Warm-Up If a triangle has two side lengths of 12 and 5, what is the range of possible values for the third side? 2.

Classifying Triangles - Sides Definitions: Scalene: A triangle with no congruent sides Isosceles: A triangle with at least two congruent sides Equilateral: A triangle with three congruent sides Fill in the blanks with one of the following options: sometimes, never, always A scalene triangle is _____________________ an equilateral triangle An equilateral is _____________________ always an isosceles triangle An isosceles triangle is __________________ an equilateral triangle

Classifying Triangles - Angles Definitions: Acute: a triangle with three acute angles Equiangular triangle: a triangle with congruent angles Right: a triangle with one right angle Obtuse: a triangle with one obtuse angles Is it possible to have more than one obtuse angle? Fill in the blanks with one of the following options: sometimes, never, always An obtuse triangle is _____________________ an acute triangle An acute triangle is ______________________ an equiangular triangle And equiangular triangle is ___________________ an acute triangle

Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

What would the Converse be? If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

What would the Converse be? If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

Proof of Pythagorean Theorem

Practice – pg. 348 #2 The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse?

Practice – pg. 348 #3 The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form

Practice 5 3

The sides of a triangle measure 2.4, 3.2, and 4. Is this triangle a right triangle?

Pythagorean Inequalities

Example Classify the triangle using the lengths of its sides. a) 6, 8, 10 b) 4, 5, 6

Pythagorean Inequalities

Example Pg. 351 #7 Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right?

Pythagorean Triples A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a 2 + b 2 = c 2 The most common Pythagorean triples are: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 Their multiples are also triples 6, 8, 10 __, __,__ __,__,__ __,__,__

Pythagorean Triples Why are these helpful to know? Less work! Super helpful on timed tests (AKA the SAT!)

Homework Pg Evens