4 2 Triangles 6 1 Classifying Introduction to

4 -2 Triangles 6 -1 Classifying Introduction to Triangles Warm Up Lesson Presentation Lesson Quiz Holt. Geometry Mc. Dougal Geometry

4 -2 Classifying Triangles Warm Up Classify each angle as acute, obtuse, or right. 1. right 3. 2. acute obtuse 4. If the perimeter is 47, find x and the lengths of the three sides. x = 5; 8; 16; 23 Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Objectives 1. Classify triangles by their angle measures and side lengths. 2. Use triangle classification to find angle measures and side lengths. 3. Identify the formulas to calculate area and perimeter and apply these to solve problems. 4. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle auxiliary line Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. Holt Mc. Dougal Geometry

4 -2 Classifying Triangles C A B AB, BC, and AC are the sides of A, B, C are the triangle's vertices. Holt Mc. Dougal Geometry ABC.

4 -2 Classifying Triangles Triangle Classification By Angle Measures Acute Triangle Three acute angles Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Angle Measures Equiangular Triangle Three congruent acute angles Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Angle Measures Right Triangle One right angle Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Angle Measures Obtuse Triangle One obtuse angle Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Check It Out! Example 1 Classify FHG by its angle measures. EHG is a right angle. Therefore m EHF +m FHG = 90°. By substitution, 30°+ m FHG = 90°. So m FHG = 60°. FHG is an equiangular triangle by definition. Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Side Lengths Equilateral Triangle Three congruent sides Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Side Lengths Isosceles Triangle At least two congruent sides Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 3 is the vertex angle. 1 and 2 are the base angles. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent. ” Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Triangle Classification By Side Lengths Scalene Triangle No congruent sides Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Remember! When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent. Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Example 2 A: Classifying Triangles by Side Lengths Classify EHF by its side lengths. From the figure, isosceles. Holt Mc. Dougal Geometry . So HF = 10, and EHF is

4 -2 Classifying Triangles Example 2 B: Classifying Triangles by Side Lengths Classify EHG by its side lengths. By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene. Holt Mc. Dougal Geometry

4 -2 Classifying Triangles Check It Out! Example 2 Classify ACD by its side lengths. From the figure, scalene. Holt Mc. Dougal Geometry . So AC = 15, and ACD is

4 -2 Classifying Triangles Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Exterior Interior Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. 4 is an exterior angle. Exterior Interior 3 is an interior angle. Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. 4 is an exterior angle. Exterior Interior The remote interior angles of 4 are 1 and 2. 3 is an interior angle. Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles Holt Mc. Dougal Geometry

4 -3 Angle Relationships in Triangles Holt Mc. Dougal Geometry