1. Introduction to Triangles
6-1
Introduction to Triangles
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry

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

3. 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.

4. Vocabulary acute triangle equiangular triangle right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
auxiliary line

5. 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.

6. C A B AB, BC, and AC are the sides of ABC.
A, B, C are the triangle's vertices.

7. Acute Triangle Three acute angles Triangle Classification
By Angle Measures
Acute Triangle
Three acute angles

8. Three congruent acute angles
Triangle Classification
By Angle Measures
Equiangular Triangle
Three congruent acute angles

9. Right Triangle One right angle Triangle Classification
By Angle Measures
Right Triangle
One right angle

10. Obtuse Triangle One obtuse angle Triangle Classification
By Angle Measures
Obtuse Triangle
One obtuse angle

11. 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.

12. Equilateral Triangle Three congruent sides Triangle Classification
By Side Lengths
Equilateral Triangle
Three congruent sides

13. The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.

15. At least two congruent sides
Triangle Classification
By Side Lengths
Isosceles Triangle
At least two congruent sides

16. 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.

18. The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.”
Reading Math

19. Scalene Triangle No congruent sides Triangle Classification
By Side Lengths
Scalene Triangle
No congruent sides

20. Remember!
When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

21. Example 2A: Classifying Triangles by Side Lengths
Classify EHF by its side lengths.
From the figure, So HF = 10, and EHF is isosceles.

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

23. Check It Out! Example 2
Classify ACD by its side lengths.
From the figure, So AC = 15, and ACD is scalene.

25. 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

26. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

27. 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

28. 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.

29. 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.