2. 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

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

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

5. 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

6. 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

7. 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. 2x 7th angle + x 8th angle = 111 3x = 111 x = 37 2(37) 7th angle (37) 8th angle 74 o 7th angle 37 o 8th angle Geometry--Ch. 6 Review

8. 7) A regular convex polygon has 12 sides. Find the measure of each interior angle and each exterior angle: 360/12 = 30 o ANSWER: Since the exterior angles always have to add up to 360, each exterior angle would have to be... 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

9. 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

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

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

12. 11) Find x and y in the parallelogram shown: Geometry--Ch. 6 Review 6y+8 o 11y+1 o 5x-9 3x+4 Opposite sides must be congruent. 5x - 9 = 3x + 4 2x - 9 = 4 2x = 13 x = 6.5 If x = 6.5, then this angle would be 47 o. Since consecutive angles must be supplementary, this angle would be 133 o. 11y + 1 = 133 11y = 132 y = 12

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

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

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

16. 17) Given the following trapezoid and its midsegment, find the value of x: Geometry--Ch. 6 Review 2x + 8 6x + 3 8x + 5 (2x+8) + (8x+5) = 2 (6x+3) 10x + 13 = 12x + 6 13 = 2x + 6 7 = 2x 15 24 33 By plugging the x = 3.5 back in, we can see that we’re correct. x = 3.5 9 units apart

17. 18) Find the missing angles in the following kite: Geometry--Ch. 6 Review G E M T 12 3 4 107 o 42 o Kites have one pair of opposite angles ≅. 107 o So angles T & E are both 107 o. Since the angle sum of a triangle is 180 o, m ∠ 2 = 31 o. 31 o 42 o The two triangles in the kite are ≅ ( by SSS). Therefore, we know the other missing angles as well.

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