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题目

解题步骤

1. Understand the question

   interior & exterinterior & exterior of any polygon referencsum of the interior angle measures:$s=(n-2)\\cdot180$s: sum of anglesn: number of sidesangle measure:$360^\\circ$interior & exterior angles of regular polygons reference:interior angle measureof a regular polygon:$\\frac{(n-2)\\cdot180}{n}$exterior angle measureof a regular polygon:$\\frac{360}{n}$the number of sidesof a regular polygon:practice questions1. what is the sum of the measures of the interior angles of a pentagon?2. what is the sum of the measures of the interior angles of a 27-gon?3. what is the measure of each interior angle of a regular octagon?4. what is the measure of each interior angle of a regular 20-gon?5. five angles of a hexagon measure $119^\\circ$, $129^\\circ$, $104^\\circ$, $139^\\circ$, and $95^\\circ$. what is the measure of the sixth angle?6. the sum of the interior angles of a polygon is $1620^\\circ$. how many sides does the polygon have?7. the sum of the interior angles of a polygon is $3960^\\circ$. how many sides does the polygon have?8. what is the sum of the measures of the exterior angles of a nonagon?9. what is the measure of each exterior angle of a regular 20-gon?

2. Explanation

   Step1: Solve Q1: Pentagon interior sum

   Use formula $S=(n-2)\times180^\circ$, $n=5$
   $S=(5-2)\times180^\circ=3\times180^\circ=540^\circ$

   Step2: Solve Q2: 27-gon interior sum

   Use formula $S=(n-2)\times180^\circ$, $n=27$
   $S=(27-2)\times180^\circ=25\times180^\circ=4500^\circ$

   Step3: Solve Q3: Regular octagon interior angle

   Use formula $\frac{(n-2)\times180^\circ}{n}$, $n=8$
   $\frac{(8-2)\times180^\circ}{8}=\frac{6\times180^\circ}{8}=135^\circ$

   Step4: Solve Q4: Regular 20-gon interior angle

   Use formula $\frac{(n-2)\times180^\circ}{n}$, $n=20$
   $\frac{(20-2)\times180^\circ}{20}=\frac{18\times180^\circ}{20}=162^\circ$

   Step5: Solve Q5: Hexagon sixth angle

   First find total interior sum: $n=6$, $S=(6-2)\times180^\circ=720^\circ$
   Sum given angles: $119^\circ+129^\circ+104^\circ+139^\circ+95^\circ=586^\circ$
   Sixth angle: $720^\circ-586^\circ=134^\circ$

   Step6: Solve Q6: Sides for $1620^\circ$ sum

   Rearrange $S=(n-2)\times180^\circ$ to $n=\frac{S}{180^\circ}+2$
   $n=\frac{1620^\circ}{180^\circ}+2=9+2=11$

   Step7: Solve Q7: Sides for $3960^\circ$ sum

   Use $n=\frac{S}{180^\circ}+2$
   $n=\frac{3960^\circ}{180^\circ}+2=22+2=24$

   Step8: Solve Q8: Nonagon exterior sum

   Sum of exterior angles of any polygon is $360^\circ$

   Step9: Solve Q9: Regular 20-gon exterior angle

   Use formula $\frac{360^\circ}{n}$, $n=20$
   $\frac{360^\circ}{20}=18^\circ$

3. Final answer
   1. $540^\circ$
   2. $4500^\circ$
   3. $135^\circ$
   4. $162^\circ$
   5. $134^\circ$
   6. 11 sides
   7. 24 sides
   8. $360^\circ$
   9. $18^\circ$

答案

Question Analysis

OCR Text

interior & exterinterior & exterior of any polygon referencsum of the interior angle measures:$s=(n-2)\\cdot180$s: sum of anglesn: number of sidesangle measure:$360^\\circ$interior & exterior angles of regular polygons reference:interior angle measureof a regular polygon:$\\frac{(n-2)\\cdot180}{n}$exterior angle measureof a regular polygon:$\\frac{360}{n}$the number of sidesof a regular polygon:practice questions1. what is the sum of the measures of the interior angles of a pentagon?2. what is the sum of the measures of the interior angles of a 27-gon?3. what is the measure of each interior angle of a regular octagon?4. what is the measure of each interior angle of a regular 20-gon?5. five angles of a hexagon measure $119^\\circ$, $129^\\circ$, $104^\\circ$, $139^\\circ$, and $95^\\circ$. what is the measure of the sixth angle?6. the sum of the interior angles of a polygon is $1620^\\circ$. how many sides does the polygon have?7. the sum of the interior angles of a polygon is $3960^\\circ$. how many sides does the polygon have?8. what is the sum of the measures of the exterior angles of a nonagon?9. what is the measure of each exterior angle of a regular 20-gon?

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