Triangles & Angles

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

How do you find the missing angle of a triangle? Remember the interior angles add up to 180˚. If you know two angles, add them up and then subtract from 180˚. 34˚ C B A Find the measure of A.  C is 34˚ and B is right so =____. 90˚ 34˚+ 90˚ = ______ 124˚ 180˚ - 124˚ = A A = 56˚ Same answer if you write an equation and solve. 90 ˚ + 34 ˚ + x = 180 ˚ 124˚ + x = 180 ˚ A = 56˚ x = 56˚

Find the missing angle measurement. 25˚ 16 + 25 = _______ 41˚ 139˚ 180 – 41 = _______ The missing angle is 139˚. ? 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 The angles will be congruent (equal measures), but the side lengths will not A   X C B B   Y C   Z Z Y

Find the missing angle measurement: ∆HJK ~ ∆ MNP M N P 22˚ 85˚ Find  J H J K ? Since the triangles are similar, the corresponding angles must be congruent (and will have equal angle measures) H  M and K  P and  J  N H and M = 22 ˚ and K and P = 85 ˚ and  J and N = ?? 22 + 85 = ______ 107˚ 73˚ 180 – 107 = ______ Angle N is 73˚, and  J  N so  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

Non-Adjacent Interior Angles of Triangles non-adjacent interior angles are the two angles that are not adjacent to (next to) the exterior angle also known as remote-interior angles Non-adjacent interior angles 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 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 = ________

Find the missing angle measurement: 103 = 74 + ? ? 103˚ 74˚

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

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˚