Geometry--Ch. 6 Review

Classify each polygon as regular/irregular, concave/convex,and name it based on its number of sides:

1) 2)

irregularconcavedecagon

regularconvexpentagon

3) Find the value of x:

Geometry--Ch. 6 Review

xo

87o

108o148o

112o

Since the figure is a pentagon,the interior angle sum must be 540o.

Angle Sum = (5-2)(180)

= 540o

These four angles add up to 455o.

x = 540 - 455x = 85o

4) Find the value of x:

Geometry--Ch. 6 Review

4xo

128o

116o

118o

3x+20o4x-3o

Since the figure is a hexagon,the interior angle sum is 720o.

These angles add up to 362o.

Therefore, the remaining angles must add up to 358o.

(4x-3) + (3x+20) + 4x = 358

11x + 17 = 358

11x = 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

6) The sum of the measures of six angles in a convex octagon is 969o. The 7th angle is twice as large as the 8th 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 969o. The 7th angle is twice as large as the 8th angle. Find the measures of both missing angles:The remaining 111o must be divided into 3 equal parts.The reason for this is because one angle is twice as large as the other.

2x7th angle

+ x8th angle

= 111

3x = 111

x = 37

2(37)

7th angle

(37) 8th angle

74o7th angle

37o 8th angle

Geometry--Ch. 6 Review

7) A regular convex polygon has 12 sides. Find the measure of each interior angle and each exterior angle:

360/12 = 30o

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 = 150o

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:

14 cm

14 cm14 cmANSWER: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 3Area = (½)(14)( 7 3 )

cm2Area = 49 3

Geometry--Ch. 6 Review

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.

11) Find x and y in the parallelogram shown:

Geometry--Ch. 6 Review

6y+8o 11y+1o

5x-9

3x+4Opposite 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 47o.

Since consecutive angles must be supplementary,this angle would be 133o.

11y + 1 = 133

11y = 132

y = 12

12) Find x, y, and z in the parallelogram shown:

Geometry--Ch. 6 Review

2yo

3x+5o

53o2x+11o

zo

Opposite angles must be ≅.

3x + 5 = 53

3x = 48

x = 16

If x = 16, then this angle is 43o.

In a parallelogram, alternate interior angles are ≅.

z = 43

Like all triangles, this one’sangles add up to 180o.

43o53o

53 + 43 + 2y = 180

2y = 84 y = 42

Do the following quadrilaterals have to be parallelograms? If so, why?

Geometry--Ch. 6 Review

15)

13) 14)

YES; Both pairs ofopposite sides are ≅.

YES; The same pair

of opposite sides is parallel and ≅.

NO; We need BOTH pairs of opposite angles to be ≅.

16) Find the missing angles in the following rhombus:

Geometry--Ch. 6 Review

68o

12

34

5Opposite angles are ≅.68o

Since consecutive angles are supplementary, these large angles are each 112o.

56o

56o

56o

56o

In a rhombus, diagonals bisect the opposite angles. Therefore, both 112o angles get split into

four different 56o angles.

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 = 3.5back in, we can see that

we’re correct.x = 3.5

9 units apart

18) Find the missing angles in the following kite:

Geometry--Ch. 6 Review

G

E

M

T

1 2

3

4

107o

42o

Kites have one pair of opposite angles ≅.

107o

So angles T & E are both 107o.

Since the angle sum of a triangleis 180o, m∠2 = 31o.

31o31o

42o

The two triangles in the kiteare ≅ (by SSS). Therefore, weknow the other missing angles as well.

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.)