TRIANGLES DISCOVERY LESSON: EXTERIOR ANGLES OF A TRIANGLE EQUAL THE SUM OF THE OPPOSITE INTERIOR ANGLES

WARM UP 120 x y Alternate angles are _______________________ Corresponding angles are ___________________________ Same side interior angles/co-interior angles are _____________________ Congruent means: ___________________________ Supplementary means: _________________________ Find the measure of angle x and y: 30 120 x y

Today we are going to prove that the exterior angle of a triangle is equal to the sum of opposite interior angles. Before we can do that, we must prove that the sum of the angles of a triangle equals 180º. Spend a few minutes in your groups trying to prove the theorem that states that the angles of a triangle equal 180º. Use your prior knowledge of lines and parallel lines to help you. 1 2 a b c

Statements Reasons angle 1 = angle a alternate angles are congruent angle 2 = angle b alternate angles are congruent angle 1 + angle c + angle 2 = 180 angles on a straight line angle a + angle c + angle b = 180 substitution Therefore the sum of the interior angles of a triangle equal 180º. 1 2 a b c

Now, using parallel lines, and what you have just proved, prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. 1 2 3 4 c b a

Statements Reasons angle 1 = a alternate angles congruent angle 2 = b alternate angles congruent angle 4 = a + c corresponding angles congruent angle 3 = angle 4 vertical angles congruent angle 3 = a + c substitution a + c = interior opposite angles of triangle angle 3 = exterior angle of a triangle Therefore the interior opposite angles of a triangle equal the exterior angle. c b 3 a 1 2 4

Watch this short video explaining the exterior angle theorem. https://www.youtube.com/watch?v=yeV037nPvdc

Let’s apply what we have just proved. Always justify your reasoning. Find the missing angles.

Time to reflect….. Take a few minutes to think about what you have learned today. Write a short paragraph explaining why the exterior angle of a triangle is equal to the opposite interior angles. Use the diagram to help you.

Complete the following worksheet. You may work with a partner.