of triangle. Find the area of any type of quadrilateral.

74 Postulate 22: Area of a Square Postulate The area of a square is the square of the length of its side. A = s 2 s

75 Area Postulates Postulate 23: Area Congruence Postulate If two polygons are congruent, then they have the same area. Postulate 24: Area Addition Postulate The area of a region is the sum of the areas of its nonoverlapping parts.

76 Theorem 6.20: Area of a Rectangle The area of a rectangle is the product of a base and its corresponding height. Corresponding height indicates a segment perpendicular to the base to the opposite side. A = bh or A = length x width h b

77 Theorem 6.21: Area of a Parallelogram The area of a parallelogram is the product of a base and its corresponding height. Remember the height must be perpendicular to one of the bases. The height will be given to you or you will need to find it. To find it, use Pythagorean Theorem a 2 + b 2 = c 2 A = bh h b

78 Theorem 6.22: Area of a Triangle The area of a triangle is one half the product of the base and its corresponding height. The base for this formula is the side that is perpendicular to the height. h h h b b b

79 Theorem 6.23: Area of a Trapezoid The area of a trapezoid is one half the product of the height and the sum of the bases. The height is the perpendicular segment between the bases of the trapezoid. A = ½ h (b 1 +b 2 ) b 1 h b 2

80 Theorem 6.24: Area of a Kite The area of a kite is one half the product of the lengths of the diagonals. A = ½ d 1 d 2 d 1 d 2

81 Theorem 6.25: Area of a Rhombus The area of a rhombus is equal to one half the product of the lengths of the diagonals. A = ½ d 1 d 2 d 1 d 2

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