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Chapter 8 Lesson 4 Triangle Sum Rule The sum of the measures of the angles in a triangle is 180˚ m∠1 + m∠2 + m∠3 = 180˚ The small letter "m" in front of ∠1 says "the measure of angle 1" (m∠1) To find the sum of angle measures in any 4-sided figure you can divide the figure in two by drawing a diagonal. A diagonal is a line segment that connects two nonadjacent vertices of a polygon. The diagonal creates two triangles. Since the sum of each triangle is 180˚ the sum of the four sided figure is 2•180˚= 360˚ Sum of the angle of a quadrilateral The sum of the measures of the angles in a quadrilateral is 360˚ m∠1 + m∠2 + m∠3 +m∠4 = 360˚ Examples: 1. Find the unknown angle measures in the triangle. 2. Find the unknown angle measure in the quadrilateral. In a convex polygon, all diagonals can be drawn within the interior of the figure. When you divide any convex polygon into triangles you can find the sum of its interior angle measures. 5 triangles so... 5•180˚= 900˚ Diagonals: When we figure out the sum of the angles we only want to use diagonals to make triangles. We can really draw more diagonals than that. diagonals for sum of angles total number of diagonals But in this unit we will only be concerned with drawing enough diagonals to make triangle. Examples: Divide the polygon into triangles to find the sum of its angle measures. 1. 2. If a 3-sided figure has 1• 180 =180˚ 4-sided figure has 2• 180 = 360˚ 5-sided figure has 3• 180 = 540˚ 6-sided figure has 4• 180 = 720˚ Can you figure out... 7-sided figure 8-sided figure What about a rule that works for any sided figure? Does it work for Concave Polygons? 4-sided 5-sided 7-sided Word Problems: 1. A truss bridge is supported by triangular frames. If every triangular frame in a truss bridge is an isosceles right triangle, what is the measure of each angle in one of the frames? (what is an isosceles right triangle?) 2. Each outer wall of the Pentagon in Washington D.C. measures 921 feet. What is the measure of each angle made by the Pentagon's outer walls? 3. The angle between the lines of sight from a lighthouse to a tugboat and to a cargo ship is 27˚. The angle between the lines of sight at the cargo ship is twice the angle between the lines of sight at the tugboat. What are the angles at the tugboat and at the cargo ship?