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 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°