Triangles & Their Angles Common Core Investigation 4: Geometry

Learning Goal 3(8.G.A.5): The student will understand and use informal arguments to prove congruency and similarity using physical models, transparencies or geometry software. 4 3 2 1 In addition to level 3.0 and above and beyond what was taught in class, I may: Make connection with real-world situations Make connection with other concepts in math Make connection with other content areas.  I understand and use informal arguments to prove congruency and similarity using physical models, transparencies or geometry software. understand the relationship between the measure of an exterior angle and the other angles of a triangle find and prove the measure of a missing angle of a triangle explain the relationship of the angles formed when parallel lines are cut by a transversal use vocabulary associated with parallel lines cut by a transversal informally prove similarity of triangles I understand congruency and similarity using physical models, transparencies or geometry software. construct triangles find the measure of a missing angle in triangle find missing angles of parallel lines cut by a transversal   With help from the teacher, I have partial success with the unit content. Even with help, I have no success with the unit content.

What do you know about triangles? Has 3 sides. Some triangles are right, acute or obtuse. Some triangles are equilateral, isosceles or scalene. The angles of a triangle add up to 180˚.

How do you find the missing angle of a triangle? Remember all triangles add up to 180˚. If you know two angles, add them up and then subtract from 180˚. A Find the measure of A.  C is 34˚ and B is ________. 124˚ 34˚+ 90˚ = ______ 34˚ 180˚ - 124˚ = A B C A = 56˚

Find the missing angle measurement. 41˚ 25˚ 16 + 25 = _______ 139˚ 180 – 41 = _______ ? 16˚

Similar Triangles Similar means same shape but not the same size. Similar triangles are the same shape but different sizes. Corresponding angles in similar triangles are congruent. Triangle ABC is similar to Triangle XYZ. (∆ABC ~ ∆XYZ) A X A   X B   Y C   Z C B Z Y

Angle-Angle Criterion for Similarity of Triangles How do we know that all of the angles of the two triangles really are congruent? Let’s look at ∆ABC & ∆XYZ again. A X If A is 40˚ and X is 40 ˚ they are . If B is 60˚ and Y is 60 ˚ they are . What is the measure of C? 40 + 60 = 100 180 – 100 = 80˚ C B Z Y Since C and Z have the same measure, we can conclude they are . If all three angles are , ∆ABC & ∆XYZ are similar triangles. What is the measure of Z? 40 + 60 = 100 180 – 100 = 80˚

Find the missing angle measurement: H  M and K  P  J  N J How can you find  J? ∆HJK ~ ∆ MNP ? N H K 22 + 85 = ______ 107˚ 22˚ 85˚ 73˚ M 180 – 107 = ______ P  J = 73˚

Exterior Angles of Triangles An exterior angle is formed by one side of a triangle and the extension of an adjacent side of the triangle. (adjacent means next to) Exterior Angle

Exterior Angles of Triangles The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles (also known as remote-interior angles). Non-adjacent interior angles 98˚ The exterior angle adds up to the measure of 98 + 26. It is 124˚. 26˚ What is the measure of the missing angle in the triangle? Name two ways that you could figure that out. 56˚ 180 – 124 = ________

Exterior Angles of Triangles Use the link below to see how the exterior angle is related to the 2 non-adjacent interior angles. http://www.mathopenref.com/triangleextangletheorem.html

Find the missing angle measurement: 103 = 74 + ? -74 -74 29 = ? The missing angle is 29˚. ? 103˚ 74˚