Students will be able to: ◦ Find measures of angles of triangles ◦ Use parallel lines to prove theorems about triangles ◦ Define and apply the triangle sum and triangle exterior angle theorems
The sum of the angles of a triangle is 180 degrees
List some things you know about triangles The sum of the angles add to be 180 There are obtuse, acute, and right triangles There are scalene, equilateral, and isosceles triangles In a right triangle, the two acute angles add to be 90 degrees
Through a point not on a line, there is one and only one line parallel to the given line.
Sometimes when proving things we need to add lines to the diagram. Auxiliary Line: a line you add to a diagram to help explain relationships in proofs. In the picture below the red line was added to help with the following proof.
Statements Reasons 1. ΔABC 1. Given 2. Draw line PR through B, 2. Parallel Postulate parallel to AC 3.<1, <2, <3 are Suppl 3. Def. of Linear Pairs 4. Def. of Suppl. <s 4.<1 + <2 + <3 = 180 5. Alt. Interior Angles 5. <1 ≅<A, <3 ≅<C 6. m<1 = m<A, m<3 = m<C 6. Def. of Congruence 7. m<A + m<2 + m<C = 180 7. Substitution
The sum of the measures of the angles of a triangle is 180. If you know the measures of two angles of a triangle, you can use this theorem to find the third measure.
What are the values of x, y, and z? X = 102 Y = 78 Z = 66
Exterior Angle of a Polygon ◦ An angle formed by a side and an extension of an adjacent side. Remote Interior Angles ◦ The two nonadjacent interior angles to the exterior angle Below <1 is the exterior angle, <2 and <3 are the remote interior angles
The measure of the exterior angle equals the sum of the two remote interior angles. mExterior Angle = Remote Int. + Remote Int m<1 = m<2 + m<3