 4.1 Triangles and Angles.

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4.1 Triangles and Angles

Definition of a triangle
A triangle is three segments joined at three noncollinear end points.

Types of Triangles by Sides
3 Sides congruent → Equilateral

Types of Triangles by Sides
2 Sides congruent → Isosceles Part of the Isosceles Triangle

Types of Triangles by Sides
No Sides congruent → Scalene

Types of Triangles by Sides
3 Sides congruent → Equilateral 2 Sides congruent → Isosceles No Sides congruent → Scalene

Types of Triangle by Angles
All Angles less than 90 degrees → Acute

Types of Triangle by Angles
One Angle greater than 90 degrees, but less than 180° → Obtuse

Types of Triangle by Angles
One Angle equal to 90 degrees → Right

How to classify a triangle
Choose one from each category Sides Angles____ Scalene Acute Isosceles Right Equilateral Obtuse

Equiangluar All the angles are Equal
This will ALWAYS be paired up with Equilateral

Parts of the Right Triangle
Across from the right angle is the hypotenuse.

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

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°

Triangle Sum Theorem Solve for x

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

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

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

Solve for x

Corollary for the fact that interior angles add to 180º
The acute angles of a Right triangle are complementary.

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

Example 6 Find the missing measures 80° 53°

Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90° Statement
Reason 1. mC = 90° 2. mA + mB + mC = 180° 3. mA + mB + 90° = 180° 4. mA + mB = 90°