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1 Bell Work

2 Geometry Day 36 Topic: Chapter 4 Congruent Figures Chapter 6 Polygons & Quads

3 Chapter 4 Big Ideas Visualization Visualization can help you connect properties of real objects with two-dimensional drawings of these objects. Reasoning and Proof Definitions establish meanings and remove possible misunderstanding. Other truths are more complex and difficult to see. It is often possible to verify complex truths by reasoning from simpler ones by using deductive reasoning.

4 Chapter 8 Essential Understanding 4-1 Comparing the corresponding parts of two figures can show whether the figures are congruent. 4-2 & 4-3 Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent. 4-4 If two Triangles are congruent, then every pair of their corresponding parts is congruent.

5 Common Core State Standards Geometry (GM) Similarity, Right Triangles, and Trigonometry GM: G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

6 Student Objectives I can recognize congruent figures and their corresponding parts. (4-1) I can prove two triangles congruent using the SSS and SAS Postulates. (4-2) I can prove two triangles congruent using the ASA Postulate and the AAS Theorem. (4-3) I can use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent. (4-4)

7 Chapter 4-1 Congruent Figures Congruent Polygons (p.219) Congruent Polygons have congruent corresponding parts their matching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order.

8 Chapter 4-1 Congruent Figures Theorem 4-1 Third Angles Theorem (p.220) If two angles of one triangle are congruent in two angles of another triangle, then the third angles are congruent.

9 Chapter 4-2 Triangle Congruence Postulate 4-1 Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

10 Chapter 4-2 Triangle Congruence Postulate 4-2 Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

11 Chapter 4-4 Triangle Congruence Postulate 4-3 Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

12 Chapter 4-4 Triangle Congruence Theorem 4-2 Angle-Angle-Side (AAS) Theorem If two angles and a nonincluded side of one triangle congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent.

13 Chapter 4-5 Isosceles and Equilateral Triangles Key Concepts Legs of an Isosceles Triangle (p250) Base of an Isosceles Triangle (p250) Vertex angle of an Isosceles Triangle (p250) Base Angles of an Isosceles Triangle (p250) Corollary (p252) Theorem 4-3 Isosceles Triangle Theorem (p250) Theorem 4-4 Converse of the Isosceles Triangle Theorem (p251) Theorem 4-5 (p252) Corollary to Theorem 4-3 (p252) Corollary to Theorem 4-4 (p252)

14 Chapter 4-5 Isosceles and Equilateral Triangles Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

15 Chapter 4-5 Isosceles and Equilateral Triangles Theorem 4-4 Converse to the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

16 Chapter 4-5 Isosceles and Equilateral Triangles Theorem 4-5 If a line bisects the vertex angle of an Isosceles Triangle, then the line is also the perpendicular bisector of the base.

17 Corollaries to Theorem 4-3 & 4-4 Corollary A theorem that can be proved easily using another theorem. Since a corollary is a theorem, you can use it as a reason in a proof. Corollary to Theorem 4-3 If a triangle is equilateral, then the triangle is equiangular. Corollary to Theorem 4-4 If a triangle is equiangular, then the triangle is equilateral.

18 Chapter Review Open your books to page 273 You will be given 30 minutes to work on the Chapter Review (1-33) on pages You may work with a partner if you do so quietly I will be walking around answering questions as needed. We will review your answers once your time is up.

19 To be continued Tomorrow, Wednesday 08 Mar 2017, we will continue working with Congruent Figures. Homework: Worksheet on Congruent Triangles Your next test (1.7) on Chapter 4 & 6 will be Tuesday 03/14/2017. Visit our website for resources to help prepare you for your test.

20 We are a TEAM Remember what it mean to be part of a team Together Everyone Achieves More Mr. Matthew J Ford

21 Following rules means being HAPPY Being HAPPY means being: Honest Accountable Proactive Positive Yourself Mr. Matthew J Ford

22 Any Questions???

23 You guy Rock!!! I Love all you crazy kiddos You guys are real Mathletes in Training!

24 Chapter 6 Big Ideas Measurement Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes. Reasoning and Proof Definitions establish meanings and remove possible misunderstandings. Other truths are more complex and difficult to see. It is often possible to verify complex truths by reasoning from simpler ones using deductive reasoning. In this chapter, you will: Explore Polygons and Quadrilaterals Measurement Reasoning and Proof Coordinate Geometry

25 Chapter 2 Essential Understanding 6-1 The sum of the angle measures of a polygon depends on the number of sides the polygon has. 6-2 Parallelograms have special properties regarding their sides, angles, and diagonals. 6-3 If a quadrilateral s sides. Angles, and diagonals have certain properties, it can be shown that the quadrilateral is a parallelogram. 6-4 & 6-5 The special parallelograms, rhombus, rectangle, and square, have basic properties of their sides, angles, and diagonals that help identify them. 6-6 The angles, sides, and diagonals of a trapezoid have certain properties.

26 Common Core State Standards Geometry (GM) Similarity, Right Triangles, and Trigonometry GM: CO.C.11 Prove and apply theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. GM: G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

27 Student Objectives I can define and classify special types of parallelograms. (6-4) I can use properties of diagonals of rhombuses and rectangles. (6-4) I can determine whether a parallelogram is a rhombus or rectangle. (6-5) I can verify and use properties of Trapezoids and Kites. (6-6)

28 Review of Polygons

29 Properties of Parallelograms Theorem 6-1 The sum of the measures of the interior angles of an n-gon is (n-2)180 Corollary to Theorem 6-1 The measure of each interior angle of a regular n-gon is n n Theorem 6-2 The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. 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. Theorem 6-7 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

30 Concave or Convex Polygons

31 Equilateral Polygons Polygons with all sides congruent

32 Equiangular Polygons Polygons with all angles congruent

33 Regular Polygons

34 6-1 Polygon Angle-Sum Theorem

35 Corollary to the Polygon Angle-Sum Theorem

36 Regular Polygons

37 6-2 Polygon Exterior Angle-Sum Theorem

38 Regular Polygon Angle Examples

39 Recall

40 Khan Academy / Special Parallelograms

41 Quadrilaterals Convex Trapezoids Parallelograms are comprised of three different types of quadrilaterals Rectangle Square Rhombus Concave

42 It s that time again

43 Special Paralellograms Rhombus a parallelogram with four (4) congruent sides Rectangle A parallelogram with four right angles (90 ) Square A parallelogram with four congruent sides and four right angles

44 Chapter 6-6 Vocabulary Trapezoid A quadrilateral with exactly one pair of parallel sides. Base The parallel sides of a Trapezoid. Leg The nonparallel sides of a Trapezoid Base Angle The two angles that share a base of a Trapezoid. Isosceles Trapezoid A Trapezoid with legs that are congruent. Midsegment of a Trapezoid The segment that joins the midpoints of its legs. Kite A quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

45 Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel. a four-sided plane rectilinear figure with opposite sides parallel.

46 Parallelogram There are six important properties of parallelograms to know: Opposite sides are congruent (AB = DC). Opposite angels are congruent (D = B). Consecutive angles are supplementary (A + D = 180 ). If one angle is right, then all angles are right. The diagonals of a parallelogram bisect each other. Each diagonal of a parallelogram separates it into two congruent triangles.

47 Opposite Sides In a quadrilateral, opposite sides do not share a vertex.

48 Opposite Angles In a quadrilateral, opposite angles do not share a side.

49 Consecutive Angles Angles of a polygon that share a side are consecutive angles.

50 Quadrilaterals / Parallelograms

51 Theorem 6-8 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a Parallelogram.

52 Theorem 6-9 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a Parallelogram.

53 Theorem 6-10 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a Parallelogram.

54 Theorem 6-11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

55 Theorem 6-12 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a Parallelogram.

56 Theorem 6-13 p.376 If a parallelogram is a Rhombus, then its diagonals are perpendicular.

57 Theorem 6-14 p.376 If a parallelogram is a Rhombus, then each diagonal bisects a pair of opposite angles.

58 Theorem 6-15 p.378 If a parallelogram is a Rectangle, then it s diagonals are congruent.

59 Theorem 6-16 p.383 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

60 Theorem 6-17 p.384 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

61 Theorem 6-18 p.384 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

62 Theorem 6-19 p.389 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

63 Theorem 6-20 p.391 If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

64 Theorem 6-21 p.391 If a quadrilateral is a trapezoid, then: (1) the midsegment is parallel to the bases, and (2) the length of the midsegment is half of the sum of lengths of the bases.

65 Theorem 6-22 p.392 If a quadrilateral is a kite, then its diagonals are perpendicular.

66 Concept Summary Relationships Among Quadrilaterals Turn to page 393 in your books. Read along with me as we review the Concept Summary