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### Geometry: Unit 4 Congruent Triangles

### Triangle Basics

### Triangle Proofs I

### Triangle Proofs II

### Triangle Proofs III

Virginia SOL Standard G.6### Equilateral & Isosceles

Virginia SOL Standard G.6### Congruence Transformations

Virginia SOL Standard G.6### Coordinate Plane Triangle Proofs

Virginia SOL Standard G.6### Special Segments

Virginia SOL Standard G.6### Closing

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

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!

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!

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!

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.

New Definitions

- None Today

New Postulates

- None Today

New Theorems

- None Today

Homework

- Homework Set 4.4
- DUE TOMORROW so DO TODAY!

Lesson 4.5

Lesson 4.5 Objectives

The student will be able to…

- Use properties of equilateral and isosceles triangles.

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!

Lesson 4.6

Lesson 4.6 Objectives

The student will be able to…

- Identify congruence transformations.
- Verify congruence after a transformation.

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!

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.

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!

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.

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!

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

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