Examkin

Polygon Angles and Properties

35 questions

All types35Multiple choice19Numeric16
Question 21Question →

A regular polygon has an interior angle that is 140∘140^\circ greater than its exterior angle. How many sides does this polygon have?

Show answer & explanation

Answer: 18

Answer

The regular polygon has 18 sides.
The correct answer is 18. By setting up the equations for the interior angle II and exterior angle EE, we have I=E+140∘I = E + 140^\circ and I+E=180∘I + E = 180^\circ. Substituting the first equation into the second gives (E+140∘)+E=180∘(E + 140^\circ) + E = 180^\circ, which simplifies to 2E+140∘=180∘  ⟹  2E=40∘  ⟹  E=20∘2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ. The number of sides nn of a regular polygon is given by n=360∘/En = 360^\circ / E. Thus, n=360∘/20∘=18n = 360^\circ / 20^\circ = 18.

Step-by-Step Solution

1
Set up the algebraic relationship between the interior angle (II) and the exterior angle (EE).
I=E+140∘I = E + 140^\circ
The problem states that the interior angle is 140∘140^\circ greater than the exterior angle.
2
Use the supplementary relationship between any interior angle and its corresponding exterior angle.
I+E=180∘I + E = 180^\circ
An interior angle and its adjacent exterior angle form a linear pair, which sums to 180∘180^\circ.
3
Substitute the expression for II from Step 1 into the equation from Step 2 and solve for EE.
(E+140∘)+E=180∘  ⟹  2E+140∘=180∘  ⟹  2E=40∘  ⟹  E=20∘(E + 140^\circ) + E = 180^\circ \implies 2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ
Substituting allows us to solve for a single variable representing the exterior angle.
4
Calculate the number of sides (nn) using the sum of the exterior angles of a convex polygon.
n=360∘E=360∘20∘=18n = \frac{360^\circ}{E} = \frac{360^\circ}{20^\circ} = 18
The sum of the exterior angles of any convex polygon is 360∘360^\circ, and in a regular polygon, all nn exterior angles are equal.

Key Concept

Relationship between interior and exterior angles of a regular polygon
Estimated Time:1m 30s
Question 22Question →

A convex polygon has nn sides. The interior angles of the polygon form an arithmetic progression with a common difference of d∘d^\circ, where dd is a positive integer. If the smallest interior angle of the polygon measures 100∘100^\circ, what is the maximum possible value of nn?

Show answer & explanation

Answer: 8

Answer

The maximum possible value of nn is 8.
The correct answer is 8 because we set the sum of the interior angles (n−2)×180∘(n-2) \times 180^\circ equal to the sum of the arithmetic progression n2[200+(n−1)d]\frac{n}{2}[200 + (n-1)d]. Solving for (n−1)d(n-1)d gives (n−1)d=160−720n(n-1)d = 160 - \frac{720}{n}. Since the polygon is convex, the largest angle 100+(n−1)d100 + (n-1)d must be strictly less than 180∘180^\circ, which means (n−1)d<80∘(n-1)d < 80^\circ. Substituting the expression yields 160−720n<80160 - \frac{720}{n} < 80, which simplifies to n<9n < 9. Since nn must be an integer, the maximum possible value of nn is 8. For n=8n = 8, the common difference d=10d = 10 is a positive integer, satisfying all conditions.

Step-by-Step Solution

1
Express the sum of the interior angles using the polygon angle sum formula and the arithmetic progression formula.
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. The sum of the angles in arithmetic progression with first term 100∘100^\circ and common difference d∘d^\circ is n2[2(100)+(n−1)d]=100n+n(n−1)d2\frac{n}{2}[2(100) + (n-1)d] = 100n + \frac{n(n-1)d}{2}.
This sets up the system relating the geometry of the polygon to the algebraic progression of its angles.
2
Equate the two expressions and solve for the quantity (n−1)d(n-1)d.
100n+n(n−1)d2=180n−360  ⟹  n(n−1)d2=80n−360  ⟹  (n−1)d=160−720n100n + \frac{n(n-1)d}{2} = 180n - 360 \implies \frac{n(n-1)d}{2} = 80n - 360 \implies (n-1)d = 160 - \frac{720}{n}.
This isolates the quantity (n−1)d(n-1)d, which represents the difference between the largest and smallest angles.
3
Apply the convexity constraint that every interior angle must be strictly less than 180∘180^\circ.
The largest angle is the last term of the progression: 100+(n−1)d100 + (n-1)d. For the polygon to be convex, this angle must be strictly less than 180∘180^\circ. Therefore, 100+(n−1)d<180  ⟹  (n−1)d<80100 + (n-1)d < 180 \implies (n-1)d < 80. Substituting (n−1)d=160−720n(n-1)d = 160 - \frac{720}{n} gives 160−720n<80  ⟹  80<720n  ⟹  n<9160 - \frac{720}{n} < 80 \implies 80 < \frac{720}{n} \implies n < 9.
A convex polygon cannot have any interior angles greater than or equal to 180∘180^\circ.
4
Identify the maximum integer value of nn and verify that a positive integer common difference dd exists.
Since n<9n < 9 and nn must be an integer, the maximum possible value is n=8n = 8. For n=8n = 8, we calculate (8−1)d=160−7208  ⟹  7d=70  ⟹  d=10(8-1)d = 160 - \frac{720}{8} \implies 7d = 70 \implies d = 10. Since d=10d = 10 is a positive integer, the solution is valid.
This ensures the result satisfies all constraints, including that the common difference is a positive integer.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and all interior angles of a convex polygon must be strictly less than 180∘180^\circ.
Question 23Question →

An irregular convex octagon has five interior angles that each measure 144∘144^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the measure, in degrees, of the largest interior angle of this octagon?

Show answer & explanation

Answer: 150

Answer

The measure of the largest interior angle of the octagon is 150∘150^\circ.
The total sum of the interior angles of an octagon is (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. The sum of the five given angles is 5×144∘=720∘5 \times 144^\circ = 720^\circ, leaving a sum of 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ for the remaining three angles. Since these three angles are in the ratio 3:4:53:4:5, we set 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, yielding 12x=360∘12x = 360^\circ and x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ. Comparing 150∘150^\circ to the other angles of the octagon (which are 144∘144^\circ, 90∘90^\circ, and 120∘120^\circ), the largest interior angle is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of the octagon.
The total sum is 1080∘1080^\circ.
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For an octagon, n=8n = 8, so the sum is (8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
2
Calculate the sum of the five given congruent angles.
The sum of these five angles is 720∘720^\circ.
Since five angles each measure 144∘144^\circ, their combined sum is 5×144∘=720∘5 \times 144^\circ = 720^\circ.
3
Determine the sum of the remaining three interior angles.
The sum of the remaining angles is 360∘360^\circ.
Subtracting the sum of the five congruent angles from the total sum of the octagon's interior angles yields the sum of the remaining three angles: 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ.
4
Use the ratio 3:4:53:4:5 to find the measures of the remaining three angles.
The measures of the three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ.
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 3x+4x+5x=12x=360∘3x + 4x + 5x = 12x = 360^\circ, which gives x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ.
5
Compare the measures of all interior angles of the octagon to find the largest one.
The largest angle is 150∘150^\circ.
The octagon's interior angles consist of five angles of 144∘144^\circ, and three angles of 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. Comparing these values, 150∘150^\circ is the largest measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratios can be used to partition a total sum into specific parts.
Estimated Time:2m 0s
Question 24Question →

The measures of the five exterior angles of a convex pentagon are in the ratio 2:3:4:4:52:3:4:4:5. What is the measure of the largest interior angle of this pentagon?

Show answer & explanation

Answer: 140∘140^\circ

Answer

The measure of the largest interior angle of the pentagon is 140∘140^\circ.
The correct answer is 140∘140^\circ. The sum of the exterior angles of any convex polygon is 360∘360^\circ. For a pentagon with exterior angles in the ratio 2:3:4:4:52:3:4:4:5, the sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18 parts. Each part is equal to 360∘/18=20∘360^\circ / 18 = 20^\circ. The smallest exterior angle has 22 parts, which measures 2×20∘=40∘2 \times 20^\circ = 40^\circ. Since each interior angle is supplementary to its corresponding exterior angle, the largest interior angle corresponds to the smallest exterior angle. Thus, the largest interior angle measures 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.

Step-by-Step Solution

1
Determine the sum of the exterior angles of a convex pentagon.
The sum of the exterior angles of any convex polygon is 360∘360^\circ.
This is a fundamental property of convex polygons and provides the total value to distribute among the ratio parts.
2
Calculate the measure of the smallest exterior angle using the given ratio of 2:3:4:4:52:3:4:4:5.
The sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18. The smallest exterior angle has 22 parts, so its measure is 218×360∘=40∘\frac{2}{18} \times 360^\circ = 40^\circ.
The largest interior angle will be adjacent (and supplementary) to the smallest exterior angle.
3
Calculate the measure of the largest interior angle by subtracting the smallest exterior angle from 180∘180^\circ.
The largest interior angle is 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.
An interior angle and its adjacent exterior angle are supplementary and add up to 180∘180^\circ.

Key Concept

The sum of the exterior angles of any convex polygon is 360∘360^\circ. An interior angle and its adjacent exterior angle are supplementary, meaning they add up to 180∘180^\circ. Consequently, the largest interior angle corresponds to the smallest exterior angle.
Estimated Time:1m 30s
Question 25Question →

An irregular convex hexagon has two interior angles that are right angles. The remaining four interior angles have measures in the ratio 4:5:5:64:5:5:6. What is the measure, in degrees, of the largest interior angle of this hexagon?

Show answer & explanation

Answer: 162

Answer

The correct answer is 162162 degrees. The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two right angles (180∘180^\circ total) leaves a remaining sum of 540∘540^\circ. The ratio of the remaining four angles is 4:5:5:64:5:5:6, which can be represented as 4x,5x,5x,6x4x, 5x, 5x, 6x, summing to 20x=54020x = 540. Solving for the multiplier gives x=27x = 27. The largest angle is 6(27)=162∘6(27) = 162^\circ, which is also greater than the two 90∘90^\circ angles.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the two right angles (180∘180^\circ) gives a remaining sum of 540∘540^\circ for the other four angles. Let these four angles be 4x,5x,5x,4x, 5x, 5x, and 6x6x. Their sum is 20x=54020x = 540, which solves to x=27x = 27. The largest angle is 6x=6(27)=162∘6x = 6(27) = 162^\circ, which is also larger than the two 90∘90^\circ angles.

Step-by-Step Solution

1
Calculate the sum of all interior angles of a convex hexagon.
The sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ.
The sum of the interior angles of any convex nn-gon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Subtract the sum of the two right angles from the total sum.
The remaining sum is 720∘−180∘=540∘720^\circ - 180^\circ = 540^\circ.
Two right angles contribute 90∘+90∘=180∘90^\circ + 90^\circ = 180^\circ to the total.
3
Set up a linear equation representing the ratio of the remaining four angles.
The equation is 4x+5x+5x+6x=5404x + 5x + 5x + 6x = 540, which simplifies to 20x=54020x = 540, yielding x=27x = 27.
The angles are proportional to the parts of the ratio, and their sum must equal the remaining 540∘540^\circ.
4
Calculate the largest angle from the ratio and compare with the right angles.
The largest angle is 6×27=162∘6 \times 27 = 162^\circ.
The largest term in the ratio is 66, and the resulting angle 162∘162^\circ is larger than both 90∘90^\circ and the other calculated angles (108∘108^\circ and 135∘135^\circ).

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratio relationships can be solved using algebraic multipliers.
Question 26Question →

An irregular convex decagon (10-sided polygon) has four interior angles that each measure 150∘150^\circ. The remaining six interior angles are congruent to each other. What is the degree measure of each of these remaining six angles?

Show answer & explanation

Answer: 140

Answer

The measure of each of the remaining six interior angles is 140∘140^\circ.
The total sum of the interior angles of a 10-sided convex polygon is (10−2)×180∘=1,440∘(10-2) \times 180^\circ = 1,440^\circ. Subtracting the sum of the four angles that each measure 150∘150^\circ (4×150∘=600∘4 \times 150^\circ = 600^\circ) leaves 840∘840^\circ for the remaining six angles. Since these remaining six angles are congruent, each measures 840∘÷6=140∘840^\circ \div 6 = 140^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of a convex decagon.
1,440∘1,440^\circ
The interior angle sum of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and a decagon has 10 sides.
2
Find the sum of the four angles measuring 150∘150^\circ.
600∘600^\circ
Multiply the number of angles by their given degree measure.
3
Determine the sum of the remaining six congruent angles.
840∘840^\circ
Subtract the sum of the four known angles from the total interior angle sum of the decagon.
4
Divide the remaining sum by the number of congruent angles.
140∘140^\circ
Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
Question 27Question →

An irregular convex hexagon has three interior angles that each measure 110∘110^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the degree measure of the exterior angle corresponding to the smallest interior angle of this hexagon?

Show answer & explanation

Answer: 82.5∘82.5^\circ

Answer

82.5∘82.5^\circ
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the three known angles (330∘330^\circ) leaves 390∘390^\circ for the remaining three angles. Using the ratio 3:4:53:4:5, the smallest of these remaining angles is calculated as 312×390∘=97.5∘\frac{3}{12} \times 390^\circ = 97.5^\circ. Since 97.5∘97.5^\circ is smaller than the other angles (110∘110^\circ, 130∘130^\circ, 162.5∘162.5^\circ), it is the smallest interior angle of the hexagon. The exterior angle is supplementary to the interior angle, so it measures 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.

Step-by-Step Solution

1
Calculate the sum of all interior angles of the hexagon.
The sum is (6−2)×180∘=4×180∘=720∘(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
This establishes the total measure of the interior angles of a hexagon.
2
Subtract the sum of the three known interior angles from the total sum.
The remaining sum is 720∘−(3×110∘)=720∘−330∘=390∘720^\circ - (3 \times 110^\circ) = 720^\circ - 330^\circ = 390^\circ.
This determines the combined measure of the three remaining angles.
3
Determine the measures of the remaining three angles using the ratio.
Setting up the equation 3y+4y+5y=390∘3y + 4y + 5y = 390^\circ gives 12y=390∘12y = 390^\circ, or y=32.5∘y = 32.5^\circ. The individual angles are 3×32.5∘=97.5∘3 \times 32.5^\circ = 97.5^\circ, 4×32.5∘=130∘4 \times 32.5^\circ = 130^\circ, and 5×32.5∘=162.5∘5 \times 32.5^\circ = 162.5^\circ.
This finds each of the remaining interior angle measures.
4
Find the smallest interior angle and compute its supplementary exterior angle.
The smallest interior angle of the hexagon is 97.5∘97.5^\circ. The corresponding exterior angle is 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.
An interior angle and its adjacent exterior angle sum to 180∘180^\circ.

Key Concept

Polygon Angles and Properties
Question 28Question →

A convex pentagon has one interior angle that measures 108∘108^\circ. The remaining four interior angles have measures in the ratio 3:4:5:63:4:5:6. What is the measure of the largest interior angle of this pentagon?

Show answer & explanation

Answer: 144∘144^\circ

Answer

The measure of the largest interior angle of the pentagon is 144∘144^\circ.
The correct answer is the option stating 144∘144^\circ. To find this, we calculate the sum of all interior angles of a pentagon using (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtracting the known angle of 108∘108^\circ leaves 432∘432^\circ for the remaining four angles. The ratio 3:4:5:63:4:5:6 implies these angles can be represented as 3x3x, 4x4x, 5x5x, and 6x6x, summing to 18x18x. Solving 18x=43218x = 432 gives x=24x = 24. The largest angle is 6x=6×24∘=144∘6x = 6 \times 24^\circ = 144^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a pentagon.
The sum of the interior angles of any convex pentagon (n=5n = 5) is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Before finding individual angles, we must determine the total sum of all interior angles in the polygon.
2
Determine the sum of the remaining four angles.
The remaining four angles sum to 540∘−108∘=432∘540^\circ - 108^\circ = 432^\circ.
We subtract the measure of the known angle from the total sum to find the portion allocated to the remaining angles.
3
Set up and solve an algebraic equation using the given ratio.
Let the four remaining angles be represented as 3x3x, 4x4x, 5x5x, and 6x6x. Their sum is 3x+4x+5x+6x=18x3x + 4x + 5x + 6x = 18x. Setting this equal to the remaining sum gives 18x=43218x = 432, which simplifies to x=24x = 24.
Using the ratio allows us to define the relative sizes of the remaining angles in terms of a single variable, which we can solve for using their sum.
4
Find the measure of the largest interior angle.
The largest of these four angles is represented by 6x6x, which is 6×24∘=144∘6 \times 24^\circ = 144^\circ. Since 144∘144^\circ is also larger than the given angle of 108∘108^\circ, it is the largest interior angle of the pentagon.
We multiply the value of xx by the largest coefficient in the ratio and compare it to the other given angle to identify the maximum measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For an irregular polygon, when some angles are known and others are in a ratio, we subtract the known angles from the total sum and distribute the remaining sum proportionally according to the ratio.
Estimated Time:1m 30s
Question 29Question →

A convex hexagon has interior angles with measures of 2x∘2x^\circ, (3x−15)∘(3x - 15)^\circ, (2x+25)∘(2x + 25)^\circ, (3x+10)∘(3x + 10)^\circ, (4x−50)∘(4x - 50)^\circ, and (x+30)∘(x + 30)^\circ. What is the degree measure of the smallest interior angle of this hexagon?

Show answer & explanation

Answer: 78

Answer

The degree measure of the smallest interior angle is 78.
The sum of the interior angles of a hexagon is calculated as (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ. Adding the algebraic expressions for the six angles yields 15x=72015x = 720, which gives x=48x = 48. Substituting x=48x = 48 back into the expressions gives the angle measures of 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest angle is 78∘78^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of the hexagon.
Sum of interior angles = 720 degrees
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For a hexagon (n=6n = 6), the sum is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ.
2
Set up an algebraic equation by summing all the given angle expressions and equating them to 720.
2x+(3x−15)+(2x+25)+(3x+10)+(4x−50)+(x+30)=7202x + (3x - 15) + (2x + 25) + (3x + 10) + (4x - 50) + (x + 30) = 720
The sum of the actual measures of the interior angles must equal the calculated sum of 720 degrees.
3
Combine like terms to simplify the equation.
15x=72015x = 720
Grouping the xx terms (2x+3x+2x+3x+4x+x=15x2x + 3x + 2x + 3x + 4x + x = 15x) and the constant terms (−15+25+10−50+30=0-15 + 25 + 10 - 50 + 30 = 0) simplifies the expression.
4
Solve for xx.
x=48x = 48
Dividing both sides of the equation 15x=72015x = 720 by 15 isolates the variable xx.
5
Substitute the value of xx back into the angle expressions to identify the smallest angle.
The angles are 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest measure is 78∘78^\circ.
Evaluating each expression at x=48x = 48 determines the actual angle measures, from which the smallest can be chosen. Evaluating (x+30)∘(x + 30)^\circ gives 48+30=78∘48 + 30 = 78^\circ, which is the minimum value.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Setting up and solving linear algebraic equations is required to determine unknown angle measures.
Estimated Time:1m 30s
Question 30Question →

An irregular convex heptagon (7-sided polygon) has three interior angles that each measure 140∘140^\circ. The remaining four interior angles have measures in the ratio 2:3:3:42:3:3:4. What is the measure of the largest interior angle of this heptagon?

Show answer & explanation

Answer: 160∘160^\circ

Answer

160∘160^\circ
The total sum of the interior angles of a convex heptagon is (7−2)×180∘=900∘(7-2) \times 180^\circ = 900^\circ. Subtracting the three angles that each measure 140∘140^\circ (3×140∘=420∘3 \times 140^\circ = 420^\circ) leaves 480∘480^\circ for the remaining four angles. Using the ratio 2:3:3:42:3:3:4, we represent the angles as 2x2x, 3x3x, 3x3x, and 4x4x, giving the equation 2x+3x+3x+4x=480∘2x + 3x + 3x + 4x = 480^\circ, which simplifies to 12x=480∘12x = 480^\circ and yields x=40∘x = 40^\circ. The largest of these four angles is 4x=4(40∘)=160∘4x = 4(40^\circ) = 160^\circ. Since 160∘160^\circ is larger than 140∘140^\circ, the largest interior angle of the heptagon is 160∘160^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a heptagon.
900∘900^\circ
The sum of the interior angles of a convex polygon with nn sides is given by (n−2)×180∘(n-2) \times 180^\circ. For a heptagon (n=7n=7), the sum is (7−2)×180∘=5×180∘=900∘(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ.
2
Find the sum of the three known interior angles.
420∘420^\circ
Since three interior angles each measure 140∘140^\circ, their sum is 3×140∘=420∘3 \times 140^\circ = 420^\circ.
3
Calculate the sum of the remaining four interior angles.
480∘480^\circ
Subtracting the sum of the three known angles from the total interior sum gives 900∘−420∘=480∘900^\circ - 420^\circ = 480^\circ.
4
Set up and solve an equation for the remaining four angles using the given ratio.
x=40∘x = 40^\circ
Let the measures of the remaining four angles be 2x2x, 3x3x, 3x3x, and 4x4x. Their sum is 2x+3x+3x+4x=12x2x + 3x + 3x + 4x = 12x. Setting this equal to the remaining sum gives 12x=480∘12x = 480^\circ, which simplifies to x=40∘x = 40^\circ.
5
Determine the measures of the remaining angles and identify the largest angle.
160∘160^\circ
The remaining angles measure 2(40∘)=80∘2(40^\circ) = 80^\circ, 3(40∘)=120∘3(40^\circ) = 120^\circ, 3(40∘)=120∘3(40^\circ) = 120^\circ, and 4(40∘)=160∘4(40^\circ) = 160^\circ. The largest of these is 160∘160^\circ. Comparing this with the three 140∘140^\circ angles, the largest interior angle of the entire heptagon is 160∘160^\circ.

Key Concept

Interior angle sum of polygons and ratio division
Question 31Question →

A convex polygon has nn sides. The sum of the measures of n−1n-1 of its interior angles is 2030∘2030^\circ. What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 130

Answer

The measure of the remaining interior angle is 130 degrees.
The sum of the interior angles of a convex polygon must be a multiple of 180∘180^\circ. The multiple of 180∘180^\circ immediately greater than 2030∘2030^\circ is 2160∘2160^\circ (which is 12×180∘12 \times 180^\circ). The difference between the total sum and the sum of the n−1n-1 angles is 2160∘−2030∘=130∘2160^\circ - 2030^\circ = 130^\circ. Since 130∘130^\circ is less than 180∘180^\circ, this is a valid interior angle for a convex polygon.

Step-by-Step Solution

1
Write the formula for the total sum of the interior angles of a convex polygon.
The sum of the interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
This formula relates the number of sides to the total sum of the interior angles.
2
Apply the properties of convex polygons to set up an inequality for the total sum.
Since the remaining interior angle must be greater than 0∘0^\circ and less than 180∘180^\circ, the total sum of all nn angles must satisfy 2030∘<(n−2)×180∘<2030∘+180∘2030^\circ < (n-2) \times 180^\circ < 2030^\circ + 180^\circ, which simplifies to 2030∘<(n−2)×180∘<2210∘2030^\circ < (n-2) \times 180^\circ < 2210^\circ.
By definition, every interior angle of a convex polygon is strictly less than 180∘180^\circ.
3
Solve the inequality to find the integer value of n−2n-2.
Dividing the entire inequality by 180∘180^\circ gives 11.28<n−2<12.2811.28 < n-2 < 12.28. Since the number of sides nn must be an integer, n−2n-2 must be the integer 1212.
A polygon must have a whole number of sides.
4
Calculate the total sum of all interior angles and find the remaining angle.
The total sum is 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. Subtracting the sum of the other n−1n-1 angles gives the remaining angle: 2160∘−2030∘=130∘2160^\circ - 2030^\circ = 130^\circ.
The difference between the total sum of all interior angles and the sum of the n−1n-1 angles is the measure of the final angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, where each interior angle is strictly less than 180∘180^\circ.
Question 32Question →

For a certain convex polygon, the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1. How many sides does this polygon have?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12 sides.
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n-2) \times 180^\circ. The sum of the exterior angles of any convex polygon is always 360∘360^\circ. Since the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1, we can set up the equation: (n−2)×180∘360∘=51\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}. Simplifying the fraction on the left gives n−22=5\frac{n-2}{2} = 5. Multiplying both sides by 2 gives n−2=10n - 2 = 10, and adding 2 to both sides gives n=12n = 12. Therefore, the polygon has 12 sides.

Step-by-Step Solution

1
Write the formulas for the sum of the interior angles and the sum of the exterior angles of a convex polygon.
The sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ, where nn is the number of sides. The sum of the exterior angles is always 360∘360^\circ for any convex polygon.
These formulas represent the components of the given ratio.
2
Set up the ratio equation using the given information.
(n−2)×180∘360∘=51\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}
The problem states that the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1.
3
Simplify the equation and solve for the number of sides, nn.
n−22=5  ⟹  n−2=10  ⟹  n=12\frac{n-2}{2} = 5 \implies n-2 = 10 \implies n = 12
Simplifying 180360\frac{180}{360} to 12\frac{1}{2} makes it easier to solve the algebraic equation for nn.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, while the sum of the exterior angles is always 360∘360^\circ.
Estimated Time:1m 30s
Question 33Question →

A convex octagon has interior angles whose measures, in degrees, are eight consecutive even integers. What is the measure, in degrees, of the largest interior angle of this octagon?

Show answer & explanation

Answer: 142

Answer

The correct answer is 142.
The sum of the interior angles of a convex octagon is (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. If we represent the eight consecutive even integer angle measures as x,x+2,x+4,x+6,x+8,x+10,x+12,x, x+2, x+4, x+6, x+8, x+10, x+12, and x+14x+14, their sum is 8x+568x + 56. Setting this equal to 1080∘1080^\circ and solving for xx yields x=128x = 128. The largest angle is x+14x + 14, which equals 128+14=142∘128 + 14 = 142^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex octagon.
The sum of the interior angles is 1080∘1080^\circ.
The formula for the sum of the interior angles of an nn-sided polygon is (n−2)×180∘(n-2) \times 180^\circ. For an octagon (n=8n=8), the sum is (8−2)×180∘=6×180∘=1080∘(8-2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
2
Set up an equation representing the sum of the eight consecutive even integer angle measures.
The equation is 8x+56=10808x + 56 = 1080.
Letting the smallest angle measure be xx, the eight consecutive even integer angle measures are x,x+2,x+4,x+6,x+8,x+10,x+12,x, x+2, x+4, x+6, x+8, x+10, x+12, and x+14x+14. Their sum is 8x+568x + 56, which must equal the total sum of the interior angles (1080∘1080^\circ).
3
Solve the equation for the smallest angle measure, xx.
x=128x = 128
Subtracting 56 from both sides of the equation yields 8x=10248x = 1024. Dividing both sides by 8 gives x=128x = 128.
4
Calculate the measure of the largest interior angle.
The measure of the largest angle is 142∘142^\circ.
The largest angle is represented by the expression x+14x + 14. Substituting 128128 for xx gives 128+14=142128 + 14 = 142.

Key Concept

Calculating the sum of the interior angles of a convex polygon and using algebraic methods to find unknown angle measures.
Question 34Question →

The measures of the six interior angles of a convex hexagon are in the ratio 3:4:5:5:6:73:4:5:5:6:7. What is the measure, in degrees, of the largest interior angle of this hexagon?

Show answer & explanation

Answer: 168168

Answer

168
The sum of the interior angles of any convex nn-gon is given by (n−2)×180∘(n-2) \times 180^\circ. For a hexagon (n=6n=6), this sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. The angles are in the ratio 3:4:5:5:6:73:4:5:5:6:7, which sum to 3+4+5+5+6+7=303+4+5+5+6+7 = 30 parts. Each part corresponds to 720∘÷30=24∘720^\circ \div 30 = 24^\circ. The largest angle is represented by the largest part of the ratio, 7, which equals 7×24∘=168∘7 \times 24^\circ = 168^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
720∘720^\circ
The sum of the interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For a hexagon (n=6n=6), the sum is (6−2)×180∘=4×180∘=720∘(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Determine the total number of parts in the given ratio.
30 parts
Sum the components of the ratio: 3+4+5+5+6+7=303 + 4 + 5 + 5 + 6 + 7 = 30.
3
Find the value of one part in degrees.
24∘24^\circ per part
Divide the total interior angle sum by the total number of ratio parts: 720∘÷30=24∘720^\circ \div 30 = 24^\circ.
4
Calculate the measure of the largest interior angle.
168∘168^\circ
The largest angle corresponds to the largest component in the ratio, which is 7. Multiply the value of one part by 7: 7×24∘=168∘7 \times 24^\circ = 168^\circ.

Key Concept

Calculating the interior angle measures of an irregular convex polygon using the polygon interior angle sum formula and a given ratio.
Estimated Time:1m 30s
Question 35Question →

In a convex polygon, the measures of the interior angles form an arithmetic progression. The smallest interior angle measures 120∘120^\circ, and the common difference between consecutive interior angles is 5∘5^\circ. What is the number of sides of this polygon?

Show answer & explanation

Answer: 9

Answer

The number of sides of the polygon is 9.
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. Since the angles form an arithmetic progression with the first term a=120∘a = 120^\circ and common difference d=5∘d = 5^\circ, their sum is also given by the arithmetic series formula: Sn=n2[2(120)+(n−1)5]=n2(5n+235)S_n = \frac{n}{2}[2(120) + (n-1)5] = \frac{n}{2}(5n + 235). Setting the two sums equal yields n2(5n+235)=180(n−2)\frac{n}{2}(5n + 235) = 180(n - 2), which simplifies to the quadratic equation n2−25n+144=0n^2 - 25n + 144 = 0. Solving this gives n=9n = 9 or n=16n = 16. Because the polygon is convex, every interior angle must be less than 180∘180^\circ. If n=16n = 16, the largest angle would be 120∘+15(5∘)=195∘120^\circ + 15(5^\circ) = 195^\circ, which is impossible for a convex polygon. If n=9n = 9, the largest angle is 120∘+8(5∘)=160∘120^\circ + 8(5^\circ) = 160^\circ, which is valid. Therefore, the number of sides must be 9.

Step-by-Step Solution

1
Set up the equation equating the geometric sum of interior angles to the arithmetic series sum.
The sum of the interior angles of a convex nn-gon is (n−2)×180∘(n - 2) \times 180^\circ. The sum of the arithmetic sequence of angles is Sn=n2[2(120∘)+(n−1)5∘]S_n = \frac{n}{2}[2(120^\circ) + (n - 1)5^\circ]. Setting them equal gives: n2(5n+235)=180(n−2)\frac{n}{2}(5n + 235) = 180(n - 2).
This establishes the algebraic relationship between the polygon's geometric properties and the given sequence of angle measures.
2
Simplify the equation and solve the resulting quadratic equation for nn.
5n2+235n=360n−720⇒5n2−125n+720=0⇒n2−25n+144=05n^2 + 235n = 360n - 720 \Rightarrow 5n^2 - 125n + 720 = 0 \Rightarrow n^2 - 25n + 144 = 0. Factoring the quadratic yields (n−9)(n−16)=0(n - 9)(n - 16) = 0, so n=9n = 9 or n=16n = 16.
Solving the quadratic equation yields all mathematically possible values for the number of sides.
3
Apply the convexity constraint to determine the valid number of sides.
For a polygon to be convex, every interior angle must be less than 180∘180^\circ. For n=16n = 16, the largest angle is 120∘+15(5∘)=195∘120^\circ + 15(5^\circ) = 195^\circ, which is impossible. For n=9n = 9, the largest angle is 120∘+8(5∘)=160∘120^\circ + 8(5^\circ) = 160^\circ, which is valid.
The definition of a convex polygon requires all interior angles to be strictly less than 180∘180^\circ, which eliminates the extraneous solution of 16.

Key Concept

Sum of interior angles of a convex polygon and arithmetic progressions
PreviousPage 2 / 2