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4.1 Triangles and Angles

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**Definition of a triangle**

A triangle is three segments joined at three noncollinear end points.

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**Types of Triangles by Sides**

3 Sides congruent → Equilateral

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**Types of Triangles by Sides**

2 Sides congruent → Isosceles Part of the Isosceles Triangle

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**Types of Triangles by Sides**

No Sides congruent → Scalene

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**Types of Triangles by Sides**

3 Sides congruent → Equilateral 2 Sides congruent → Isosceles No Sides congruent → Scalene

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**Types of Triangle by Angles**

All Angles less than 90 degrees → Acute

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**Types of Triangle by Angles**

One Angle greater than 90 degrees, but less than 180° → Obtuse

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**Types of Triangle by Angles**

One Angle equal to 90 degrees → Right

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**How to classify a triangle**

Choose one from each category Sides Angles____ Scalene Acute Isosceles Right Equilateral Obtuse

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**Equiangluar All the angles are Equal**

This will ALWAYS be paired up with Equilateral

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**Parts of the Right Triangle**

Across from the right angle is the hypotenuse.

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**Interior Angles vs. Exterior Angles**

M a N b c P Interior angles: <a, <b, <c Exterior angles: <M, <N, <P

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**m<a + m<b + m<c = 180°**

Triangle Sum Theorem The sum of the three interior angles of a triangle is 180º a b c m<a + m<b + m<c = 180°

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Triangle Sum Theorem Solve for x

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Example 2 Find the measure of each angle. 2x + 10 x x + 2

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**Exterior Angle Theorem**

The measure of an exterior angle equals the measure of the two nonadjecent interior angles.

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**Example 3 Given that ∠ A is 50º and ∠B is 34º, what is the measure of**

∠BCD? What is the measure of ∠ACB? D A B C

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Solve for x

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**Corollary for the fact that interior angles add to 180º**

The acute angles of a Right triangle are complementary.

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Example 4 A. Given the following triangle, what is the length of the hypotenuse? B. What are the length of the legs? C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement? 13 12 5

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Example 6 Find the missing measures 80° 53°

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**Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90° Statement**

Reason 1. mC = 90° 2. mA + mB + mC = 180° 3. mA + mB + 90° = 180° 4. mA + mB = 90°

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