Lesson 18 Triangle Theorems

Consider the following diagram What 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 diagram What 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 diagram What 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 Theorem The 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∠C 90 + 37 + m∠C = 180 37 + m∠C = 90 What does this mean about ∠B & ∠C? They are complementary m∠C = 53° 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 Corollaries A corollary to a theorem is a statement that follows directly from that theorem by applying previous definitions, postulates, and/or theorems Corollary 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 Angles Remote 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 Theorem The measure of each exterior angle of a triangle is equal to the sum of the measure of 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 or 57 + 56 + m∠3 = 180, Δ sum thm m∠3 = 67°, subtraction

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