4.2 Angles of Triangles Theorem 4-1: Angle Sum Theorem The sum of the measures of the angles of a triangle is 180. x + y + z = 180 Use this theorem to find a missing angle measure. Find the value of X X +Y +Z = 180 X + 80 + 45 = 180 X + 125 = 180 X = 55 Y 80 45 Z X y x z

Theorem 4.2 Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Remote Interior Angles Exterior Angle Exterior Angle is formed by one side of a triangle and the extension of another side. Remote Interior Angle: The interior angles of the triangle not adjacent to a given exterior angle. Remote Interior Angles Exterior Angle 2 1 3

Theorem 4.3 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. 2 1 3 m1 = m2 + m3

Flow Proof Flow Proof: Organizes a series of statements in logical order, starting with the given statements. Each statement is written in a box with the reason verifying the statement written below the box. Arrows are used to indicate how the statements relate to each other. Refer to page 212 and follow the flow proof shown

What does complementary mean? together they = 90 How many degrees are there in a triangle? 180 So if you have a right triangle, what is special about one of the angles? It is 90 So how many degrees do you have left for the 2 acute angles? 90 So…Corollary 4.1 says: The acute angles of a right triangle are complementary. x + y = 90 Corollary: A statement that can easily be proved using a theorem. Find mJ and mK in right triangle JKL. mJ + mK = 90 x + 15 + x + 9 = 90 2x + 24 = 90 2x = 66 x = 33 So mJ = 48 & mK = 42 J x + 15 x + 9 L K x y

Corollary 4.2 There can be at most one right or obtuse angle in a triangle. Try page 214 #1-7 1. 43 2. 99 3. 55 4. 33 5. 147 6. 65 7. 25