GeometryCh 6 Review Classify each polygon as regularirregular

Geometry--Ch. 6 Review Classify each polygon as regular/irregular, concave/convex, and name it based on its number of sides: 1) 2) irregular concave decagon regular convex pentagon

Geometry--Ch. 6 Review 3) Find the value of x: 148 o 108 o xo 112 o Since the figure is a pentagon, the interior angle sum must be 540 o. Angle Sum = (5 -2)(180) = 540 87 o o These four angles add up to 455 o. x = 540 - 455 x = 85 o

Geometry--Ch. 6 Review 4) Find the value of x: 128 o 4 x-3 o Since the figure is a hexagon, the interior angle sum is 720 o. o These angles add up to 362. 4 x 3 x+20 o 116 o o 118 o Therefore, the remaining angles must add up to 358 o. (4 x-3) + (3 x+20) + 4 x = 358 11 x + 17 = 358 11 x = 341 x = 31

Geometry--Ch. 6 Review 5) Find the interior angle sum for a convex septagon: ANSWER: Since a septagon has seven sides, we insert a 7 into the interior angle sum formula. Angle Sum = (n - 2)(180) Angle Sum = (7 -2)(180) Angle Sum = (5)(180) 900 degrees

Geometry--Ch. 6 Review 6) The sum of the measures of six angles in a convex octagon is 969 o. The 7 th angle is twice as large as the 8 th angle. Find the measures of both missing angles: ANSWER: Since an octagon has eight sides, we know that the sum of its interior angles should be 1080 degrees. Angle Sum = (8 -2)(180) = 1080 Since six of the angles add up to 969 degrees, the remaining two angles must add up to 111 degrees. 1080 - 969 = 111

Geometry--Ch. 6 Review 6) The sum of the measures of six angles in a convex octagon is 969 o. The 7 th angle is twice as large as the 8 th angle. Find the measures of both missing angles: The remaining 111 o must be divided into 3 equal parts. The reason for this is because one angle is twice as large as the other. 7 th angle 2 x 8 th angle x + 3 x x = 111 = 37 7 th angle 8 th angle 2(37) 74 o (37) 37 o

Geometry--Ch. 6 Review 7) A regular convex polygon has 12 sides. Find the measure of each interior angle and each exterior angle: ANSWER: Since the exterior angles always have to add up to 360, each exterior angle would have to be. . . 360/12 = 30 o Since the interior and exterior angles always combine to form linear pairs, each interior angle would have to be. . . 180 - 30 = 150 o

Geometry--Ch. 6 Review 8) Each interior angle of a regular convex polygon measures 144 degrees. How many sides does the polygon have? ANSWER: If each interior angle is 144 degrees, then each exterior angle would have to be 36 degrees. 180 - 144 = 36 If each exterior angle is 36 degrees, then the polygon is a decagon with 10 sides. 360/36 = 10 sides

Geometry--Ch. 6 Review 9) Find the area of an equilateral triangle with sides of 14 cm: ANSWER: If you drop an altitude down from the vertex angle, two 30/60/90 triangles are formed. 14 cm From last chapter, we know the length of the altitude is 7 3. 14 cm Area = (½)(14)( 7 3 ) Area = 49 3 cm 2 73 7 cm

Geometry--Ch. 6 Review 10) Name the four properties of all parallelograms: ~Both pairs of opposite sides are congruent. ~Both pairs of opposite angles are congruent. ~Consecutive angles are supplementary. ~Diagonals bisect each other.

Geometry--Ch. 6 Review 11) Find x and y in the parallelogram shown: 3 x+4 Opposite sides must be congruent. 5 x - 9 = 3 x + 4 2 x - 9 = 4 2 x = 13 6 y+8 o 11 y+1 o 5 x-9 x = 6. 5 o If 11 y x = +6. 5, then this angle would be 47. 1 = 133 Since consecutive angles must be supplementary, 11 y = 132 this angle would be 133 o. y = 12

Geometry--Ch. 6 Review 12) Find x, y, and z in the parallelogram shown: Like all triangles, this one’s Opposite angles must beo ≅. angles add up to 180. 3 x + 5 = 53 53 + 43 + 2 y = 180 3 x = 48 x = 16 2 y 3 x+5 o 53 o o 53 o 2 x+11 o zo 43 o o If x = 16, then this angle is 43. y = 42 2 y = 84 In a parallelogram, alternate interior angles are ≅. z = 43

Geometry--Ch. 6 Review Do the following quadrilaterals have to be parallelograms? If so, why? 13) YES; Both pairs of opposite sides are ≅. 15) 14) NO; We need BOTH pairs of opposite angles to be ≅. YES; The same pair of opposite sides is parallel and ≅.

Geometry--Ch. 6 Review 16) Find the missing angles in the following rhombus: Opposite angles are ≅. Since consecutive angles are supplementary, these large angles are each 112 o. In a rhombus, diagonals bisect the opposite angles. Therefore, both 112 o angles get split into four different 56 o angles. 56 o 1 2 68 o 56 o 5 3 56 o 4 56 o 68 o

Geometry--Ch. 6 Review 17) Given the following trapezoid and its midsegment, find the value of x: 2 x + 8 15 9 units (2 x+8) + (8 x+5) = 2 (6 x+3) 6 x + 3 8 x + 5 24 apart 33 10 x + 13 = 12 x + 6 13 = 2 x + 6 7 = 2 x x = 3. 5 By plugging the x = 3. 5 back in, we can see that we’re correct.

Geometry--Ch. 6 Review 18) Find the missing angles in the following kite: Kites have one pair of opposite angles ≅. So angles T & E are both 107 o. Since the angle sum of a triangle is 180 o, m∠ 2 = 31 o. G 42 o 4 T The two triangles in the kite are ≅ (by SSS). Therefore, we know the other missing angles as well. 42 o E 107 o 3 107 o o 31 o 1 2 31 M

Geometry--Ch. 6 Review TRUE or FALSE? 19) The diagonals of a rectangle are congruent. TRUE 20) The diagonals of a trapezoid bisect each other. FALSE 21) All rhombuses are squares. (The converse is true. ) FALSE 22) All parallelograms are quadrilaterals. TRUE