unit 4 notes: triangles 4-1 triangle angle-sum theorem...

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Unit 4 Notes: Triangles

4-1 Triangle Angle-Sum Theorem

Angle review, label each angle with the correct classification:

___________________ _____________________ _______________________

Triangle – a polygon with three sides.

There are two ways to classify triangles: by angles and by sides

There are four ways to classify a triangle by its angles:

________________ __________________ _________________ __________________

There are three ways to classify triangles based on sides

____________ _________________ _______________

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Can triangles be named in the following ways? Yes/No, Why?

Acute Scalene ___________________________________

Isosceles Right __________________________________

Acute Equilateral ____________________________

Obtuse Equilateral ________________________________

Right Equilateral _________________________________

Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

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Triangle Exterior Angle Theorem

side side

Remote

remote

exterior

2

m<3

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Practice, find the measure of each variable

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4-2 Triangle Congruence & Third Angle Theorem

Congruent Polygons – If two polygons are congruent, then their corresponding parts are congruent.

The converse can be used to prove two figures are congruent.

Examples:

Third-Angle-Theorem

If two angles of one triangle are congruent to two angles in another triangle, then the third angles in the

triangles are also congruent.

Examples:

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B

H

M A

T

Given:

B is the midpoint of Prove:

Statement Reason

1. _______________________________________ 1. _____________________________________

2. _______________________________________ 2. _____________________________________

3. _______________________________________ 3. _____________________________________

4. _______________________________________ 4. _____________________________________

5. _______________________________________ 5. _____________________________________

6. _______________________________________ 6. _____________________________________

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4-3 SSS & SAS Congruence Postulates

Side – Side – Side Triangle Congruence Postulate (SSS)

If all three sides of one triangle are congruent to three sides of another triangle, then the triangles are

congruent.

Examples:

Given: , B is the midpoint of AC

Prove: CBDABD

BA C

D

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Side – Angle – Side Triangle Congruence Postulate (SAS)

If two sides and an included angle of one triangle are congruent to two sides and an included angle

of another triangle, then the two triangles are congruent.

Included Angle - The angle between two sides of a triangle.

Examples:

Given: C is the midpoint of AE and BD

Prove:

D

C

A

E

B

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4-4 ASA & AAS Conguence

Angle – Side – Angle Triangle Congruence Postulate (ASA)

If two consecutive angles and the included side of a triangle are congruent to two consecutive angles and the

included side of another triangle, then the triangles are congruent.

Examples:

Given: C is the midpoint of AE , A E

Prove:

D

C

A

E

B

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Angle – Angle – Side Triangle Congruence Theorem (AAS)

If two consecutive angles and the corresponding side of one triangle are congruent to two consecutive

angles and the corresponding side of another triangle, then the triangles are congruent.

*This theorem stems from the third angle theorem.

Examples:

Given: ACDB ,

Prove: CBDABD

BA C

D

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4-5 Corresponding parts of congruent triangles are congruent CPCTC

Corresponding parts of congruent figures / triangles are congruent – CPCFC or CPCTC

If two figures are congruent, then all corresponding parts of those Figures/Triangles are congruent.

Examples:

Given: The figure with AB AD , DC AD , AB = CD, and E is the midpoint of AD .

Prove:

C

EA D

B

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4-6 Base Angle Theorem

Base angle theorem for Isosceles triangles –

If two sides of a triangle are congruent, then the base angles are congruent.

Converse to the base angle theorem for isosceles triangles –

If the base angles of a triangle are congruent, then the sides opposite the base angles are congruent.

Isosceles vertex angle bisector theorem –

The bisector of a vertex angle of an isosceles triangle is perpendicular to the base of the triangle.

Corollaries- If a triangle is equilateral, then it is equiangular.

If a triangle is equiangular, then it is equilateral.

Examples:

Given: EBC ECB , AB = CD, and E is the midpoint of AD

Prove:

C

EA D

B

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4-7 Hypotenuse Leg (HL) Theorem & Leg Leg (LL) Theorem

Hypotenuse Leg (HL) Congruence Theorem -

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right

triangle, then the triangles are congruent.

Leg Leg (LL) Congruence Theorem -

If the leg and leg of one right triangle are congruent to the leg and leg of another right triangle, then the

triangles are congruent. These is not used often because of the SAS Postulate

Examples:

Given: BCAB , BCDC , and DBAC

Prove: BAC CDB

E

D

B C

A