Polygon Angles and Properties Alıştırma Soruları — ACT — Sayfa 2

Soru 21Soru →

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

8

9

12

18

36

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The regular polygon has 18 sides.

The correct answer is 18. By setting up the equations for the interior angle

I

and exterior angle

E

, we have

I = E + 140^\circ

and

I + E = 180^\circ

. Substituting the first equation into the second gives

(E + 140^\circ) + E = 180^\circ

, which simplifies to

2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

. The number of sides

n

of a regular polygon is given by

n = 360^\circ / E

. Thus,

n = 360^\circ / 20^\circ = 18

.

Adım Adım Çözüm

1

Set up the algebraic relationship between the interior angle (

I

) and the exterior angle (

E

).

I = E + 140^\circ

The problem states that the interior angle is

140^\circ

greater than the exterior angle.

2

Use the supplementary relationship between any interior angle and its corresponding exterior angle.

I + E = 180^\circ

An interior angle and its adjacent exterior angle form a linear pair, which sums to

180^\circ

.

3

Substitute the expression for

I

from Step 1 into the equation from Step 2 and solve for

E

.

(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 (

n

) using the sum of the exterior angles of a convex polygon.

n = \frac{360^\circ}{E} = \frac{360^\circ}{20^\circ} = 18

The sum of the exterior angles of any convex polygon is

360^\circ

, and in a regular polygon, all

n

exterior angles are equal.

Anahtar Kavram

Relationship between interior and exterior angles of a regular polygon

Tahmini Süre:1m 30s

Soru 22Soru →

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

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The maximum possible value of

n

is 8.

The correct answer is 8 because we set the sum of the interior angles

(n-2) \times 180^\circ

equal to the sum of the arithmetic progression

\frac{n}{2}[200 + (n-1)d]

. Solving for

(n-1)d

gives

(n-1)d = 160 - \frac{720}{n}

. Since the polygon is convex, the largest angle

100 + (n-1)d

must be strictly less than

180^\circ

, which means

(n-1)d < 80^\circ

. Substituting the expression yields

160 - \frac{720}{n} < 80

, which simplifies to

n < 9

. Since

n

must be an integer, the maximum possible value of

n

is 8. For

n = 8

, the common difference

d = 10

is a positive integer, satisfying all conditions.

Adım Adım Çözüm

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

n

sides is

(n-2) \times 180^\circ

. The sum of the angles in arithmetic progression with first term

100^\circ

and common difference

d^\circ

is

\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

.

100n + \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

, 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^\circ

.

The largest angle is the last term of the progression:

100 + (n-1)d

. For the polygon to be convex, this angle must be strictly less than

180^\circ

. Therefore,

100 + (n-1)d < 180 \implies (n-1)d < 80

. Substituting

(n-1)d = 160 - \frac{720}{n}

gives

160 - \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^\circ

.

4

Identify the maximum integer value of

n

and verify that a positive integer common difference

d

exists.

Since

n < 9

and

n

must be an integer, the maximum possible value is

n = 8

. For

n = 8

, we calculate

(8-1)d = 160 - \frac{720}{8} \implies 7d = 70 \implies d = 10

. Since

d = 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.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and all interior angles of a convex polygon must be strictly less than

180^\circ

.

Soru 23Soru →

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

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The measure of the largest interior angle of the octagon is

150^\circ

.

The total sum of the interior angles of an octagon is

(8-2) \times 180^\circ = 1080^\circ

. The sum of the five given angles is

5 \times 144^\circ = 720^\circ

, leaving a sum of

1080^\circ - 720^\circ = 360^\circ

for the remaining three angles. Since these three angles are in the ratio

3:4:5

, we set

3x + 4x + 5x = 360^\circ

, yielding

12x = 360^\circ

and

x = 30^\circ

. The largest of these three angles is

5 \times 30^\circ = 150^\circ

. Comparing

150^\circ

to the other angles of the octagon (which are

144^\circ

,

90^\circ

, and

120^\circ

), the largest interior angle is

150^\circ

.

Adım Adım Çözüm

1

Calculate the total sum of the interior angles of the octagon.

The total sum is

1080^\circ

.

The sum of the interior angles of any convex polygon with

n

sides is given by

(n - 2) \times 180^\circ

. For an octagon,

n = 8

, so the sum is

(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^\circ

.

Since five angles each measure

144^\circ

, their combined sum is

5 \times 144^\circ = 720^\circ

.

3

Determine the sum of the remaining three interior angles.

The sum of the remaining angles is

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^\circ - 720^\circ = 360^\circ

.

4

Use the ratio

3:4:5

to find the measures of the remaining three angles.

The measures of the three angles are

90^\circ

,

120^\circ

, and

150^\circ

.

Let the measures of the remaining three angles be

3x

,

4x

, and

5x

. Their sum is

3x + 4x + 5x = 12x = 360^\circ

, which gives

x = 30^\circ

. The largest of these three angles is

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^\circ

.

The octagon's interior angles consist of five angles of

144^\circ

, and three angles of

90^\circ

,

120^\circ

, and

150^\circ

. Comparing these values,

150^\circ

is the largest measure.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratios can be used to partition a total sum into specific parts.

Tahmini Süre:2m 0s

Soru 24Soru →

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

80^\circ

100^\circ

120^\circ

140^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $140^\circ$

Cevap

The measure of the largest interior angle of the pentagon is

140^\circ

.

The correct answer is

140^\circ

. The sum of the exterior angles of any convex polygon is

360^\circ

. For a pentagon with exterior angles in the ratio

2:3:4:4:5

, the sum of the ratio parts is

2+3+4+4+5=18

parts. Each part is equal to

360^\circ / 18 = 20^\circ

. The smallest exterior angle has

2

parts, which measures

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^\circ - 40^\circ = 140^\circ

.

Adım Adım Çözüm

1

Determine the sum of the exterior angles of a convex pentagon.

The sum of the exterior angles of any convex polygon is

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

2:3:4:4:5

.

The sum of the ratio parts is

2+3+4+4+5=18

. The smallest exterior angle has

2

parts, so its measure is

\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^\circ

.

The largest interior angle is

180^\circ - 40^\circ = 140^\circ

.

An interior angle and its adjacent exterior angle are supplementary and add up to

180^\circ

.

Anahtar Kavram

The sum of the exterior angles of any convex polygon is

360^\circ

. An interior angle and its adjacent exterior angle are supplementary, meaning they add up to

180^\circ

. Consequently, the largest interior angle corresponds to the smallest exterior angle.

Tahmini Süre:1m 30s

Soru 25Soru →

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

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

The correct answer is

162

degrees. The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the two right angles (

180^\circ

total) leaves a remaining sum of

540^\circ

. The ratio of the remaining four angles is

4:5:5:6

, which can be represented as

4x, 5x, 5x, 6x

, summing to

20x = 540

. Solving for the multiplier gives

x = 27

. The largest angle is

6(27) = 162^\circ

, which is also greater than the two

90^\circ

angles.

The sum of the interior angles of a hexagon is

(6-2) \times 180^\circ = 720^\circ

. Subtracting the two right angles (

180^\circ

) gives a remaining sum of

540^\circ

for the other four angles. Let these four angles be

4x, 5x, 5x,

and

6x

. Their sum is

20x = 540

, which solves to

x = 27

. The largest angle is

6x = 6(27) = 162^\circ

, which is also larger than the two

90^\circ

angles.

Adım Adım Çözüm

1

Calculate the sum of all interior angles of a convex hexagon.

The sum is

(6-2) \times 180^\circ = 720^\circ

.

The sum of the interior angles of any convex

n

-gon is given by

(n-2) \times 180^\circ

.

2

Subtract the sum of the two right angles from the total sum.

The remaining sum is

720^\circ - 180^\circ = 540^\circ

.

Two right angles contribute

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 = 540

, which simplifies to

20x = 540

, yielding

x = 27

.

The angles are proportional to the parts of the ratio, and their sum must equal the remaining

540^\circ

.

4

Calculate the largest angle from the ratio and compare with the right angles.

The largest angle is

6 \times 27 = 162^\circ

.

The largest term in the ratio is

6

, and the resulting angle

162^\circ

is larger than both

90^\circ

and the other calculated angles (

108^\circ

and

135^\circ

).

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratio relationships can be solved using algebraic multipliers.

Soru 26Soru →

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

Cevabı ve açıklamayı göster

Cevap: 140

Cevap

The measure of each of the remaining six interior angles is

140^\circ

.

The total sum of the interior angles of a 10-sided convex polygon is

(10-2) \times 180^\circ = 1,440^\circ

. Subtracting the sum of the four angles that each measure

150^\circ

(

4 \times 150^\circ = 600^\circ

) leaves

840^\circ

for the remaining six angles. Since these remaining six angles are congruent, each measures

840^\circ \div 6 = 140^\circ

.

Adım Adım Çözüm

1

Calculate the total sum of the interior angles of a convex decagon.

1,440^\circ

The interior angle sum of a polygon with

n

sides is

(n-2) \times 180^\circ

, and a decagon has 10 sides.

2

Find the sum of the four angles measuring

150^\circ

.

600^\circ

Multiply the number of angles by their given degree measure.

3

Determine the sum of the remaining six congruent angles.

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^\circ

Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

.

Soru 27Soru →

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

17.5^\circ

27.5^\circ

50.0^\circ

82.5^\circ

97.5^\circ

Cevabı ve açıklamayı göster

Cevap: $82.5^\circ$

Cevap

82.5^\circ

The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the three known angles (

330^\circ

) leaves

390^\circ

for the remaining three angles. Using the ratio

3:4:5

, the smallest of these remaining angles is calculated as

\frac{3}{12} \times 390^\circ = 97.5^\circ

. Since

97.5^\circ

is smaller than the other angles (

110^\circ

,

130^\circ

,

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^\circ - 97.5^\circ = 82.5^\circ

.

Adım Adım Çözüm

1

Calculate the sum of all interior angles of the hexagon.

The sum is

(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^\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^\circ

gives

12y = 390^\circ

, or

y = 32.5^\circ

. The individual angles are

3 \times 32.5^\circ = 97.5^\circ

,

4 \times 32.5^\circ = 130^\circ

, and

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^\circ

. The corresponding exterior angle is

180^\circ - 97.5^\circ = 82.5^\circ

.

An interior angle and its adjacent exterior angle sum to

180^\circ

.

Anahtar Kavram

Polygon Angles and Properties

Soru 28Soru →

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

84^\circ

120^\circ

144^\circ

162^\circ

180^\circ

Cevabı ve açıklamayı göster

Cevap: $144^\circ$

Cevap

The measure of the largest interior angle of the pentagon is

144^\circ

.

The correct answer is the option stating

144^\circ

. To find this, we calculate the sum of all interior angles of a pentagon using

(5-2) \times 180^\circ = 540^\circ

. Subtracting the known angle of

108^\circ

leaves

432^\circ

for the remaining four angles. The ratio

3:4:5:6

implies these angles can be represented as

3x

,

4x

,

5x

, and

6x

, summing to

18x

. Solving

18x = 432

gives

x = 24

. The largest angle is

6x = 6 \times 24^\circ = 144^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a pentagon.

The sum of the interior angles of any convex pentagon (

n = 5

) is

(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^\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

3x

,

4x

,

5x

, and

6x

. Their sum is

3x + 4x + 5x + 6x = 18x

. Setting this equal to the remaining sum gives

18x = 432

, which simplifies to

x = 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

6x

, which is

6 \times 24^\circ = 144^\circ

. Since

144^\circ

is also larger than the given angle of

108^\circ

, it is the largest interior angle of the pentagon.

We multiply the value of

x

by the largest coefficient in the ratio and compare it to the other given angle to identify the maximum measure.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(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.

Tahmini Süre:1m 30s

Soru 29Soru →

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

Cevabı ve açıklamayı göster

Cevap: 78

Cevap

The degree measure of the smallest interior angle is 78.

The sum of the interior angles of a hexagon is calculated as

(6 - 2) \times 180^\circ = 720^\circ

. Adding the algebraic expressions for the six angles yields

15x = 720

, which gives

x = 48

. Substituting

x = 48

back into the expressions gives the angle measures of

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest angle is

78^\circ

.

Adım Adım Çözüm

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

n

sides is given by

(n - 2) \times 180^\circ

. For a hexagon (

n = 6

), the sum is

(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) = 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 = 720

Grouping the

x

terms (

2x + 3x + 2x + 3x + 4x + x = 15x

) and the constant terms (

-15 + 25 + 10 - 50 + 30 = 0

) simplifies the expression.

4

Solve for

x

.

x = 48

Dividing both sides of the equation

15x = 720

by 15 isolates the variable

x

.

5

Substitute the value of

x

back into the angle expressions to identify the smallest angle.

The angles are

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest measure is

78^\circ

.

Evaluating each expression at

x = 48

determines the actual angle measures, from which the smallest can be chosen. Evaluating

(x + 30)^\circ

gives

48 + 30 = 78^\circ

, which is the minimum value.

Anahtar Kavram

The sum of the interior angles of a convex

n

-sided polygon is

(n - 2) \times 180^\circ

. Setting up and solving linear algebraic equations is required to determine unknown angle measures.

Tahmini Süre:1m 30s

Soru 30Soru →

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

80^\circ

120^\circ

140^\circ

160^\circ

300^\circ

Cevabı ve açıklamayı göster

Cevap: $160^\circ$

Cevap

160^\circ

The total sum of the interior angles of a convex heptagon is

(7-2) \times 180^\circ = 900^\circ

. Subtracting the three angles that each measure

140^\circ

(

3 \times 140^\circ = 420^\circ

) leaves

480^\circ

for the remaining four angles. Using the ratio

2:3:3:4

, we represent the angles as

2x

,

3x

,

3x

, and

4x

, giving the equation

2x + 3x + 3x + 4x = 480^\circ

, which simplifies to

12x = 480^\circ

and yields

x = 40^\circ

. The largest of these four angles is

4x = 4(40^\circ) = 160^\circ

. Since

160^\circ

is larger than

140^\circ

, the largest interior angle of the heptagon is

160^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a heptagon.

900^\circ

The sum of the interior angles of a convex polygon with

n

sides is given by

(n-2) \times 180^\circ

. For a heptagon (

n=7

), the sum is

(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ

.

2

Find the sum of the three known interior angles.

420^\circ

Since three interior angles each measure

140^\circ

, their sum is

3 \times 140^\circ = 420^\circ

.

3

Calculate the sum of the remaining four interior angles.

480^\circ

Subtracting the sum of the three known angles from the total interior sum gives

900^\circ - 420^\circ = 480^\circ

.

4

Set up and solve an equation for the remaining four angles using the given ratio.

x = 40^\circ

Let the measures of the remaining four angles be

2x

,

3x

,

3x

, and

4x

. Their sum is

2x + 3x + 3x + 4x = 12x

. Setting this equal to the remaining sum gives

12x = 480^\circ

, which simplifies to

x = 40^\circ

.

5

Determine the measures of the remaining angles and identify the largest angle.

160^\circ

The remaining angles measure

2(40^\circ) = 80^\circ

,

3(40^\circ) = 120^\circ

,

3(40^\circ) = 120^\circ

, and

4(40^\circ) = 160^\circ

. The largest of these is

160^\circ

. Comparing this with the three

140^\circ

angles, the largest interior angle of the entire heptagon is

160^\circ

.

Anahtar Kavram

Interior angle sum of polygons and ratio division

Soru 31Soru →

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

Cevabı ve açıklamayı göster

Cevap: 130

Cevap

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^\circ

. The multiple of

180^\circ

immediately greater than

2030^\circ

is

2160^\circ

(which is

12 \times 180^\circ

). The difference between the total sum and the sum of the

n-1

angles is

2160^\circ - 2030^\circ = 130^\circ

. Since

130^\circ

is less than

180^\circ

, this is a valid interior angle for a convex polygon.

Adım Adım Çözüm

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

n

sides is

(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^\circ

and less than

180^\circ

, the total sum of all

n

angles must satisfy

2030^\circ < (n-2) \times 180^\circ < 2030^\circ + 180^\circ

, which simplifies to

2030^\circ < (n-2) \times 180^\circ < 2210^\circ

.

By definition, every interior angle of a convex polygon is strictly less than

180^\circ

.

3

Solve the inequality to find the integer value of

n-2

.

Dividing the entire inequality by

180^\circ

gives

11.28 < n-2 < 12.28

. Since the number of sides

n

must be an integer,

n-2

must be the integer

12

.

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 \times 180^\circ = 2160^\circ

. Subtracting the sum of the other

n-1

angles gives the remaining angle:

2160^\circ - 2030^\circ = 130^\circ

.

The difference between the total sum of all interior angles and the sum of the

n-1

angles is the measure of the final angle.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, where each interior angle is strictly less than

180^\circ

.

Soru 32Soru →

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

7

8

10

12

14

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The correct answer is 12 sides.

The sum of the interior angles of any convex polygon with

n

sides is given by

(n-2) \times 180^\circ

. The sum of the exterior angles of any convex polygon is always

360^\circ

. Since the ratio of the sum of the interior angles to the sum of the exterior angles is

5:1

, we can set up the equation:

\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}

. Simplifying the fraction on the left gives

\frac{n-2}{2} = 5

. Multiplying both sides by 2 gives

n - 2 = 10

, and adding 2 to both sides gives

n = 12

. Therefore, the polygon has 12 sides.

Adım Adım Çözüm

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) \times 180^\circ

, where

n

is the number of sides. The sum of the exterior angles is always

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.

\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

5:1

.

3

Simplify the equation and solve for the number of sides,

n

.

\frac{n-2}{2} = 5 \implies n-2 = 10 \implies n = 12

Simplifying

\frac{180}{360}

to

\frac{1}{2}

makes it easier to solve the algebraic equation for

n

.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, while the sum of the exterior angles is always

360^\circ

.

Tahmini Süre:1m 30s

Soru 33Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 142

Cevap

The correct answer is 142.

The sum of the interior angles of a convex octagon is

(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,

and

x+14

, their sum is

8x + 56

. Setting this equal to

1080^\circ

and solving for

x

yields

x = 128

. The largest angle is

x + 14

, which equals

128 + 14 = 142^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a convex octagon.

The sum of the interior angles is

1080^\circ

.

The formula for the sum of the interior angles of an

n

-sided polygon is

(n-2) \times 180^\circ

. For an octagon (

n=8

), the sum is

(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 = 1080

.

Letting the smallest angle measure be

x

, the eight consecutive even integer angle measures are

x, x+2, x+4, x+6, x+8, x+10, x+12,

and

x+14

. Their sum is

8x + 56

, which must equal the total sum of the interior angles (

1080^\circ

).

3

Solve the equation for the smallest angle measure,

x

.

x = 128

Subtracting 56 from both sides of the equation yields

8x = 1024

. Dividing both sides by 8 gives

x = 128

.

4

Calculate the measure of the largest interior angle.

The measure of the largest angle is

142^\circ

.

The largest angle is represented by the expression

x + 14

. Substituting

128

for

x

gives

128 + 14 = 142

.

Anahtar Kavram

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

Soru 34Soru →

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

72

84

126

144

168

Cevabı ve açıklamayı göster

Cevap: $168$

Cevap

168

The sum of the interior angles of any convex

n

-gon is given by

(n-2) \times 180^\circ

. For a hexagon (

n=6

), this sum is

(6-2) \times 180^\circ = 720^\circ

. The angles are in the ratio

3:4:5:5:6:7

, which sum to

3+4+5+5+6+7 = 30

parts. Each part corresponds to

720^\circ \div 30 = 24^\circ

. The largest angle is represented by the largest part of the ratio, 7, which equals

7 \times 24^\circ = 168^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a convex hexagon.

720^\circ

The sum of the interior angles of a polygon with

n

sides is

(n-2) \times 180^\circ

. For a hexagon (

n=6

), the sum is

(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 = 30

.

3

Find the value of one part in degrees.

24^\circ

per part

Divide the total interior angle sum by the total number of ratio parts:

720^\circ \div 30 = 24^\circ

.

4

Calculate the measure of the largest interior angle.

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 \times 24^\circ = 168^\circ

.

Anahtar Kavram

Calculating the interior angle measures of an irregular convex polygon using the polygon interior angle sum formula and a given ratio.

Tahmini Süre:1m 30s

Soru 35Soru →

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

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The number of sides of the polygon is 9.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. Since the angles form an arithmetic progression with the first term

a = 120^\circ

and common difference

d = 5^\circ

, their sum is also given by the arithmetic series formula:

S_n = \frac{n}{2}[2(120) + (n-1)5] = \frac{n}{2}(5n + 235)

. Setting the two sums equal yields

\frac{n}{2}(5n + 235) = 180(n - 2)

, which simplifies to the quadratic equation

n^2 - 25n + 144 = 0

. Solving this gives

n = 9

or

n = 16

. Because the polygon is convex, every interior angle must be less than

180^\circ

. If

n = 16

, the largest angle would be

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible for a convex polygon. If

n = 9

, the largest angle is

120^\circ + 8(5^\circ) = 160^\circ

, which is valid. Therefore, the number of sides must be 9.

Adım Adım Çözüm

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

n

-gon is

(n - 2) \times 180^\circ

. The sum of the arithmetic sequence of angles is

S_n = \frac{n}{2}[2(120^\circ) + (n - 1)5^\circ]

. Setting them equal gives:

\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

n

.

5n^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

, so

n = 9

or

n = 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^\circ

. For

n = 16

, the largest angle is

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible. For

n = 9

, the largest angle is

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^\circ

, which eliminates the extraneous solution of 16.

Anahtar Kavram

Sum of interior angles of a convex polygon and arithmetic progressions

Polygon Angles and Properties

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