Lesson 18Triangle Theorems

Consider the following diagramWhat do you think is special about m 3, m 4, ∠ ∠& m 5?∠m 3 + m 4 + m 5 = 180°∠ ∠ ∠If lines u and d are parallel, then what is special about 1 & 4? Justify your response.∠ ∠

Consider the following diagramWhat do you think is special about m 3, m 4, ∠ ∠& m 5?∠m 3 + m 4 + m 5 = 180°∠ ∠ ∠If lines u and d are parallel, then what is special about 1 & 4? Justify your response.∠ ∠∠1 4, alt. int. ’s ≅ ∠ ∠ ≅What is special about 2 & 5?∠ ∠

Consider the following diagramWhat do you think is special about m 3, m 4, ∠ ∠& m 5?∠m 3 + m 4 + m 5 = 180°∠ ∠ ∠If lines u and d are parallel, then what is special about 1 & 4? Justify your response.∠ ∠∠1 4, alt. int. ’s ≅ ∠ ∠ ≅What is special about 2 & 5?∠ ∠∠2 5, alt. int. ’s ≅ ∠ ∠ ≅Therefore by the def. of ’s m 1 = m 4 ≅ ∠ ∠ ∠and m 2 = m 5 ∠ ∠By substitution m 3 + m 1 + m 2 = 180°∠ ∠ ∠

Theorem 18-1: Triangle Angle Sum TheoremThe sum of the measures of the angles of a triangle is equal to 180°.m A + m B + m C = 180°∠ ∠ ∠If m A = 56° & m C = 63°, then ∠ ∠find m B.∠56 + m B + 63 = 180∠m B + 119 = 180∠m B = 61°∠

m∠B = 37°, find m∠C90 + 37 + m C = 180∠37 + m C = 90∠What does this mean about B & C?∠ ∠They are complementarym C = 57°∠How many right angles can a triangle have? Why?Only one because two right angles is 180° and you still need another angle for a triangle

Triangle Angle Sum Theorem CorollariesA corollary to a theorem is a statement that follows directly from that theorem by applying previous definitions, postulates, and/or theoremsCorollary 18-1-1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. (No Choice Thm.)Corollary 18-1-2: The acute angles of a right triangle are complementary.Corollary 18-1-3: The measure of each angle of an equiangular triangle is 60°.Corollary 18-1-4: A triangle can have at most one right or one obtuse angle.

Remote Interior AnglesRemote interior angles are the interior angles that are not adjacent to the exterior angle∠1 & 2 are the remote interior ∠angles for 4∠What is special about m 1, m 2 & ∠ ∠m 3?∠What is special about m 3 & m 4?∠ ∠m 3 + m 4 = m 1 + m 2 + m 3∠ ∠ ∠ ∠ ∠What can you subtract from both sides?

Theorem 18-2: Exterior Angle TheoremThe measure of each exterior angle of a triangle is equal to the sum of the measure of the its two remote interior angles.m 4 = m 1 + m 2∠ ∠ ∠

m∠1 = 57° & m∠4 = 113°, find m∠2

m 1 + m 2 = m 4 ∠ ∠ ∠57 + m 2 = 113∠m 2 = 56°∠Now find m 3 justify your steps∠113 + m 3 = 180, linear pair thm∠

or57 + 56 + m 3 = 180, ∠ Δ sum thmm 3 = 67°, subtraction∠

Questions/ReviewThe title to this lesson is misleading, since there is much more we will learn about triangles this yearSo there will be many more theorems about trianglesDo not get caught up with the term corollary, just be able to apply what they say & you will be fine