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Polygon Angles and Properties

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Question 1Question →

The interior angles of a quadrilateral are in the ratio 2:3:4:62:3:4:6. What is the degree measure of the largest interior angle of the quadrilateral?

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Answer: 144

Answer

144
The sum of the interior angles of a quadrilateral is 360∘360^\circ. Given the ratio 2:3:4:62:3:4:6, the sum of the parts is 2+3+4+6=152 + 3 + 4 + 6 = 15. The value of one part is 360∘÷15=24∘360^\circ \div 15 = 24^\circ. The largest angle corresponds to the largest part of the ratio, which is 66. Therefore, the largest angle measure is 6×24∘=144∘6 \times 24^\circ = 144^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a quadrilateral.
The sum of the interior angles of any quadrilateral is 360∘360^\circ.
This is a fundamental property of quadrilaterals, which can also be derived using the formula (n−2)×180∘(n - 2) \times 180^\circ with n=4n = 4.
2
Calculate the total number of parts in the given ratio.
The sum of the ratio parts is 2+3+4+6=152 + 3 + 4 + 6 = 15.
Adding the parts of the ratio allows us to find the size of a single share of the total angle sum.
3
Find the measure of one part of the ratio.
One part is equal to 360∘÷15=24∘360^\circ \div 15 = 24^\circ.
Dividing the total sum of the angles by the sum of the ratio parts determines the angle measure per ratio unit.
4
Multiply the largest ratio part by the value of one part to find the largest angle.
The largest angle is 6×24∘=144∘6 \times 24^\circ = 144^\circ.
The largest interior angle corresponds to the largest number in the ratio, which is 66.

Key Concept

Using ratios to find angle measures in a polygon.
Estimated Time:45s
Question 2Question →

A convex pentagon has four interior angles that measure 80∘80^\circ, 110∘110^\circ, 120∘120^\circ, and 130∘130^\circ. What is the degree measure of the fifth interior angle?

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Answer: 100

Answer

The degree measure of the fifth interior angle is 100∘100^\circ.
The sum of the interior angles of a pentagon (n=5n = 5) is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. The sum of the four given angles is 80∘+110∘+120∘+130∘=440∘80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ. The measure of the fifth angle is the difference between these two values: 540∘−440∘=100∘540^\circ - 440^\circ = 100^\circ.

Step-by-Step Solution

1
Find the sum of the interior angles of a convex pentagon.
The sum of the interior angles is 540∘540^\circ.
The sum of the interior angles of any nn-sided convex polygon is calculated using the formula (n−2)×180∘(n - 2) \times 180^\circ. For a pentagon, n=5n = 5, which gives (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
2
Sum the measures of the four given interior angles.
The sum of the four given angles is 440∘440^\circ.
Adding the given measures: 80∘+110∘+120∘+130∘=440∘80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ.
3
Subtract the sum of the four given angles from the total sum of the interior angles.
The measure of the fifth interior angle is 100∘100^\circ.
Subtracting the sum of the known angles from the total pentagon interior angle sum yields 540∘−440∘=100∘540^\circ - 440^\circ = 100^\circ.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ.
Question 3Question →

A regular octagon has 8 congruent interior angles. What is the degree measure of one of these interior angles?

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Answer: 135

Answer

The degree measure of one interior angle of a regular octagon is 135 degrees.
The interior angles of a regular octagon sum to 1080∘1080^\circ because (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. Since a regular octagon has 8 equal angles, dividing 1080∘1080^\circ by 88 results in 135∘135^\circ per interior angle.

Step-by-Step Solution

1
Identify the number of sides in a regular octagon.
The number of sides, nn, is 88.
An octagon is defined as a polygon with 8 sides and 8 angles.
2
Calculate the sum of the interior angles of the octagon.
The sum of the interior angles is 1080∘1080^\circ.
The sum of the interior angles of any convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For n=8n = 8, this is (8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
3
Find the measure of one interior angle by dividing the sum by the number of angles.
Each interior angle measures 135∘135^\circ.
A regular polygon has congruent interior angles. Therefore, dividing the total sum of 1080∘1080^\circ by the 88 congruent angles yields the measure of each individual interior angle: 1080∘÷8=135∘1080^\circ \div 8 = 135^\circ.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ. For a regular polygon, each interior angle has a measure of (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}.
Question 4Question →

For a certain four-sided convex polygon, the ratio of its interior angle measures is 1:2:3:41:2:3:4. What is the degree measure of the smallest angle?

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Answer: 36∘36^\circ

Answer

36∘36^\circ
The correct answer is 36∘36^\circ because the sum of the interior angles of a four-sided polygon is 360∘360^\circ. The ratio 1:2:3:41:2:3:4 means the angles can be represented as xx, 2x2x, 3x3x, and 4x4x. Their sum is 10x=360∘10x = 360^\circ, which yields x=36∘x = 36^\circ. The smallest angle is xx, which is 36∘36^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a four-sided convex polygon.
The sum of the interior angles of a four-sided polygon is 360∘360^\circ.
The sum of the interior angles of a polygon with nn sides is given by the formula (n−2)×180∘(n-2) \times 180^\circ. For a four-sided polygon, (4−2)×180∘=360∘(4-2) \times 180^\circ = 360^\circ.
2
Find the sum of the parts of the given ratio.
1+2+3+4=101 + 2 + 3 + 4 = 10 parts
To distribute the total angle measure proportionally, the individual parts of the ratio 1:2:3:41:2:3:4 must be summed.
3
Calculate the measure of one part of the ratio.
360∘÷10=36∘360^\circ \div 10 = 36^\circ
Dividing the total sum of the interior angles by the total number of parts determines the degree measure of a single part.
4
Find the measure of the smallest angle.
1×36∘=36∘1 \times 36^\circ = 36^\circ
The smallest angle corresponds to the smallest part of the ratio, which is 1.

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 this total can be divided proportionally using a given ratio to find individual angle measures.
Estimated Time:1m 0s
Question 5Question →

A convex hexagon has four interior angles that measure 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ. The remaining two interior angles are in the ratio 3:53:5. What is the degree measure of the largest interior angle of this hexagon?

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Answer: 150∘150^\circ

Answer

The degree measure of the largest interior angle is 150∘150^\circ.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the four given angles (100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ) from 720∘720^\circ leaves a remaining sum of 240∘240^\circ. Since the two remaining angles are in the ratio 3:53:5, they can be written as 3x3x and 5x5x, where 3x+5x=240∘3x + 5x = 240^\circ. Solving for xx gives 8x=240∘  ⟹  x=30∘8x = 240^\circ \implies x = 30^\circ. The two remaining angles are 90∘90^\circ and 150∘150^\circ. Comparing all six interior angles of the hexagon, the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a 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-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Find the sum of the four given interior angles.
100∘+115∘+125∘+140∘=480∘100^\circ + 115^\circ + 125^\circ + 140^\circ = 480^\circ.
This determines the total measure of the known angles.
3
Determine the remaining sum of the two unknown angles.
720∘−480∘=240∘720^\circ - 480^\circ = 240^\circ.
Subtracting the sum of the known angles from the total sum yields the combined measure of the remaining two angles.
4
Set up an equation using the ratio 3:53:5 to find the measures of the two remaining angles.
3x+5x=240∘  ⟹  8x=240∘  ⟹  x=30∘3x + 5x = 240^\circ \implies 8x = 240^\circ \implies x = 30^\circ. The two angles are 3(30∘)=90∘3(30^\circ) = 90^\circ and 5(30∘)=150∘5(30^\circ) = 150^\circ.
Representing the two angles in terms of a common variable xx allows us to solve for their individual measures.
5
Identify the largest interior angle among all six angles of the hexagon.
The largest angle is 150∘150^\circ.
Comparing all six angles (90∘90^\circ, 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, 140∘140^\circ, and 150∘150^\circ), the maximum value is 150∘150^\circ.

Key Concept

Calculating the interior angle sum of a polygon and solving for unknown angles using ratios.
Question 6Question →

An irregular convex hexagon has three interior angles that are congruent to each other, and the remaining three interior angles have measures in the ratio 3:4:53:4:5. If the sum of the measures of the three congruent angles is 360∘360^\circ, what is the degree measure of the largest interior angle of the hexagon?

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Answer: 150∘150^\circ

Answer

The degree measure of the largest interior angle of the hexagon is 150∘150^\circ.
The correct answer is 150∘150^\circ. The total sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the sum of the three congruent angles (360∘360^\circ) leaves 360∘360^\circ for the remaining three angles. With their measures in the ratio 3:4:53:4:5, we can write the equation 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, which simplifies to 12x=360∘12x = 360^\circ, giving x=30∘x = 30^\circ. The measures of these three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The three congruent angles each measure 360∘/3=120∘360^\circ / 3 = 120^\circ. Comparing all six angles (120∘,120∘,120∘,90∘,120∘,150∘120^\circ, 120^\circ, 120^\circ, 90^\circ, 120^\circ, 150^\circ), the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
720∘720^\circ
Using the formula (n−2)×180∘(n - 2) \times 180^\circ for a polygon with n=6n = 6 sides, the sum is (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Determine the sum of the remaining three interior angles.
360∘360^\circ
We subtract the sum of the three congruent angles (360∘360^\circ) from the total sum of the hexagon's interior angles (720∘720^\circ): 720∘−360∘=360∘720^\circ - 360^\circ = 360^\circ.
3
Set up and solve an equation using the ratio of the remaining three angles.
x=30∘x = 30^\circ
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 360∘360^\circ, so 3x+4x+5x=360∘  ⟹  12x=360∘  ⟹  x=30∘3x + 4x + 5x = 360^\circ \implies 12x = 360^\circ \implies x = 30^\circ.
4
Calculate the measures of all six interior angles to identify the largest one.
The angles are 120∘120^\circ, 120∘120^\circ, 120∘120^\circ, 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The largest is 150∘150^\circ.
Each of the three congruent angles measures 360∘/3=120∘360^\circ / 3 = 120^\circ. The other three angles measure 3(30∘)=90∘3(30^\circ) = 90^\circ, 4(30∘)=120∘4(30^\circ) = 120^\circ, and 5(30∘)=150∘5(30^\circ) = 150^\circ. Comparing all these values, the maximum is 150∘150^\circ.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ. Irregular polygons share this total sum, and ratio relationships can be solved by introducing a variable multiplier.
Question 7Question →

The measures of five of the interior angles of a convex hexagon are in the ratio 3:4:5:6:73:4:5:6:7. The measure of the sixth interior angle is 30∘30^\circ less than the average measure of the other five angles. What is the degree measure of the largest interior angle of the hexagon?

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Answer: 175∘175^\circ

Answer

The correct answer is 175^\circ because solving the equation for the sum of the hexagon's interior angles yields a multiplier of x=25x = 25, making the largest angle 7x=175∘7x = 175^\circ.
The correct answer is 175° because the sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Representing the five ratio-based angles as 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x gives their sum as 25x25x and their average as 5x5x. The sixth angle is therefore 5x−305x - 30. Setting up the sum of all six angles yields 25x+(5x−30)=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x = 750 \implies x = 25. The largest angle is 7x=7(25)=175∘7x = 7(25) = 175^\circ.

Step-by-Step Solution

1
Find the sum of the interior angles of a hexagon.
Sum = 720∘720^\circ
Using the interior angle sum formula S=(n−2)×180∘S = (n - 2) \times 180^\circ for a hexagon where n=6n = 6, we get S=(6−2)×180∘=4×180∘=720∘S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Express the five ratio-based angles in terms of a variable xx.
The angles are 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x.
Since the measures of five angles are in the ratio 3:4:5:6:73:4:5:6:7, we can define them as multiples of a common scale factor xx.
3
Compute the average measure of these five angles in terms of xx.
Average = 5x5x
The sum of the five angles is 3x+4x+5x+6x+7x=25x3x + 4x + 5x + 6x + 7x = 25x. The average is the sum divided by the count: 25x5=5x\frac{25x}{5} = 5x.
4
Express the measure of the sixth angle in terms of xx.
Sixth angle = 5x−305x - 30
The sixth angle is described as being 30∘30^\circ less than the average of the other five angles, which we found to be 5x5x.
5
Write and solve the equation for the sum of all six interior angles.
x=25x = 25
The sum of all six angles must equal the total sum of 720∘720^\circ: 25x+(5x−30)=720  ⟹  30x−30=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x - 30 = 720 \implies 30x = 750 \implies x = 25.
6
Calculate the measure of the largest interior angle.
175∘175^\circ
The largest interior angle corresponds to the largest term in the ratio, which is 7x7x. Substituting x=25x = 25 yields 7×25=175∘7 \times 25 = 175^\circ.

Key Concept

Applying the interior angle sum formula for polygons combined with ratio and algebraic translation properties.

Alternative Method

Instead of using the sum of interior angles, one could use the sum of exterior angles, which is always 360∘360^\circ. The exterior angles corresponding to the five angles in ratio 3:4:5:6:73:4:5:6:7 would be 180−3x180 - 3x, 180−4x180 - 4x, 180−5x180 - 5x, 180−6x180 - 6x, and 180−7x180 - 7x. The sixth exterior angle is 180−(5x−30)=210−5x180 - (5x - 30) = 210 - 5x. Summing these six exterior angles: (180−3x)+(180−4x)+(180−5x)+(180−6x)+(180−7x)+(210−5x)=1110−30x=360(180 - 3x) + (180 - 4x) + (180 - 5x) + (180 - 6x) + (180 - 7x) + (210 - 5x) = 1110 - 30x = 360. Solving for xx yields 30x=750  ⟹  x=2530x = 750 \implies x = 25. The largest interior angle corresponds to the smallest exterior angle, which is 180−7x=180−7(25)=5∘180 - 7x = 180 - 7(25) = 5^\circ, giving the largest interior angle as 180−5=175∘180 - 5 = 175^\circ.
Estimated Time:3m 0s
Question 8Question →

A convex polygon has nn sides. The sum of the measures of its interior angles is 33 times the sum of the measures of its exterior angles. What is the value of nn?

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Answer: 8

Answer

8
The sum of the interior angle measures of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ. The sum of the exterior angle measures is always 360∘360^\circ. According to the problem, the sum of the interior angles is 33 times the sum of the exterior angles, which can be written as the equation (n−2)×180=3×360(n-2) \times 180 = 3 \times 360. Simplifying the right side gives (n−2)×180=1080(n-2) \times 180 = 1080. Dividing both sides by 180180 results in n−2=6n - 2 = 6. Adding 22 to both sides yields n=8n = 8.

Step-by-Step Solution

1
Use the formula for the sum of the interior angle measures of a convex polygon with nn sides.
The sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ.
By the polygon interior angle sum theorem, the sum of the interior angles of any convex nn-gon is (n−2)×180∘(n-2) \times 180^\circ.
2
Identify the sum of the exterior angle measures of a convex polygon.
The sum of the exterior angles is 360∘360^\circ.
The sum of the exterior angles of any convex polygon is always constant and equals 360∘360^\circ, regardless of the number of sides.
3
Set up the equation relating the two sums as given in the problem statement.
(n−2)×180∘=3×360∘(n-2) \times 180^\circ = 3 \times 360^\circ
The problem states that the sum of the interior angles is 33 times the sum of the exterior angles.
4
Solve the equation for nn.
n=8n = 8
Divide both sides by 180∘180^\circ to get n−2=6n - 2 = 6, then add 22 to both sides to find n=8n = 8.

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 the sum of its exterior angles is 360∘360^\circ.
Question 9Question →

The sum of the measures of the interior angles of a convex polygon is 900∘900^\circ. How many sides does this polygon have?

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Answer: 7

Answer

The polygon has 7 sides.
The correct answer is 77 because the sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. Setting this equal to 900∘900^\circ yields (n−2)×180=900(n - 2) \times 180 = 900. Dividing both sides by 180180 gives n−2=5n - 2 = 5, and adding 22 to both sides results in n=7n = 7.

Step-by-Step Solution

1
Set up the equation for the sum of the interior angles of a polygon.
(n−2)×180∘=900∘(n - 2) \times 180^\circ = 900^\circ
The sum of the interior angles of a convex polygon with nn sides is always (n−2)×180∘(n - 2) \times 180^\circ.
2
Divide both sides of the equation by 180∘180^\circ.
n−2=5n - 2 = 5
To isolate the term with nn, we divide 900900 by 180180.
3
Solve for nn by adding 22 to both sides.
n=7n = 7
Adding 22 to both sides isolates the variable nn.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n - 2) \times 180^\circ.
Estimated Time:45s
Question 10Question →

A regular decagon has 1010 sides of equal length. What is the measure, in degrees, of one exterior angle of this decagon?

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Answer: 36

Answer

The correct answer is 36.
The sum of the exterior angles of any convex polygon is always 360∘360^\circ. A regular decagon has 1010 congruent sides and therefore 1010 congruent exterior angles. Dividing 360∘360^\circ by 1010 yields 36∘36^\circ for each exterior angle.

Step-by-Step Solution

1
Determine the number of exterior angles in a regular decagon.
A decagon has 10 sides, so it has 10 exterior angles.
A polygon has the same number of exterior angles as its number of sides.
2
State the sum of the exterior angles for a convex polygon.
The sum of the exterior angles is 360 degrees.
The exterior angles of any convex polygon always sum to 360 degrees regardless of the number of sides.
3
Calculate the measure of one exterior angle.
360 / 10 = 36
Since the decagon is regular, all of its exterior angles are equal in measure, so we divide the total sum by the number of angles.

Key Concept

The sum of the exterior angles of any convex polygon is 360∘360^\circ. For a regular polygon with nn sides, the measure of each exterior angle is 360∘n\frac{360^\circ}{n}.
Question 11Question →

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

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Answer: 60∘60^\circ

Answer

60∘60^\circ
The sum of the interior angles of a convex pentagon is calculated as (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Representing the five angle measures in the given ratio as 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x, their sum is 18x=540∘18x = 540^\circ, which gives x=30∘x = 30^\circ. The smallest angle corresponds to the smallest term in the ratio, which is 2x=2×30∘=60∘2x = 2 \times 30^\circ = 60^\circ. Therefore, the correct measure is 60∘60^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a convex pentagon.
The sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
The sum of the interior angles of any convex nn-gon is given by the formula (n−2)×180∘(n - 2) \times 180^\circ, and a pentagon has 55 sides.
2
Set up an algebraic equation using the given ratio of the angle measures.
Let the angle measures be 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x. Their sum is 2x+3x+4x+4x+5x=18x=540∘2x + 3x + 4x + 4x + 5x = 18x = 540^\circ.
The sum of the actual angle measures must equal the total interior angle sum of the pentagon.
3
Solve for the variable xx.
x=540∘/18=30∘x = 540^\circ / 18 = 30^\circ.
Dividing the total sum by the sum of the ratio parts gives the value of a single ratio unit.
4
Calculate the measure of the smallest interior angle.
Smallest angle =2x=2×30∘=60∘= 2x = 2 \times 30^\circ = 60^\circ.
The smallest term in the ratio is 22, so multiplying this term by xx gives the smallest angle 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 with angles in a given ratio, the individual angle measures can be found by setting up a linear equation where the sum of the ratio parts multiplied by a variable equals the total sum.
Question 12Question →

An irregular convex polygon has nn sides. The measures of its interior angles, in degrees, are all distinct integers. If all of the interior angles are obtuse, what is the maximum possible value of nn?

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Answer: 26

Answer

The maximum possible value of nn is 26.
For a convex 2626-gon, the sum of the interior angles is 24×180∘=4320∘24 \times 180^\circ = 4320^\circ. We can choose 2626 distinct integer angles from the range [91∘,179∘][91^\circ, 179^\circ] that sum to exactly 4320∘4320^\circ because the maximum possible sum of 2626 distinct integers in this range is 4329∘4329^\circ, which is greater than 4320∘4320^\circ. For n=27n = 27, the sum of the interior angles must be 25×180∘=4500∘25 \times 180^\circ = 4500^\circ, but the maximum possible sum of 2727 distinct integers in the range is only 4482∘4482^\circ, which is less than 4500∘4500^\circ. Therefore, 2626 is the maximum value of nn.

Step-by-Step Solution

1
Determine the set of possible angle measures.
The angles must be integers in the range [91∘,179∘][91^\circ, 179^\circ].
Interior angles of a convex polygon must be less than 180∘180^\circ. Since they are obtuse and distinct integers, they must be strictly greater than 90∘90^\circ, giving the range [91,179][91, 179].
2
Write the sum of the interior angles of a convex nn-gon.
Sum =(n−2)×180∘= (n - 2) \times 180^\circ.
This is the standard formula for the sum of the interior angles of any convex nn-gon.
3
Find the maximum possible sum of nn distinct angles in the range [91,179][91, 179].
Maximum Sum =179n−n(n−1)2= 179n - \frac{n(n - 1)}{2}.
The maximum sum is achieved by selecting the largest nn integers from the set: 179,178,…,179−(n−1)179, 178, \dots, 179 - (n - 1).
4
Set up the inequality and simplify.
n2+n−720≤0n^2 + n - 720 \le 0.
Since the sum of the angles must be less than or equal to the maximum possible sum, we have (n−2)×180≤179n−n(n−1)2(n-2) \times 180 \le 179n - \frac{n(n-1)}{2}. Multiplying by 2 and simplifying yields the quadratic inequality.
5
Solve the quadratic inequality for the largest integer nn.
n=26n = 26.
Evaluating the quadratic expression for consecutive integers: for n=26n = 26, 262+26−720=−18≤026^2 + 26 - 720 = -18 \le 0; for n=27n = 27, 272+27−720=36>027^2 + 27 - 720 = 36 > 0. Thus, 26 is the maximum possible integer value.

Key Concept

Sum of interior angles of a convex polygon combined with algebraic optimization.
Question 13Question →

A convex polygon has nn sides. The sum of the measures of its interior angles is 66 times the sum of the measures of its exterior angles (one at each vertex). What is the value of nn?

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Answer: 14

Answer

The number of sides, nn, of the convex polygon is 14.
The sum of the interior angles of a convex polygon with nn sides is given by the formula (n−2)×180∘(n-2) \times 180^\circ, and the sum of its exterior angles is always 360∘360^\circ. According to the problem, the sum of the interior angles is 66 times the sum of the exterior angles, yielding the equation (n−2)×180=6×360(n - 2) \times 180 = 6 \times 360. Dividing both sides of the equation by 180180 gives n−2=12n - 2 = 12. Adding 22 to both sides results in n=14n = 14.

Step-by-Step Solution

1
State the sum of interior and exterior angles formulas.
Interior sum = (n−2)×180∘(n-2) \times 180^\circ, Exterior sum = 360∘360^\circ.
To represent the geometric properties of the polygon algebraically.
2
Set up the equation based on the given ratio.
(n−2)×180=6×360(n-2) \times 180 = 6 \times 360.
The problem states the interior sum is 6 times the exterior sum.
3
Solve the equation for nn.
n=14n = 14.
Divide by 180 to get n−2=12n - 2 = 12, then add 2 to both sides.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ, and the sum of the exterior angles (one per vertex) is always 360∘360^\circ.
Question 14Question →

An irregular convex hexagon has two interior angles measuring 90∘90^\circ and 130∘130^\circ, respectively. The remaining four interior angles have measures in the ratio 5:6:7:75:6:7:7. What is the degree measure of the largest interior angle in this hexagon?

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Answer: 140∘140^\circ

Answer

The correct answer is 140∘140^\circ.
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two given angles (90∘90^\circ and 130∘130^\circ) leaves 500∘500^\circ for the remaining four angles. Since these angles are in the ratio 5:6:7:75:6:7:7, we represent them as 5x5x, 6x6x, 7x7x, and 7x7x. Their sum is 25x=500∘25x = 500^\circ, which gives x=20∘x = 20^\circ. The largest of the remaining angles is 7x=7(20∘)=140∘7x = 7(20^\circ) = 140^\circ. Since 140∘140^\circ is greater than both 90∘90^\circ and 130∘130^\circ, it is the largest interior angle of the hexagon.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a hexagon.
Using the formula for the sum of the interior angles of a polygon with nn sides, (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.
To establish the total sum of all interior angles of the polygon.
2
Subtract the two known angle measures from the total sum.
720∘−90∘−130∘=500∘720^\circ - 90^\circ - 130^\circ = 500^\circ.
To find the sum of the remaining four interior angles.
3
Set up an algebraic equation to find the value of one ratio unit, xx.
5x+6x+7x+7x=500∘  ⟹  25x=500∘  ⟹  x=20∘5x + 6x + 7x + 7x = 500^\circ \implies 25x = 500^\circ \implies x = 20^\circ.
To determine the constant multiplier for the ratio of the remaining angles.
4
Calculate the measures of the remaining angles and determine the largest angle.
The remaining angles are 5(20∘)=100∘5(20^\circ) = 100^\circ, 6(20∘)=120∘6(20^\circ) = 120^\circ, 7(20∘)=140∘7(20^\circ) = 140^\circ, and 7(20∘)=140∘7(20^\circ) = 140^\circ. Comparing all six angles of the hexagon (90∘,100∘,120∘,130∘,140∘,140∘90^\circ, 100^\circ, 120^\circ, 130^\circ, 140^\circ, 140^\circ), the largest angle is 140∘140^\circ.
To identify the maximum interior angle measure of the hexagon.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. The individual angle measures in an irregular polygon can be determined using algebraic representations of their relationships or ratios.

Alternative Method

Once the value of the ratio unit x=20∘x = 20^\circ is determined, you can quickly find the largest candidate angle by multiplying the largest ratio component (77) by xx to get 7(20∘)=140∘7(20^\circ) = 140^\circ, and then compare it to the given angles (90∘90^\circ and 130∘130^\circ) to verify that it is indeed the largest.
Estimated Time:1m 30s
Question 15Question →

The ratio of the measure of an interior angle of a regular polygon to the measure of its exterior angle is 3:13:1. How many sides does this polygon have?

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Answer: 8

Answer

8
The interior angle and exterior angle of a polygon at any vertex are supplementary, meaning they add up to 180∘180^\circ. Given the ratio of the interior angle to the exterior angle is 3:13:1, we can express their measures as 3x3x and xx. Solving the equation 3x+x=180∘3x + x = 180^\circ gives 4x=180∘4x = 180^\circ, which means x=45∘x = 45^\circ. Therefore, the measure of each exterior angle of the regular polygon is 45∘45^\circ. Since the sum of the exterior angles of any convex polygon is always 360∘360^\circ, the number of sides nn is calculated by dividing 360∘360^\circ by the measure of one exterior angle: n=360∘45∘=8n = \frac{360^\circ}{45^\circ} = 8.

Step-by-Step Solution

1
Set up an equation for the interior and exterior angles using the given ratio.
Let the measure of the exterior angle be xx and the measure of the interior angle be 3x3x.
The ratio of the interior angle to the exterior angle is 3:13:1, so their measures can be represented as 3x3x and xx respectively.
2
Use the fact that the interior angle and exterior angle at any vertex of a polygon are supplementary (form a linear pair).
3x+x=180∘  ⟹  4x=180∘  ⟹  x=45∘3x + x = 180^\circ \implies 4x = 180^\circ \implies x = 45^\circ.
An interior angle and its adjacent exterior angle always lie on a straight line and sum to 180∘180^\circ.
3
Calculate the number of sides of the regular polygon using the measure of one exterior angle.
n=360∘45∘=8n = \frac{360^\circ}{45^\circ} = 8.
The sum of the exterior angles of any convex polygon is always 360∘360^\circ. For a regular polygon with nn sides, each exterior angle measures 360∘n\frac{360^\circ}{n}.

Key Concept

The relationship between the interior and exterior angles of a regular polygon, and the formula relating the number of sides to the sum of the exterior angles.
Question 16Question →

The sum of the measures of all but one of the interior angles of a convex polygon is 2010∘2010^\circ. What is the measure, in degrees, of the remaining interior angle?

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Answer: 150

Answer

The measure of the remaining interior angle is 150 degrees.
The sum of the interior angles of any convex polygon with nn sides is a multiple of 180∘180^\circ given by (n−2)×180∘(n-2) \times 180^\circ. Because the polygon is convex, the measure of the remaining angle must be strictly less than 180∘180^\circ. Thus, the total sum of all interior angles must be the smallest multiple of 180∘180^\circ that is strictly greater than the given sum of 2010∘2010^\circ. Since 11×180∘=1980∘11 \times 180^\circ = 1980^\circ (which is less than 2010∘2010^\circ), the total sum must be at least 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. Subtracting the given sum of the other angles from this total gives 2160∘−2010∘=150∘2160^\circ - 2010^\circ = 150^\circ. Since 150∘150^\circ is less than 180∘180^\circ, this is a mathematically valid remaining angle for a convex polygon.

Step-by-Step Solution

1
Set up the inequality for the sum of the interior angles.
The total sum S=(n−2)×180∘S = (n-2) \times 180^\circ must satisfy 2010∘<S<2010∘+180∘2010^\circ < S < 2010^\circ + 180^\circ, which simplifies to 2010∘<S<2190∘2010^\circ < S < 2190^\circ.
Since the polygon is convex, the remaining interior angle must have a measure strictly between 0∘0^\circ and 180∘180^\circ.
2
Determine the value of n−2n-2 by finding the unique integer multiple.
Dividing the inequality by 180∘180^\circ gives 11.17<n−2<12.1711.17 < n-2 < 12.17. Since nn must be an integer, n−2=12n-2 = 12, which means the polygon has n=14n = 14 sides.
The number of sides of a polygon must be a whole number, so n−2n-2 must be an integer.
3
Calculate the measure of the remaining interior angle.
x=(12×180∘)−2010∘=2160∘−2010∘=150∘x = (12 \times 180^\circ) - 2010^\circ = 2160^\circ - 2010^\circ = 150^\circ.
Subtract the sum of the other interior angles from the total sum of the interior angles of a 14-gon.

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 each interior angle of a convex polygon must measure strictly less than 180∘180^\circ.
Question 17Question →

A designer is creating a custom floor tile in the shape of an irregular convex pentagon. The tile has one right angle. The remaining four interior angles are in the ratio 2:3:3:42:3:3:4. What is the measure, in degrees, of the largest interior angle of this tile?

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Answer: 150∘150^\circ

Answer

The largest interior angle of the tile measures 150∘150^\circ.
The correct answer is 150∘150^\circ. The sum of the interior angles of a pentagon is 540∘540^\circ. Subtracting the right angle (90∘90^\circ) leaves 450∘450^\circ for the remaining four angles. Since they are in the ratio 2:3:3:42:3:3:4, their sum can be represented as 12y=450∘12y = 450^\circ, which yields y=37.5∘y = 37.5^\circ. The largest of these angles is 4y=150∘4y = 150^\circ, which is greater than the other angles (75∘75^\circ, 112.5∘112.5^\circ, and 90∘90^\circ).

Step-by-Step Solution

1
Calculate the sum of the interior angles of a pentagon.
The sum of the interior angles is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
A pentagon has 5 sides, and the sum of the interior angles of any convex nn-gon is given by the formula (n−2)×180∘(n - 2) \times 180^\circ.
2
Subtract the right angle to find the sum of the remaining four interior angles.
The sum of the remaining angles is 540∘−90∘=450∘540^\circ - 90^\circ = 450^\circ.
One of the angles is a right angle, which measures 90∘90^\circ.
3
Set up an equation using the given ratio to find the value of one part of the ratio.
Let the four remaining angles be 2y2y, 3y3y, 3y3y, and 4y4y. Their sum is 2y+3y+3y+4y=12y=450∘2y + 3y + 3y + 4y = 12y = 450^\circ, which gives y=37.5∘y = 37.5^\circ.
The remaining angles are in the ratio 2:3:3:42:3:3:4, so their measures are proportional to these values.
4
Calculate the measure of the largest interior angle.
The largest angle corresponds to the largest term in the ratio, which is 4y4y. Thus, the largest angle is 4×37.5∘=150∘4 \times 37.5^\circ = 150^\circ.
The question asks for the measure of the largest interior 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, and ratios can be used to partition a total quantity into proportional parts.
Question 18Question →

A convex polygon has nn sides. The sum of the measures of all but one of its interior angles is 2020∘2020^\circ. What is the measure, in degrees, of the remaining interior angle?

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Answer: 140

Answer

The measure of the remaining interior angle is 140∘140^\circ.
For a convex polygon with nn sides, the sum of all interior angles is (n−2)×180∘(n-2) \times 180^\circ. If we represent the remaining interior angle as xx, then the sum of all interior angles can be written as 2020∘+x2020^\circ + x. Because the polygon is convex, the measure of the remaining interior angle must satisfy the inequality 0∘<x<180∘0^\circ < x < 180^\circ. Substituting this into the sum expression gives the inequality 2020∘<(n−2)×180∘<2200∘2020^\circ < (n-2) \times 180^\circ < 2200^\circ. Dividing by 180∘180^\circ, we find 11.22<n−2<12.2211.22 < n-2 < 12.22. Since n−2n-2 must be an integer, n−2n-2 must equal 1212, which means the total sum of the interior angles is 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. The remaining angle is found by subtracting 2020∘2020^\circ from 2160∘2160^\circ, resulting in 140∘140^\circ.

Step-by-Step Solution

1
Use the polygon interior angle sum formula for an nn-sided polygon.
The sum of all interior angles is (n−2)×180∘(n-2) \times 180^\circ.
This formula connects the number of sides of a convex polygon to the total sum of its interior angles.
2
Set up an equation containing the sum of all but one angle (2020∘2020^\circ) and the remaining angle (xx).
(n−2)×180∘=2020∘+x(n-2) \times 180^\circ = 2020^\circ + x
The total sum of all interior angles is equal to the sum of the n−1n-1 known angles plus the remaining angle.
3
Apply the convexity constraint 0∘<x<180∘0^\circ < x < 180^\circ to construct an inequality for the total sum of the interior angles.
2020∘<(n−2)×180∘<2200∘2020^\circ < (n-2) \times 180^\circ < 2200^\circ
Since the remaining angle must be strictly between 0∘0^\circ and 180∘180^\circ for a convex polygon, adding 2020∘2020^\circ gives the boundaries for the total sum.
4
Divide the inequality by 180∘180^\circ to isolate the term n−2n-2.
11.22<n−2<12.2211.22 < n-2 < 12.22
This determines the numerical boundaries for the integer value of n−2n-2.
5
Find the unique integer value for n−2n-2 and calculate the exact total sum of the interior angles.
n−2=12n-2 = 12, which gives a total sum of 12×180∘=2160∘12 \times 180^\circ = 2160^\circ.
Because nn must be an integer, n−2n-2 must be an integer. The only integer in the interval (11.22,12.22)(11.22, 12.22) is 1212.
6
Subtract the sum of the other angles from the total sum of the interior angles to solve for xx.
x=2160∘−2020∘=140∘x = 2160^\circ - 2020^\circ = 140^\circ.
This yields the exact value of the remaining interior angle.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ, where each interior angle is strictly between 0∘0^\circ and 180∘180^\circ.
Question 19Question →

A designer is creating a custom tiled floor using irregular convex pentagonal tiles. In each pentagon, the measures of three of the interior angles are in the ratio 2:3:42:3:4. The other two interior angles are congruent to each other, and each is 15∘15^\circ less than the sum of the two smallest angles in the ratio. What is the measure, in degrees, of the largest interior angle of one of these pentagonal tiles?

Show answer & explanation

Answer: 135∘135^\circ

Answer

The correct answer is 135 degrees. The largest interior angle of the pentagon is one of the two congruent angles.
The correct answer is 135 degrees. The sum of the interior angles of a pentagon is 540 degrees. Representing the three angles in the ratio as 2x, 3x, and 4x gives a sum of 9x. The remaining two angles are each equal to the sum of the two smallest ratio terms minus 15, which is 5x - 15. The sum of all five angles is 19x - 30 = 540, which yields x = 30. Evaluating the angles gives 60, 90, 120, 135, and 135 degrees. The largest of these is 135 degrees.

Step-by-Step Solution

1
Determine the sum of the interior angles of a pentagon.
The sum is 540∘540^\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 a pentagon (n=5n = 5), the sum is (5−2)×180∘=3×180∘=540∘(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
2
Set up an algebraic equation representing the sum of all five interior angles.
The equation is 2x+3x+4x+2(5x−15∘)=540∘2x + 3x + 4x + 2(5x - 15^\circ) = 540^\circ.
Let the three angles in the ratio 2:3:42:3:4 be represented as 2x2x, 3x3x, and 4x4x. The sum of the two smallest is 2x+3x=5x2x + 3x = 5x. Each of the other two congruent angles is 15∘15^\circ less than this sum, which is 5x−15∘5x - 15^\circ.
3
Solve the algebraic equation for xx.
x=30∘x = 30^\circ.
Simplify the equation: 9x+10x−30∘=540∘  ⟹  19x=570∘  ⟹  x=30∘9x + 10x - 30^\circ = 540^\circ \implies 19x = 570^\circ \implies x = 30^\circ.
4
Calculate the measures of all five interior angles and identify the largest.
The angles are 60∘60^\circ, 90∘90^\circ, 120∘120^\circ, 135∘135^\circ, and 135∘135^\circ. The largest angle is 135∘135^\circ.
Substitute x=30∘x = 30^\circ into each expression: 2(30)=60∘2(30) = 60^\circ, 3(30)=90∘3(30) = 90^\circ, 4(30)=120∘4(30) = 120^\circ, and 5(30)−15=135∘5(30) - 15 = 135^\circ for the other two. The largest value among these is 135∘135^\circ.

Key Concept

Polygon interior angle sum and algebraic representation of ratios
Question 20Question →

For a certain convex polygon with nn sides, the sum of the measures of its interior angles is exactly 2424 times the measure of one exterior angle of a regular polygon with nn sides. If this polygon is regular, what is the measure, in degrees, of each of its interior angles?

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Answer: 135

Answer

135
By translating the problem description into an algebraic relationship, we establish that the sum of the interior angles, (n−2)×180∘(n-2) \times 180^\circ, equals 2424 times the measure of a single exterior angle, 360∘n\frac{360^\circ}{n}. Simplifying this equation by dividing both sides by 180180 yields n−2=48nn - 2 = \frac{48}{n}. Multiplying by nn creates the quadratic equation n2−2n−48=0n^2 - 2n - 48 = 0, which factors as (n−8)(n+6)=0(n-8)(n+6) = 0. Since the number of sides of a polygon must be positive, n=8n = 8. For a regular octagon (n=8n=8), the measure of each interior angle is (8−2)×180∘8=135∘\frac{(8-2) \times 180^\circ}{8} = 135^\circ.

Step-by-Step Solution

1
Set up the equation based on the geometric properties of polygons.
(n−2)×180=24×360n(n-2) \times 180 = 24 \times \frac{360}{n}
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. The measure of one exterior angle of a regular polygon with nn sides is 360∘n\frac{360^\circ}{n}.
2
Simplify the equation and solve for the number of sides nn.
n2−2n−48=0n^2 - 2n - 48 = 0, which gives n=8n = 8.
Dividing both sides by 180180 simplifies the equation to n−2=48nn - 2 = \frac{48}{n}. Multiplying by nn yields the quadratic equation n(n−2)=48n(n - 2) = 48, or n2−2n−48=0n^2 - 2n - 48 = 0. Factoring gives (n−8)(n+6)=0(n - 8)(n + 6) = 0. Since the number of sides must be a positive integer, n=8n = 8.
3
Calculate the measure of each interior angle of a regular octagon (n=8n = 8).
135∘135^\circ
The measure of each interior angle of a regular polygon is given by (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}. Substituting n=8n = 8 yields (8−2)×180∘8=6×180∘8=135∘\frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ.

Key Concept

The relationship between the sum of interior angles, the number of sides, and the exterior angles of regular polygons.
Estimated Time:2m 30s
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