Geometry: Unit 4 Congruent Triangles

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# Geometry: Unit 4 Congruent Triangles - PowerPoint PPT Presentation

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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### 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
• SSS, SAS, ASA, AAS
• CPCTC
• Equilateral and Isosceles Triangles
• Coordinate Plane Triangles
• Bisectors, Medians, and Altitudes

### Triangle Basics

Lesson 4.1

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
• 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
• 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
• 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
• Triangle-Sum Theorem
• Exterior Angle Theorem
• Triangle Angle-Sum Corollaries
• Third Angles Theorem
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
• 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 of a polygon can be drawn using an auxiliary line.
Triangle-Sum Theorem
• The sum of the measures of the angles of a triangle is 180.
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
• 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
• 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 Set 4.1
• DUE TOMORROW so DO TODAY!

### Triangle Proofs I

Lesson 4.2

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.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
• Included Angle
• Included Side
New Postulates
• Side-Side-Side (SSS) Congruence Postulate
• Side-Angle-Side (SAS) Congruence Postulate
• Angle-Side-Angle (ASA) Congruence Postulate
New Theorems
• Angle-Angle-Side (AAS) Congruence Theorem
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
• 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
• 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
• 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.
Homework
• Homework Set 4.2
• DUE TOMORROW so DO TODAY!

### Triangle Proofs II

Lesson 4.3

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.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
• None Today
New Postulates
• None Today
New Theorems
• None Today
Homework
• Homework Set 4.3
• DUE TOMORROW so DO TODAY!

### Triangle Proofs III

Lesson 4.4

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.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
• None Today
New Postulates
• None Today
New Theorems
• None Today
Homework
• Homework Set 4.4
• DUE TOMORROW so DO TODAY!

### Equilateral & Isosceles

Lesson 4.5

Lesson 4.5 Objectives

The student will be able to…

• Use properties of equilateral and isosceles triangles.
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
• Vertex Angle
• Base Angles
• Legs of Isosceles Triangle
New Postulates
• CPCTC - Corresponding Parts of Congruent Triangles are Congruent
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
• 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.

Homework
• Homework Set 4.5
• DUE TOMORROW so DO TODAY!

### Congruence Transformations

Lesson 4.6

Lesson 4.6 Objectives

The student will be able to…

• Identify congruence transformations.
• Verify congruence after a transformation.
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
• Transformation
• Preimage
• Image
• Congruence Transformation
• Isometry
• Reflection
• Translation
• Rotation
• Dilation
New Postulates
• None Today
New Theorems
• None Today
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
• Reflection- Flip
• Translation- Slide
• Rotation- Twist/Turn
Homework
• Homework Set 4.6
• DUE TOMORROW so DO TODAY!

### Coordinate Plane Triangle Proofs

Lesson 4.7

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.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
• Coordinate Proof
New Postulates
• None Today
New Theorems
• None Today
Coordinate Proofs
• Use distance, midpoint, or slope formulas to prove or disprove congruence algebraically.
• Distance =
• Midpoint =, )
• Slope =
Homework
• Homework Set 4.7
• DUE TOMORROW so DO TODAY!

### Special Segments

Lesson 4.8

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.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
• Angle Bisector
• Perpendicular Bisector
• Altitude
• Median
• Point of Concurrency
• Circumcenter
• Incenter
• Centroid
• Orthocenter
New Postulates
• None Today
New Theorems
• None Today
Angle Bisectors of Triangles
• Angle Bisectors of a triangle intersect at the Incenter.
Perpendicular Bisectors of Triangles
• Perpendicular Bisectors of a triangle intersect at the Circumcenter.
Medians of Triangles
• Medians of a triangle intersect at the Centroid.
Altitudes of Triangles
• Altitudes of a triangle intersect at the Orthocenter.
Homework
• Homework Set 4.8
• DUE TOMORROW so DO TODAY!

### Closing

Unit 4

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
• Identify/Classify Triangles
• SSS, SAS, ASA, AAS
• CPCTC
• Equilateral and Isosceles Triangles
• Coordinate Plane Triangles
• Bisectors, Medians, and Altitudes