Introduction to Triangles 6-1 Introduction to Triangles Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry
Warm Up Classify each angle as acute, obtuse, or right. 1. 2. 3. right 1. 2. 3. 4. If the perimeter is 47, find x and the lengths of the three sides. right acute obtuse x = 5; 8; 16; 23
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.
Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle auxiliary line
Recall that a triangle ( ) is a polygon with three sides 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.
C A B AB, BC, and AC are the sides of ABC. A, B, C are the triangle's vertices.
Acute Triangle Three acute angles Triangle Classification By Angle Measures Acute Triangle Three acute angles
Three congruent acute angles Triangle Classification By Angle Measures Equiangular Triangle Three congruent acute angles
Right Triangle One right angle Triangle Classification By Angle Measures Right Triangle One right angle
Obtuse Triangle One obtuse angle Triangle Classification By Angle Measures Obtuse Triangle One obtuse angle
Check It Out! Example 1 Classify FHG by its angle measures. EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°. FHG is an equiangular triangle by definition.
Equilateral Triangle Three congruent sides Triangle Classification By Side Lengths Equilateral Triangle Three congruent sides
The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.
At least two congruent sides Triangle Classification By Side Lengths Isosceles Triangle At least two congruent sides
Recall that an isosceles triangle has at least two congruent sides 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.
The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.” Reading Math
Scalene Triangle No congruent sides Triangle Classification By Side Lengths Scalene Triangle No congruent sides
Remember! When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.
Example 2A: Classifying Triangles by Side Lengths Classify EHF by its side lengths. From the figure, . So HF = 10, and EHF is isosceles.
Example 2B: 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.
Check It Out! Example 2 Classify ACD by its side lengths. From the figure, . So AC = 15, and ACD is scalene.
An auxiliary line used in the Triangle Sum Theorem 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
A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.
The interior is the set of all points inside the figure The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Exterior Interior
An interior angle is formed by two sides of a triangle 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.
Each exterior angle has two remote interior angles 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. The remote interior angles of 4 are 1 and 2. Exterior Interior 3 is an interior angle.