1. Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 3.3 Isosceles Triangles

2. Isosceles Triangle The two congruent sides are called legs

3. Isosceles Triangle The two congruent sides are called legs The third side is the base

4. Isosceles Triangle The two congruent sides are called legs The third side is the base The point at which the two legs meet is the vertex of the triangle.

5. Isosceles Triangle The two congruent sides are called legs The third side is the base The point at which the two legs meet is the vertex of the triangle. The angle formed by the legs is the vertex angle (opposite the base)

6. Isosceles Triangle The two congruent sides are called legs The third side is the base The point at which the two legs meet is the vertex of the triangle. The angle formed by the legs is the vertex angle (opposite the base) The angles adjacent to the base are called base angles.

7. Base Leg Vertex

8. Vertex angle Base angle

9. Base Leg Vertex Vertex angle Base angle

10. Informal Definitions Each angle of a triangle has a unique angle bisector indicated by a ray or segment from the angle’s vertex

11. Informal Definitions The median is a segment that joins one angle of a triangle to the midpoint of the opposite side

12. Informal Definitions An altitude is a line segment drawn from a vertex of a triangle to the opposite side so that the segment is perpendicular to the opposite side.

13. Informal Definitions The perpendicular bisector of a side of a triangle is a line perpendicular to the side that intersects at the midpoint of the side.

14. Theorem 3.3.1 Corresponding altitudes of congruent triangles are congruent

15. Theorem 3.3.2 The bisector of the vertex angel of an isosceles triangle separates the triangle into two congruent triangles.

16. Theorem 3.3.3 If two sides of a triangle are congruent, then the angles opposite these sides are also congruent.

17. Theorem 3.3.3 If two sides of a triangle are congruent, then the angles opposite these sides are also congruent. Alternatively: “The base angles of an isosceles triangle are congruent.”

18. Theorem 3.3.4 If two angles of a triangle are congruent, then the sides opposite these angles are congruent.

19. Theorem 3.3.4 If two angles of a triangle are congruent, then the sides opposite these angles are congruent. Note: This is the converse of Theorem 3.3.3: “If two sides of a triangle are congruent, then the angles opposite these sides are also congruent.”

20. Informal Definitions If all three sides of a triangle are congruent, the triangle is equilateral. If all three angles of a triangle are congruent, the triangle is equiangular.

21. Corollaries 3.3.5 and 3.3.6 Corollary 3.3.5: An equilateral triangle is also equiangular. Corollary 3.3.6: An equiangular triangle is also equilateral.

22. Definition The perimeter of a triangle is the sum of the lengths of its sides. Thus if a, b, and c are the lengths of the three sides, then the perimeter P is given by P = a + b + c a c b

23. Properties of Scalene Triangles Sides:No two are . Angles:Sum of  s is 180 

24. Properties of Isosceles Triangles Sides:Exactly two are  Angles:Sum of  s is 180  Two  s 

25. Properties of Equilateral (or Equiangular) Triangles Sides:All three are  Angles:Sum of  s is 180  Three   s All  s measure 60 

26. Properties of Acute Triangles Sides:Possibly two or three  sides Angles:All  s are acute Sum of  s is 180  Possibly two or three   s

27. Properties of Right Triangles Sides:Possibly two  sides Angles:One right  Sum of  s is 180  Possibly two  45   s Acute  s are complementary

28. Properties of Obtuse Triangles Sides:Possibly two  sides Angles:One obtuse  Sum of  s is 180  Possibly two acute   s