Chapter 4 Triangles Triangle Definitions A triangle has

Chapter 4 – Triangles Triangle Definitions • A triangle has three angles and three sides – Each angle is a vertex of the triangle B • – A triangle is named by listing its vertices in order – In the triangle at left, BC is the side opposite A • Triangles can be classified according to – the measure of their angles • Acute, right, obtuse, equiangular – the lengths of their sides • Scalene, isosceles, equilateral A • C •

Chapter 4 – Triangles Angle Measures of Triangles • Triangle Sum Theorem – The sum of the measures of the angles of a triangle is 180° • Corollary to the Triangle Sum Theorem – The acute angles of a right triangle are complementary • Exterior Angle Theorem – The measure of an exterior angle of a triangle is equal to the sum of the measures of the Exterior angle two nonadjacent interior angles Adjacent interior angle Nonadjacent interior angles

Chapter 4 – Triangles Isosceles and Equilateral Triangles • Isosceles Triangle Definitions – Know the names of the parts of an isosceles triangle • Isosceles Triangle Theorem and Converse – The base angles of an isosceles triangle are congruent – If two angles of a triangle are congruent then it is isosceles • Equilateral Triangle Theorem and Converse – If a triangle is equilateral, then it is equiangular • Each angle in an equilateral triangle is 60° – If a triangle is equiangular, then it is equilateral

Chapter 4 – Triangles The Pythagorean Theorem • Right Triangle Definitions – Know the names of the parts of a right triangle hypotenuse legs • The hypotenuse is opposite the right angle and is always the longest side of the triangle – If a and b are the lengths of the legs, and c is the length of the hypotenuse, then a 2 + b 2 = c 2 – In a Pythagorean triple (such as 3 -4 -5), a, b, and c are integers • The Distance Formula – The distance AB between points A and B on a coordinate plane can be found by using the Pythagorean Theorem – (AB)2 = (x 2 – x 1)2 + (y 2 – y 1)2 or AB = √(x 2 – x 1)2 + (y 2 – y 1)2

Chapter 4 – Triangles Converse of the Pythagorean Theorem • Determine whether a triangle is acute, right, or obtuse by using the Pythagorean Theorem – If the lengths of the three sides of a triangle satisfy the Pythagorean theorem, then the triangle is a right triangle • Remember that c is always the longest side of the triangle – If a 2 + b 2 = c 2 then the triangle is a right triangle – If a 2 + b 2 > c 2 then side c is too short, so the triangle is an acute triangle – If a 2 + b 2 < c 2 then side c is too long, so the triangle is an obtuse triangle

Chapter 4 – Triangles Medians of a Triangle • A median of a triangle is a segment from a vertex to the midpoint of the opposite side – The three medians of a triangle intersect at the centroid, which is the center of gravity, or balance point, of the triangle • Intersection of the medians of a triangle – The centroid of a triangle on each median is two-thirds of the distance from each vertex to the opposite side – The distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the side

Chapter 4 – Triangles Triangle Inequalities • Relation between sides and opposite angles – In any triangle the longest side is opposite the largest angle and the shortest side is opposite the smallest angle – In any triangle the largest angle is opposite the longest side and the smallest angle is opposite the shortest side • Lengths of sides of a triangle – The sum of the lengths of the two shorter sides of a triangle is greater than the length of the longest side