4-1 Classifying Triangles



4-1 Classifying Triangles

HW: p219 #3-10, 12-19, 30, 31, 41-45, 47

New Terms:

|Acute: |Obtuse: |Acute triangle: |

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|Equiangular triangle: |Right triangle: |Obtuse triangle: |

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|Equilateral triangle: |Isosceles triangle: |Scalene triangle: |

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You can classify triangles by their _____________ or their _____________.

Label the vertices of the following triangle:

List the sides:

List the angles:

Example 1:

Classify triangle BDC by its angle measures.

(B is a ______ angle.

So BDC is a _______ triangle.

ADC is a ________ triangle.

Example 2:

Classify triangle ABD by its angle measures.

(ABD and (CBD form a linear pair, so they are _______________.

Therefore m(ABD + m(CBD = ____°.

By substitution, m(ABD + 100° = 180°. So m(ABD = ____°. ABD is a ___________ triangle by definition.

Important Note: You cannot assume sides or angles are congruent just by looking at the picture. They need the proper notation or you need to prove it!

Example 3:

Classify EHF by its side lengths.

From the figure, . So HF = ____, and EHF is ________.

Example 4:

Classify EHG by its side lengths.

By the Segment Addition Postulate, EG = ___ + ___ = 10 + 4 = 14.

Since no sides are congruent, EHG is ________.

Example 5:

Find the side lengths of JKL.

Example 6:

A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?

4-2 Angle Relationships in Triangles

HW: p227 #4-22, 33-36

Complete the Lab Activity on p.222.

Answers to questions 1-4:

1.

2.

3.

4.

A _______________ is a line that is added to a figure to aid in a proof.

Example 1:

After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find m(XYZ.

A ____________ is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

Example 2:

One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?

Example 3:

The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?

Additional Vocabulary:

- Interior Angle:

- Exterior Angle:

- Remote Interior Angles:

- Exterior Angle Theorem:

Example 4:

4-3 Angle Relationships in Triangles

HW: w/o proofs #2-10, 13-18, 21-25, 31-34

w/proofs p228 #24, 25, 38

p234 #11, 12, 19, 20, 26, 27

[pic]

List the congruent sides:

[pic]

List the congruent angles:

[pic]

When you write a statement such as (ABC ( (DEF, you are also stating which parts are congruent!

In a congruence statement, the order of the vertices indicates the corresponding parts.

Example 1:

Given: ∆PQR ( ∆STW

Identify all pairs of corresponding congruent parts.

Angles: (P ( ( , (Q ( ( , (R ( (

Sides: PQ ( , QR ( , PR (

Example 2:

Given: ∆ABC ( ∆DBC

Find the value of x.

Example 3:

Given: ∆ABC ( ∆DEF

Find m(F.

Example 4:

Given:

[pic]

Prove: [pic]

|Statements |Reasons |

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Example 5:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 6:

The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent.

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 7:

Use the diagram to prove the following.

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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4-4 Triangle Congruence: SSS and SAS

HW: p.245 #1-3, 5-18, 21

[pic]

Triangle Rigidity:

Example 1:

Given: Diagram

Prove: ∆ABC ( ∆DBC

|Statements |Reasons |

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Example 2:

Given: Diagram

Prove: ∆ABC ( ∆CDA

|Statements |Reasons |

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An __________ __________ is an angle formed by two adjacent sides of a polygon.

(B is the __________ __________ between sides _____ and _____.

[pic]

Caution!!! The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

Example 3:

Given: Diagram

Prove: ∆XYZ ( ∆VWZ

|Statements |Reasons |

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Example 4:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 5:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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4-5 Triangle Congruence: ASA, AAS, and HL

HW: p256 #4-8, 11-17, 22, 23, 26-28

An __________ __________ is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

[pic]

Example 1:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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[pic]

Example 2:

Given: (X ( (V, (YZW ( (YWZ, XY ( VY

Prove: [pic]

|Statements |Reasons |

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Example 3:

Given: JL bisects (KLM, (K ( (M

Prove: [pic]

|Statements |Reasons |

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[pic]

Example 4:

Given: See picture

Prove: [pic]

|Statements |Reasons |

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|Proving Triangles are Congruent |

| |Def. [pic] |SSS |SAS |ASA |AAS |HL |

|Words | | | | | | |

|Picture | | | | | | |

4-6 Triangle Congruence: CPCTC

HW: p.262 #3, 4, 7-11, 14, 15, 17-21

CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

Example 1:

A and B are edges of the ravine. What is AB?

Example 2:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 3:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 4:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 5:

Given: [pic]

Prove: [pic]

|Statements |Reasons |

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Example 6:

Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)

Prove: (DEF ( (GHI

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Exterior

Find m(B.

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