Sum of the interior angles of a n-sided Polygon = (n-2) 180
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1 5.1 Interior angles of a polygon Sides n Number of Triangles 1 Sum of interiorangles 180 Sum of the interior angles of a n-sided Polygon = (n-2) 180 What you need to know: How to use the formula 1) The sum of the measures of the interior angles of a 25-gon is. How to find one angle in a regular polygon 2) The measure of one angle in a regular octagon is. How to use the formula in reverse 3) How many sides does a polygon have if the sum of its interior angles is 3060? Review a b a+b= a b a+b+c= c
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3 5.2 Exterior angles of a polygon. 1. Sketch the exterior angles of the octagon 2. measure each exterior angle of each polygon 3. find the sum of all to the exterior angles of each polygon The sum of all exterior angles of a polygon add up to. What you need to know. 1) The sum of the exterior angles is constant----the sum of the measures of the exterior angles of a 25-gon is 2) How to use the new knowledge backwards----if the measure of one exterior angle of a regular polygon is 24, then the polygon has sides. 1. Find all of the missing angles
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5 5.3 Kite and trapezoid properties Vertex Angle 1. There are two sets of congruent Adjacent sides. 2. Diagonals are perpendicular. 3. The line connecting the vertex angles bisects the vertex angles and the other diagonal. NonVertex Angles 4. Two isosceles triangles are formed with a kite. ---The base angles are congruent. 5. Four right triangles are formed with a kite. 6. Nonvertex angles are congruent. Vertex Angle The following are kites. 1. x = y = 2. x = y = x y
6 Isosceles Trapezoid BASE LEG LEG BASE Diagonals are congruent Diagonals create 4 sets of congruent angles. 1. Bottom base angles are congruent. Top base angles are congruent. 2. Consecutive angles between the bases are supplementary. (any trapezoid) 3. Legs are congruent. 1. x = y = 2.Perimeter = 105 cm x = y x 2 3 c m 3 0 c m x
7 Find the measures of every letter. m k l 1 l 2 l 3 l 4 j 7 0 v t s r q e n 3 0 d p i 4 0 f u g a c l 1 h b l l 3 l 4 a= b= c= d= e= f= g= h= i= j= k= m= n= p= q= r= s= t= u= v=
8 60 x Review 1. x = y = x y 2. x = y = 80 y+3 x x = y = y Draw a regular polygon. 2. Draw an equilateral polygon. 1. If the sum of the measures of two angles is 90, then the angles are. 2. In an isosceles triangle, the base angles are. 3. The sum of the measures of the angles of an octagon is. 4. Each angle of a regular hexagon measures. 5. The diagonals of a are perpendicular bisectors of each other.
9 5.4 Midsegments. A segment that connects any two midpoints of a triangle and the nonparallel sides of a trapezoid properties of midsegments for triangles. Use your ruler to show the following are true: 1. Each midsegment bisects the sides with the midpoints. 2. The midsegment is half the length of the third side. 3. The midsegment is parallel to the third side. 4. The three midsegments create four triangles. These triangles are. properties of midsegments for triangles. Use your ruler and protractor to show the following are true: 1. The midsegment bisects the legs. 2. Half the sum of the lengths of the bases is equal to the length of the midsegment. 3. The midsegment is parallel to the bases. 4. The angle formed by the leg and base is congruent to the corresponding angle formed by the same leg and midsegment.
10 1. Perimeter = 105 cm x = 2 3 c m x 3 0 c m 2 3. Find the missing values in each figure. 2. x = y = z = 1 7 x z y The figure is a trapezoid. q = q 4. The midsegment of a trapezoid is to the two bases. 5. The length of a midsegment between two sides of a triangle is the length of the third side. 6. The length of the midsegment of a trapezoid is of the lengths of the bases. 7. Draw one median in one triangle and one midsegment in the other triangle and label each as such.
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12 5.5 Parallelograms. A quadrilateral where opposite sides are parallel. Use your ruler and protractor to show the following are true: 1. Opposite angles are congruent. 2. Consecutive angles are supplementary. 3. Opposite sides are equal in measure. 4. The diagonals of a parallelogram bisect each other. 5. The diagonals form two sets of vertical angles. 1. a = b = x = y = x z 15 y z Find the missing coordinate in terms of p and q,5 p, q 8. The length of the midsegment of a trapezoid is the of the lengths of the bases. 9. The opposite angles of a parallelogram. 10. The diagonals of a parallelogram. 11. The consecutive angles of a parallelogram.
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14 5.6 Special parallelograms. Rhombus, Rectangle, and Square. Properties of a Rhombus (a square is a rhombus) Use your ruler and protractor to show the following are true: 1. These are all parallelograms, so parallelogram rules above apply. 2. The diagonals are perpendicular to each other. 3. All sides are equal in length. 4. The diagonals bisect the angles. Properties of a Rectangle (a square is a rectangle) Use your ruler and protractor to show the following are true: 1. These are all parallelograms, so same rules apply. Apply them to the Rectangle. 2. The diagonals are not perpendicular to each other, but they do bisect each other. 3. The measure of each angle is the same. 4. The diagonals are equal in length.
15 1. Lengths x= y= Angles a= b= x b 105 a y Complete each statement. None of the answers is square. 1. If the sum of the measures of two angles is 90, then the angles are. 2. In an isosceles triangle, the base angles are. 3. The sum of the measures of the angles of an octagon is. 4. Each angle of a regular hexagon measures. 5. The diagonals of a are perpendicular bisectors of each other. 6. The midsegment of a trapezoid is to the two bases. 7. The length of a between two sides of a triangle is half the length of the third side. 8. The length of the midsegment of a trapezoid is the sum of the lengths of the bases. 9. The opposite angles of a parallelogram are. 10. The diagonals of a parallelogram. 11. The consecutive angles of a parallelogram are. 12. The diagonals of a bisect the opposite angles. 13. The diagonals of a are equal in length. Given diagonal BK and B, construct rhombus BAKE. B K B
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