Unit 2 Triangles Triangle Inequalities and Isosceles Triangles

2 Triangle Inequality The shortest side is across from the smallest angle. The largest angle is across from the longest side. AB = 4.3 cm BC = 3.2 cm AC = 5.3 cm 54  37  89  B C A

3 Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. C A B 4 cm 6 cm 5 cm

4 Triangle Inequality – examples… For the triangle, list the sides in order from shortest to longest measure. 8x-10 7x+6 7x+8 C A B (7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180° 22 x + 4 = 180 ° 22x = 176 X = 8 m<C = 7x + 8 = 64 ° m<A = 7x + 6 = 62 ° m<B = 8x – 10 = 54 ° 64 °62 ° 54 °

5 The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 1: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Corollary 2: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. Converse: Converse Theorem & Corollaries

6 Triangle Inequality Rule: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a Example:Determine if it is possible to draw a triangle with side measures 12, 11, and > 17  Yes > 12  Yes > 11  Yes Therefore a triangle can be drawn.

7 Finding the range of the third side: Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides. Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side. The maximum value (if x is the largest side of the triangle)3 + 8 > x 11 > x The minimum value (if x is not that largest side of the ∆)8 – 3 > x 5> x Range of the third side is 5 < x < 11.

8 Parts of an Isosceles Triangle An isosceles triangle is a triangle with two congruent sides. The congruent sides are called legs and the third side is called the base. 3 Leg Base 21  1 and  2 are base angles  3 is the vertex angle

9 Isosceles Triangle Rule By the Isosceles Triangle Theorem, the third angle must also be x. Therefore, x + x + 50 = 180 2x + 50 = 180 2x = 130 x = 65 Example: x  50  Find the value of x. A B C If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

10 Isosceles Triangle Rule If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Example:Find the value of x.Since two angles are congruent, the sides opposite these angles must be congruent. 3x – 7 = x x = 22 X = 11 A B C 50   3x - 7 x+15 A B C