4.1- 4.2 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

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 Equilateral Triangles are ALWAYS Equilateral

Parts of the Right Triangle Legs- sides of a right triangle Hypotenuse- the side across from the right angle.

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

Corollary-a theorem with a proof that follows as a direct result of another theorem. Corollary 1.) The acute angles of a right triangle are complementary. Corollary 2.) There can be at most one right or one obtuse angle in a triangle.

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

Example 4:Solve for x

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

Example 5 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°