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Geometry: Unit 4 Congruent Triangles. VOLK SPRING 2014. Unit 4 Preview in Numbers. Instructional Days: 8 Review Days: 1 Test Days: 1 Total Homework Problems: About 120 New Theorems: 8 New Postulates: 5 New Definitions: 12. Unit 4 Preview of Topics. Identify/Classify Triangles

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unit 4 preview in numbers
Unit 4 Preview in Numbers
  • Instructional Days: 8
  • Review Days: 1
  • Test Days: 1
  • Total Homework Problems: About 120
  • New Theorems: 8
  • New Postulates: 5
  • New Definitions: 12
unit 4 preview of topics
Unit 4 Preview of Topics
  • Identify/Classify Triangles
  • SSS, SAS, ASA, AAS
  • CPCTC
  • Equilateral and Isosceles Triangles
  • Coordinate Plane Triangles
  • Bisectors, Medians, and Altitudes
triangle basics

Triangle Basics

Lesson 4.1

lesson 4 1 objectives
Lesson 4.1 Objectives

The student will be able to…

  • Identify and classify triangles by angles and sides
  • Apply the Triangle Sum Theorem.
  • Apply the Exterior Angle Theorem.
  • Name and Use CPCTC.
  • Prove triangles congruent by definition.
virginia sol standard g 6
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions
New Definitions
  • Acute Triangle
  • Obtuse Triangle
  • Right Triangle
  • Equilateral Triangle
  • Isosceles Triangle
  • Scalene Triangle
  • Auxiliary Line
  • Corollary
  • Exterior Angle
  • Remote Interior Angle
  • Congruent
  • Corresponding Parts
new postulates
New Postulates
  • Reflexive Property of Triangle Congruence - ∆ABC ≅ ∆ABC
  • Symmetric Property of Triangle Congruence - If ∆ABC ≅ ∆EFG, then ∆EFG ≅ ∆ABC.
  • Transitive Property of Triangle Congruence - If ∆ABC ≅ ∆EFG and ∆EFG ≅ ∆JKL, then ∆ABC ≅ ∆JKL.
new theorems
New Theorems
  • Triangle-Sum Theorem
  • Exterior Angle Theorem
  • Triangle Angle-Sum Corollaries
  • Third Angles Theorem
classifying triangles i
Classifying Triangles I
  • Acute Triangle: All three angles are acute.
  • Obtuse Triangle: One obtuse angle (other two acute)
  • Right Triangle: One right angle (other two acute)
classifying triangles ii
Classifying Triangles II
  • Equilateral Triangle: All three sides are congruent.
  • Isosceles Triangle: Only two sides are congruent.
  • Scalene Triangle: No two sides are congruent.
exterior angle
Exterior Angle
  • Exterior Angle of a polygon can be drawn using an auxiliary line.
triangle sum theorem
Triangle-Sum Theorem
  • The sum of the measures of the angles of a triangle is 180.
exterior angle theorem
Exterior Angle Theorem
  • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
triangle angle sum corollaries
Triangle Angle-Sum Corollaries
  • 1. The acute angles of a right triangle are complementary.
  • 2. There can be at most one right or obtuse angle in a triangle.
third angles theorem
Third Angles Theorem
  • If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
homework
Homework
  • Homework Set 4.1
  • DUE TOMORROW so DO TODAY!
lesson 4 2 objectives
Lesson 4.2 Objectives

The student will be able to…

  • Use SSS postulate to test/prove triangles congruent.
  • Use SAS postulate to test/prove triangles congruent.
  • Use ASA postulate to test/prove triangles congruent.
  • Use AAS postulate to test/prove triangles congruent.
virginia sol standard g 61
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions1
New Definitions
  • Included Angle
  • Included Side
new postulates1
New Postulates
  • Side-Side-Side (SSS) Congruence Postulate
  • Side-Angle-Side (SAS) Congruence Postulate
  • Angle-Side-Angle (ASA) Congruence Postulate
new theorems1
New Theorems
  • Angle-Angle-Side (AAS) Congruence Theorem
side side side sss congruence postulate
Side-Side-Side (SSS) Congruence Postulate
  • If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
side angle side sas congruence postulate
Side-Angle-Side (SAS) Congruence Postulate
  • If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
angle side angle asa congruence postulate
Angle-Side-Angle (ASA) Congruence 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 triangles are congruent.
angle angle side aas congruence theorem
Angle-Angle-Side (AAS) Congruence Theorem
  • If two angles and the nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the triangles are congruent.
homework1
Homework
  • Homework Set 4.2
  • DUE TOMORROW so DO TODAY!
lesson 4 3 objectives
Lesson 4.3 Objectives

The student will be able to…

  • Use SSS postulate to test/prove triangles congruent.
  • Use SAS postulate to test/prove triangles congruent.
  • Use ASA postulate to test/prove triangles congruent.
  • Use AAS postulate to test/prove triangles congruent.
virginia sol standard g 62
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions2
New Definitions
  • None Today
new postulates2
New Postulates
  • None Today
new theorems2
New Theorems
  • None Today
homework2
Homework
  • Homework Set 4.3
  • DUE TOMORROW so DO TODAY!
lesson 4 4 objectives
Lesson 4.4 Objectives

The student will be able to…

  • Use SSS postulate to test/prove triangles congruent.
  • Use SAS postulate to test/prove triangles congruent.
  • Use ASA postulate to test/prove triangles congruent.
  • Use AAS postulate to test/prove triangles congruent.
virginia sol standard g 63
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions3
New Definitions
  • None Today
new postulates3
New Postulates
  • None Today
new theorems3
New Theorems
  • None Today
homework3
Homework
  • Homework Set 4.4
  • DUE TOMORROW so DO TODAY!
lesson 4 5 objectives
Lesson 4.5 Objectives

The student will be able to…

  • Use properties of equilateral and isosceles triangles.
virginia sol standard g 64
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions4
New Definitions
  • Vertex Angle
  • Base Angles
  • Legs of Isosceles Triangle
new postulates4
New Postulates
  • CPCTC - Corresponding Parts of Congruent Triangles are Congruent
new theorems4
New Theorems
  • Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
  • Converse of Isosceles Triangle Theorem - If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
  • Equilateral Triangle Corollaries (Corollaries of the Isosceles Triangle Theorem) - 1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measure 60.
isosceles triangle theorem
Isosceles Triangle Theorem
  • If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
converse of isosceles triangle theorem
Converse of Isosceles Triangle Theorem
  • If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
equilateral triangle corollaries
Equilateral Triangle Corollaries

(Corollaries of the Isosceles Triangle Theorem)

1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measure 60.

homework4
Homework
  • Homework Set 4.5
  • DUE TOMORROW so DO TODAY!
lesson 4 6 objectives
Lesson 4.6 Objectives

The student will be able to…

  • Identify congruence transformations.
  • Verify congruence after a transformation.
virginia sol standard g 65
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions5
New Definitions
  • Transformation
  • Preimage
  • Image
  • Congruence Transformation
  • Isometry
  • Reflection
  • Translation
  • Rotation
  • Dilation
new postulates5
New Postulates
  • None Today
new theorems5
New Theorems
  • None Today
transformations
Transformations
  • A transformation transforms a preimage into a new image.
  • A congruence transformation does not change the shape of the preimage.
  • A congruence transformation is also called an isometry.
three types of isometries
Three types of Isometries
  • Reflection- Flip
  • Translation- Slide
  • Rotation- Twist/Turn
homework5
Homework
  • Homework Set 4.6
  • DUE TOMORROW so DO TODAY!
lesson 4 7 objectives
Lesson 4.7 Objectives

The student will be able to…

  • Position and label triangles in the coordinate plane.
  • Use coordinate geometry to prove triangles congruent.
virginia sol standard g 66
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions6
New Definitions
  • Coordinate Proof
new postulates6
New Postulates
  • None Today
new theorems6
New Theorems
  • None Today
coordinate proofs
Coordinate Proofs
  • Use distance, midpoint, or slope formulas to prove or disprove congruence algebraically.
  • Distance =
  • Midpoint =, )
  • Slope =
homework6
Homework
  • Homework Set 4.7
  • DUE TOMORROW so DO TODAY!
lesson 4 8 objectives
Lesson 4.8 Objectives

The student will be able to…

  • Identify and use perpendicular bisectors in triangles.
  • Identify and use angle bisectors in triangles.
  • Identify and use medians in triangles.
  • Identify and use altitudes in triangles.
virginia sol standard g 67
Virginia SOL Standard G.6
  • The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
new definitions7
New Definitions
  • Angle Bisector
  • Perpendicular Bisector
  • Altitude
  • Median
  • Point of Concurrency
  • Circumcenter
  • Incenter
  • Centroid
  • Orthocenter
new postulates7
New Postulates
  • None Today
new theorems7
New Theorems
  • None Today
angle bisectors of triangles
Angle Bisectors of Triangles
  • Angle Bisectors of a triangle intersect at the Incenter.
perpendicular bisectors of triangles
Perpendicular Bisectors of Triangles
  • Perpendicular Bisectors of a triangle intersect at the Circumcenter.
medians of triangles
Medians of Triangles
  • Medians of a triangle intersect at the Centroid.
altitudes of triangles
Altitudes of Triangles
  • Altitudes of a triangle intersect at the Orthocenter.
homework7
Homework
  • Homework Set 4.8
  • DUE TOMORROW so DO TODAY!
closing

Closing

Unit 4

unit 4 review in numbers
Unit 4 Review in Numbers
  • Instructional Days: 8
  • Review Days: 1
  • Test Days: 1
  • Total Homework Problems: About 120
  • New Theorems: 8
  • New Postulates: 5
  • New Definitions: 12
unit 4 review of topics
Unit 4 Review of Topics
  • Identify/Classify Triangles
  • SSS, SAS, ASA, AAS
  • CPCTC
  • Equilateral and Isosceles Triangles
  • Coordinate Plane Triangles
  • Bisectors, Medians, and Altitudes