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Academic institution promoting High expectations resulting in Successful students
Assumption High School
Bell Work
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Assumption High School
Geometry2016 – 2017
Day 36Topic: Chapter 4 Congruent FiguresChapter 6 Polygons & Quads
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Common Core State StandardsGeometry (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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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)
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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)
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Chapter Review
• Open your books to page 273
• You will be given 30 minutes to work on the Chapter Review (1-33) on pages 273 – 276
• 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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
http://fordmathletes.weebly.com/
Academic institution promoting High expectations resulting in Successful students
Assumption High School
We are a TEAM
• Remember what it mean to be part of a team…
•Together
•Everyone
•Achieves
•More
Mr. Matthew J Ford
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Following rules means being HAPPY
• Being HAPPY means being:
• Honest
• Accountable
• Proactive
• Positive
• Yourself
Mr. Matthew J Ford
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Any Questions???
Academic institution promoting High expectations resulting in Successful students
Assumption High School
You guy Rock!!!
•I Love all you crazy kiddos…
•You guys are real Mathletes in Training!
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Common Core State StandardsGeometry (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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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)
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Review of Polygons
Academic institution promoting High expectations resulting in Successful students
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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−2 180
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Concave or Convex Polygons
Academic institution promoting High expectations resulting in Successful students
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Equilateral Polygons
•Polygons with all sides congruent
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Equiangular Polygons
•Polygons with all angles congruent
Academic institution promoting High expectations resulting in Successful students
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Regular Polygons
Academic institution promoting High expectations resulting in Successful students
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6-1 Polygon Angle-Sum Theorem
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Corollary to the Polygon Angle-Sum Theorem
Academic institution promoting High expectations resulting in Successful students
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Regular Polygons
Academic institution promoting High expectations resulting in Successful students
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6-2 Polygon Exterior Angle-Sum Theorem
Academic institution promoting High expectations resulting in Successful students
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Regular Polygon Angle Examples
Academic institution promoting High expectations resulting in Successful students
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Recall
Academic institution promoting High expectations resulting in Successful students
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Khan Academy / Special Parallelograms
Academic institution promoting High expectations resulting in Successful students
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Quadrilaterals
• Convex
• Trapezoids
• Parallelograms are comprised of three different types of quadrilaterals• Rectangle
• Square
• Rhombus
• Concave
Academic institution promoting High expectations resulting in Successful students
Assumption High School
It’s that time again…
Academic institution promoting High expectations resulting in Successful students
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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
Academic institution promoting High expectations resulting in Successful students
Assumption High School
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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Parallelogram
•A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
•a four-sided plane rectilinear figure with opposite sides parallel.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
ParallelogramThere 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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Opposite Sides
• In a quadrilateral, opposite sides do not share a vertex.
Academic institution promoting High expectations resulting in Successful students
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Opposite Angles
• In a quadrilateral, opposite angles do not share a side.
Academic institution promoting High expectations resulting in Successful students
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Consecutive Angles
•Angles of a polygon that share a side are consecutive angles.
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Quadrilaterals / Parallelograms
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Theorem 6-8
•If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a Parallelogram.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-9
•If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a Parallelogram.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Theorem 6-10
•If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a Parallelogram.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-11
•If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-12
•If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a Parallelogram.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Theorem 6-13 p.376
•If a parallelogram is a Rhombus, then its diagonals are perpendicular.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-14 p.376
•If a parallelogram is a Rhombus, then each diagonal bisects a pair of opposite angles.
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Theorem 6-15 p.378
•If a parallelogram is a Rectangle, then it’s diagonals are congruent.
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Theorem 6-16 p.383
•If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
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Theorem 6-17 p.384
•If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-18 p.384
•If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
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Theorem 6-19 p.389
•If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.
Academic institution promoting High expectations resulting in Successful students
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Theorem 6-20 p.391
•If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.
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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.
Academic institution promoting High expectations resulting in Successful students
Assumption High School
Theorem 6-22 p.392
•If a quadrilateral is a kite, then its diagonals are perpendicular.