Question

unit 7: polygons & quadrilaterals
quiz 7 - 1: angles of polygons & parallelograms
part i: angles of polygons

1. what is the sum of the degrees of the interior angles of a 19 - gon?
2. if the sum of the interior angles of a polygon is 1800°, how many sides does it have?
3. what is the measure of an interior angle of a regular nonagon?
4. what is the sum of the exterior angles of a 25 - gon?
5. what is the the measure of each exterior angle of a regular decagon?

directions: find the value of x.

1. x =

(there is a heptagon with angles 149°, 132°, 140°, x°, 125°, 117°, 129°)

1. x =

(there is a quadrilateral with a right angle, and angles (3x + 13)°, (7x - 9)°, 95°, (7x - 4)°)

1. x =

given: the figure above is a regular polygon.
(there is a regular octagon with an interior angle (16x + 23)°)

1. x =

(there is a hexagon with angles (2x + 8)°, 41°, (8x - 5)°, (4x + 7)°, 92°, (7x - 11)°)
part ii: parallelograms

1. given jm = 27, ml = 16, jl = 46, nk = 15, m∠klm = 48°, m∠jkm = 78°, and m∠mjl = 22°, find each missing value.

(there is a parallelogram - like figure with points j, k, l, m, n)
kl =
jk =
mk =
nl =
m∠jkl =
m∠klj =
m∠kmj =
m∠kjl =

Explanation:

Step1: Sum interior angles formula

For an $n$-gon, sum $=(n-2)\times180^\circ$.
For $n=19$:
$\text{Sum}=(19-2)\times180^\circ=17\times180^\circ$

Step2: Calculate 19-gon interior sum

$17\times180^\circ=3060^\circ$

Step3: Solve for polygon sides

Set $(n-2)\times180^\circ=1800^\circ$
$n-2=\frac{1800^\circ}{180^\circ}=10$
$n=10+2=12$

Step4: Regular nonagon interior angle

First, sum of interior angles: $(9-2)\times180^\circ=1260^\circ$
Each angle $=\frac{1260^\circ}{9}=140^\circ$

Step5: Exterior angle sum rule

Sum of exterior angles of any polygon is $360^\circ$, regardless of sides.

Step6: Regular decagon exterior angle

Each exterior angle $=\frac{360^\circ}{10}=36^\circ$

Step7: Find $x$ in heptagon

Sum of interior angles: $(7-2)\times180^\circ=900^\circ$
Sum given angles: $145+133+140+125+115+117=775^\circ$
$x=900^\circ-775^\circ=125^\circ$

Step8: Find $x$ in pentagon

Sum of interior angles: $(5-2)\times180^\circ=540^\circ$
Right angle $=90^\circ$, so:
$(3x+23)+(9x-9)+(7x-4)+90+90=540$
Combine like terms: $19x+190=540$
$19x=540-190=350$
$x=\frac{350}{19}\approx18.42$

Step9: Regular octagon interior angle

Sum of interior angles: $(8-2)\times180^\circ=1080^\circ$
Each angle $=\frac{1080^\circ}{8}=135^\circ$
Set $16x+23=135$
$16x=135-23=112$
$x=\frac{112}{16}=7$

Step10: Find $x$ in hexagon

Sum of interior angles: $(6-2)\times180^\circ=720^\circ$
Combine angles:
$(3x+6)+(8x-5)+(4x+7)+62+(7x-11)+41=720$
Combine like terms: $22x+100=720$
$22x=720-100=620$
$x=\frac{620}{22}=\frac{310}{11}\approx28.18$

Step11: Parallelogram side properties

In parallelogram $JKLM$, opposite sides are equal:
$KL=JM=27$, $JK=ML=16$
Diagonals bisect each other:
$MK=2\times NK=2\times15=30$, $NL=\frac{JL}{2}=\frac{46}{2}=23$

Step12: Parallelogram angle properties

$\angle JKL = 180^\circ - m\angle KLM=180^\circ-48^\circ=132^\circ$
$\angle KLJ = m\angle MJL=22^\circ$ (alternate interior angles)
$m\angle KMJ = m\angle JKM=78^\circ$ (alternate interior angles)
$\angle KJL = 180^\circ - m\angle JKL - m\angle KLJ=180^\circ-132^\circ-22^\circ=26^\circ$

Answer:

1. $3060^\circ$
2. $12$
3. $140^\circ$
4. $360^\circ$
5. $36^\circ$
6. $125^\circ$
7. $\frac{350}{19}\approx18.42$
8. $7$
9. $\frac{310}{11}\approx28.18$

10.
$KL=27$, $m\angle JKL=132^\circ$
$JK=16$, $m\angle KLJ=22^\circ$
$MK=30$, $m\angle KMJ=78^\circ$
$NL=23$, $m\angle KJL=26^\circ$