Examkin

Quadrilaterals and Polygons

14 soru

Tüm türler14Çoktan seçmeli8Sayısal2Eşleştirme4
Soru 1Soru →

What is the measure of one interior angle of a regular octagon?

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Cevap: 135∘135^\circ

Cevap

The measure of one interior angle of a regular octagon is 135∘135^\circ.
To find a single interior angle of a regular polygon, use the formula (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}. Substituting n=8n = 8 for an octagon gives 6×180∘8=1080∘8=135∘\frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ.

Adım Adım Çözüm

1
Identify the number of sides (nn) for an octagon.
n=8n = 8
An octagon is a polygon with 8 sides.
2
Calculate the total sum of the interior angles using the formula (n−2)×180∘(n - 2) \times 180^\circ.
(8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
This formula determines the combined measure of all interior angles for any polygon.
3
Divide the total sum by the number of sides to find the measure of a single angle.
1080∘÷8=135∘1080^\circ \div 8 = 135^\circ
In a regular polygon, all interior angles are equal, so dividing the total sum by the number of angles gives the measure of one.

Anahtar Kavram

Interior Angles of Regular Polygons
Soru 2Soru →

The interior angles of a quadrilateral are represented by the algebraic expressions 2x∘2x^\circ, (3x−15)∘(3x - 15)^\circ, (x+25)∘(x + 25)^\circ, and (2x+30)∘(2x + 30)^\circ. What is the measure of the largest angle in this quadrilateral?

Cevabı ve açıklamayı göster

Cevap: 110∘110^\circ

Cevap

The largest angle measures 110∘110^\circ.
The interior angles of a quadrilateral must sum to 360∘360^\circ. By adding the four expressions together, we get 8x+40=3608x + 40 = 360. Solving for xx yields x=40x = 40. To find the largest angle, we substitute 4040 back into the original expressions: 2(40)=80∘2(40) = 80^\circ, 3(40)−15=105∘3(40) - 15 = 105^\circ, 40+25=65∘40 + 25 = 65^\circ, and 2(40)+30=110∘2(40) + 30 = 110^\circ. Comparing these four values, 110∘110^\circ is the largest.

Adım Adım Çözüm

1
Set up an equation based on the sum of interior angles.
2x+(3x−15)+(x+25)+(2x+30)=3602x + (3x - 15) + (x + 25) + (2x + 30) = 360
The sum of the interior angles of any quadrilateral is always 360∘360^\circ.
2
Combine the like terms on the left side of the equation.
8x+40=3608x + 40 = 360
Grouping the xx terms (2x+3x+x+2x=8x2x + 3x + x + 2x = 8x) and constant terms (−15+25+30=40-15 + 25 + 30 = 40) simplifies the algebraic expression.
3
Solve for the variable xx.
8x=3208x = 320, which means x=40x = 40.
Subtracting 40 from both sides and then dividing by 8 isolates xx.
4
Substitute x=40x = 40 into each of the original expressions to determine the measure of each angle.
The angles are 2(40)=80∘2(40) = 80^\circ; 3(40)−15=105∘3(40) - 15 = 105^\circ; 40+25=65∘40 + 25 = 65^\circ; and 2(40)+30=110∘2(40) + 30 = 110^\circ.
The value of xx alone does not answer the question; the specific angle measures must be evaluated.
5
Identify the largest value among the calculated angles.
The largest value is 110∘110^\circ.
The question specifically asks for the maximum angle measure in the quadrilateral.

Anahtar Kavram

The sum of the interior angles of a quadrilateral is always 360 degrees. Algebraic substitution is required to convert variable expressions back into geometric measurements.
Tahmini Süre:1m 30s
Soru 3Soru →

Match each polygon or quadrilateral with its defining geometric property.

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Öğeler

Parallelogram
Trapezoid
Rhombus
Hexagon

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

Parallelogram matches with 'two pairs of parallel opposite sides', Trapezoid matches with 'exactly one pair of parallel sides', Rhombus matches with 'four sides of equal length', and Hexagon matches with 'exactly six sides'.
Each shape is matched to its fundamental geometric definition based on the number of sides, side lengths, and parallel properties. Knowing these core definitions is essential for classifying polygons and quadrilaterals correctly.

Adım Adım Çözüm

1
Analyze the properties of a parallelogram.
It has two pairs of parallel opposite sides.
This is the fundamental geometric definition of a parallelogram.
2
Analyze the properties of a trapezoid.
It is defined by having exactly one pair of parallel sides.
This distinguishes a trapezoid from other quadrilaterals like parallelograms.
3
Analyze the properties of a rhombus.
It is a quadrilateral where all four sides are of equal length.
While it also has parallel sides, the equal side lengths are its distinguishing feature.
4
Identify the number of sides for a hexagon.
A hexagon is a polygon with exactly six sides.
The prefix 'hexa-' denotes six in geometry.

Anahtar Kavram

Basic Properties of Quadrilaterals and Polygons
Soru 4Soru →

A decorative floor tile is manufactured in the shape of a regular pentagon. What is the degree measure of each interior angle of the tile?

Cevabı ve açıklamayı göster

Cevap: 108∘108^\circ

Cevap

108∘108^\circ
The interior angles of an nn-sided polygon sum to (n−2)×180∘(n - 2) \times 180^\circ. For a 5-sided pentagon, the sum is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Since the tile is a regular pentagon, all five angles are identical. Dividing the total 540∘540^\circ by 5 yields exactly 108∘108^\circ per corner.

Adım Adım Çözüm

1
Identify the number of sides in a regular pentagon.
n=5n = 5
A pentagon is defined as a geometric polygon with exactly 5 sides.
2
Calculate the sum of all interior angles using the polygon formula (n−2)×180∘(n - 2) \times 180^\circ.
(5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
This formula determines the total degree measure for the interior angles of any convex polygon.
3
Divide the total sum by the number of sides to find the measure of a single interior angle.
540∘÷5=108∘540^\circ \div 5 = 108^\circ
Because it is a regular pentagon, all interior angles are equal in measure.

Anahtar Kavram

Interior Angles of Regular Polygons
Soru 5Soru →

Match each quadrilateral with its precise geometric defining properties.

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Öğeler

Rhombus
Rectangle
Kite
Isosceles Trapezoid

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

Rhombus matches the parallelogram with perpendicular diagonals that bisect angles. Rectangle matches the parallelogram with four right angles. Kite matches the quadrilateral with consecutive congruent sides. Isosceles Trapezoid matches the quadrilateral with exactly one pair of parallel sides.
Each quadrilateral has unique defining traits involving their sides, angles, and diagonals. A rhombus requires perpendicular diagonals and parallel opposite sides. A rectangle requires right angles and congruent diagonals. A kite requires adjacent equal sides and perpendicular diagonals. An isosceles trapezoid requires exactly one pair of parallel sides and congruent diagonals.

Adım Adım Çözüm

1
Analyze the unique properties of a Rhombus.
It is a parallelogram with all equal sides, which leads to perpendicular diagonals that bisect the opposite angles.
To distinguish it from other standard parallelograms like rectangles.
2
Analyze the unique properties of a Rectangle.
It has four right angles and, unlike a standard parallelogram, has diagonals that are identical in length.
To correctly identify the item describing congruent diagonals in a parallelogram.
3
Analyze the unique properties of a Kite.
It has adjacent sides that are equal (consecutive congruent sides), which forces the diagonals to meet at 90 degrees.
To match it with the correct non-parallelogram definition.
4
Analyze the unique properties of an Isosceles Trapezoid.
It has exactly one pair of parallel sides (bases) and the symmetry from equal non-parallel sides creates equal diagonals.
To differentiate it from general trapezoids and generic parallelograms.

Anahtar Kavram

Quadrilaterals and Polygons
Soru 6Soru →

A land surveyor is measuring a four-sided plot of land. They record three of the interior angles as 78∘78^\circ, 104∘104^\circ, and 83∘83^\circ. What is the measure of the fourth interior angle, in degrees?

Cevabı ve açıklamayı göster

Cevap: 95

Cevap

The fourth interior angle measures 95∘95^\circ.
The correct answer is found by subtracting the sum of the three known angles (265∘265^\circ) from the total interior angle sum of a quadrilateral (360∘360^\circ), which yields 95∘95^\circ.

Adım Adım Çözüm

1
Identify the total sum of interior angles for a quadrilateral.
360∘360^\circ
A quadrilateral has 4 sides, and the sum of interior angles is calculated as (n−2)×180∘(n - 2) \times 180^\circ. For n=4n = 4, this is 2×180∘=360∘2 \times 180^\circ = 360^\circ.
2
Calculate the sum of the three given angles.
78∘+104∘+83∘=265∘78^\circ + 104^\circ + 83^\circ = 265^\circ
To find the missing part, we must first determine the combined total of the known angles.
3
Subtract the sum of the given angles from the total quadrilateral angle sum.
360∘−265∘=95∘360^\circ - 265^\circ = 95^\circ
The remaining amount represents the exact measure of the fourth interior angle.

Anahtar Kavram

The sum of the interior angles of any convex quadrilateral is always exactly 360∘360^\circ.
Soru 7Soru →

In a parallelogram, the measure of one interior angle is 30∘30^\circ less than twice the measure of the consecutive interior angle. What is the measure of the larger angle?

Cevabı ve açıklamayı göster

Cevap: 110∘110^\circ

Cevap

The measure of the larger angle is 110∘110^\circ.
The correct answer correctly identifies that consecutive angles in a parallelogram are supplementary. Setting up the equation x+(2x−30)=180∘x + (2x - 30) = 180^\circ and solving for xx yields 70∘70^\circ. Substituting this back into the expression for the larger angle gives 2(70)−30=110∘2(70) - 30 = 110^\circ.

Adım Adım Çözüm

1
Identify the geometric property relating the angles.
Consecutive interior angles of a parallelogram are supplementary, meaning they add up to 180∘180^\circ.
This is a fundamental property of parallelograms formed by parallel lines cut by a transversal.
2
Translate the word problem into an algebraic equation.
Let the smaller angle be xx. The larger angle is 2x−302x - 30. Therefore, x+(2x−30)=180x + (2x - 30) = 180.
This captures both the relationship described ('30∘30^\circ less than twice') and the supplementary geometric property.
3
Solve the algebraic equation for xx.
Combine terms: 3x−30=1803x - 30 = 180. Add 3030 to both sides: 3x=2103x = 210. Divide by 33: x=70∘x = 70^\circ.
Isolating the variable determines the exact measure of the smaller angle.
4
Calculate the measure of the larger angle.
2(70)−30=140−30=110∘2(70) - 30 = 140 - 30 = 110^\circ.
The question specifically asks for the measure of the larger angle, not the smaller one.

Anahtar Kavram

Properties of Parallelograms and Algebraic Translation

Alternatif Yöntem

You can solve this by testing the answer choices. If the larger angle is 110∘110^\circ, the smaller consecutive angle must be 180∘−110∘=70∘180^\circ - 110^\circ = 70^\circ. Check if 110∘110^\circ is 30∘30^\circ less than twice 70∘70^\circ: 2(70)−30=140−30=110∘2(70) - 30 = 140 - 30 = 110^\circ. The values perfectly match the problem statement.
Tahmini Süre:1m 15s
Soru 8Soru →

The sum of the interior angles of a convex polygon is exactly three times the sum of its exterior angles. How many sides does this polygon have?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8
Applying the core polygon properties, the interior angle sum is 180∘(n−2)180^\circ(n - 2) and the exterior angle sum is 360∘360^\circ. Setting 180∘(n−2)=3(360∘)180^\circ(n - 2) = 3(360^\circ) and solving for nn yields 8.

Adım Adım Çözüm

1
Identify the interior angle sum formula.
Sum = 180∘(n−2)180^\circ(n - 2)
This formula provides the total sum of all interior angles for an nn-sided polygon.
2
Identify the exterior angle sum property.
Sum = 360∘360^\circ
A fundamental property of all convex polygons is that the sum of their exterior angles is exactly 360∘360^\circ.
3
Set up the algebraic equation.
180∘(n−2)=3×360∘180^\circ(n - 2) = 3 \times 360^\circ
The question states the interior sum is exactly three times the exterior sum.
4
Solve the equation for nn.
n=8n = 8
Dividing 1080∘1080^\circ by 180∘180^\circ yields 6, and adding 2 gives a final side count of 8.

Anahtar Kavram

Polygon Angle Sum Theorems
Soru 9Soru →

A convex hexagon has five interior angles that measure 120∘120^\circ, 135∘135^\circ, 115∘115^\circ, 140∘140^\circ, and 110∘110^\circ. If the measure of the sixth interior angle is represented by the algebraic expression (4x+16)∘(4x + 16)^\circ, what is the value of xx?

Cevabı ve açıklamayı göster

Cevap: 2121

Cevap

The correct answer is 2121.
The sum of the interior angles of a hexagon (66 sides) is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ. The five given angles add up to 620∘620^\circ. This means the sixth angle must be 720∘−620∘=100∘720^\circ - 620^\circ = 100^\circ. Setting the algebraic expression 4x+164x + 16 equal to 100100 and solving for xx by subtracting 1616 and dividing by 44 gives x=21x = 21.

Adım Adım Çözüm

1
Calculate the sum of the interior angles of a convex hexagon.
The sum is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ.
The formula for the sum of interior angles is (n−2)×180∘(n - 2) \times 180^\circ, where nn is the number of sides. A hexagon has 66 sides.
2
Find the sum of the five given angles.
120∘+135∘+115∘+140∘+110∘=620∘120^\circ + 135^\circ + 115^\circ + 140^\circ + 110^\circ = 620^\circ.
Adding the known angles allows us to find the combined measure of the known portion of the polygon.
3
Calculate the measure of the sixth angle.
720∘−620∘=100∘720^\circ - 620^\circ = 100^\circ.
Subtracting the sum of the five known angles from the total sum gives the measure of the missing angle.
4
Set up and solve the equation for xx.
4x+16=100  ⟹  4x=84  ⟹  x=214x + 16 = 100 \implies 4x = 84 \implies x = 21.
The sixth angle is represented by the expression (4x+16)∘(4x + 16)^\circ, so setting it equal to 100100 allows us to isolate and solve for xx.

Anahtar Kavram

Polygon Interior Angles and Linear Equations
Soru 10Soru →

Match each polygon description based on its angle properties to the correct geometric shape.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

A polygon where the sum of the interior angles is 540∘540^\circ.
A regular polygon where each individual interior angle measures 120∘120^\circ.
A polygon in which the sum of the interior angles equals 1080∘1080^\circ.
A polygon whose interior angle sum exactly equals its exterior angle sum.

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

Match the 540∘540^\circ sum to Pentagon, the 120∘120^\circ interior angle to Regular Hexagon, the 1080∘1080^\circ sum to Octagon, and the equal interior/exterior sums to Quadrilateral.
Each match correctly applies the sum of interior angles formula, S=(n−2)×180∘S = (n-2) \times 180^\circ, or relies on the fact that the sum of exterior angles for any convex polygon is exactly 360∘360^\circ. Solving for nn reveals the specific polygon.

Adım Adım Çözüm

1
Find the number of sides for a polygon with a 540∘540^\circ interior angle sum.
n = 5
We apply the formula (n−2)×180=540(n-2) \times 180 = 540, which simplifies to n−2=3n-2 = 3.
2
Determine the number of sides for a regular polygon with a 120∘120^\circ interior angle.
n = 6
The exterior angle is 180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ. The number of sides is 360∘/60∘=6360^\circ / 60^\circ = 6.
3
Find the number of sides for a polygon with a 1080∘1080^\circ interior angle sum.
n = 8
Applying the formula gives (n−2)×180=1080(n-2) \times 180 = 1080, which simplifies to n−2=6n-2 = 6.
4
Identify the polygon where the interior angle sum equals the exterior angle sum.
n = 4
Since the exterior sum is always 360∘360^\circ, we set (n−2)×180=360(n-2) \times 180 = 360, yielding n−2=2n-2 = 2.

Anahtar Kavram

Polygon Interior and Exterior Angle Formulas
Tahmini Süre:1m 30s
Soru 11Soru →

An architect is designing a room in the shape of a convex pentagon. The measures of the interior angles of the room are defined algebraically as (x+20)∘(x + 20)^\circ, (x+10)∘(x + 10)^\circ, (x+5)∘(x + 5)^\circ, (x−5)∘(x - 5)^\circ, and (2x)∘(2x)^\circ. What is the measure of the smallest interior angle of this room?

Cevabı ve açıklamayı göster

Cevap: 80∘80^\circ

Cevap

The smallest interior angle measures 80∘80^\circ.
A pentagon has 5 sides, so the sum of its interior angles is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Setting the sum of the algebraic expressions equal to 540∘540^\circ yields 6x+30=5406x + 30 = 540. Subtracting 3030 leaves 6x=5106x = 510, so x=85x = 85. Substituting 8585 into the smallest expression, (x−5)∘(x - 5)^\circ, gives 85−5=80∘85 - 5 = 80^\circ.

Adım Adım Çözüm

1
Determine the sum of the interior angles for a pentagon.
Using the formula (n−2)×180∘(n - 2) \times 180^\circ for a 5-sided polygon (n=5n = 5), the sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Before solving for xx, the total sum that the algebraic expressions add up to must be known.
2
Set up an equation by adding all the given angle expressions.
(x+20)+(x+10)+(x+5)+(x−5)+2x=540(x + 20) + (x + 10) + (x + 5) + (x - 5) + 2x = 540
The sum of the individual interior angles must equal the total calculated sum of the polygon's angles.
3
Combine like terms and solve for xx.
6x+30=540⇒6x=510⇒x=856x + 30 = 540 \Rightarrow 6x = 510 \Rightarrow x = 85
Simplifying the algebraic equation isolates the variable xx.
4
Substitute x=85x = 85 back into each expression to find all angle measures and identify the smallest one.
The angles are 105∘105^\circ, 95∘95^\circ, 90∘90^\circ, 80∘80^\circ, and 170∘170^\circ. The smallest of these is 80∘80^\circ.
The variable xx alone is not the answer; it must be substituted back into the expressions to find the actual angle measures.

Anahtar Kavram

The sum of the interior angles of a convex polygon is (n−2)×180∘(n-2) \times 180^\circ. This geometric principle can be combined with algebraic equations to solve for unknown angle measures.
Soru 12Soru →

A convex pentagon has three interior angles that are equal in measure. If the remaining two interior angles are supplementary to each other, what is the measure of one of the three equal angles?

Cevabı ve açıklamayı göster

Cevap: 120∘120^\circ

Cevap

The measure of one of the three equal angles is 120∘120^\circ.
A pentagon has 5 sides, so the total sum of its interior angles is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. The problem states that three angles are equal (we can call each xx) and the other two are supplementary. Supplementary angles add up to 180∘180^\circ. Setting up the algebraic equation gives 3x+180∘=540∘3x + 180^\circ = 540^\circ. Subtracting 180∘180^\circ from both sides yields 3x=360∘3x = 360^\circ. Dividing by 3, we find that x=120∘x = 120^\circ.

Adım Adım Çözüm

1
Calculate the total sum of the interior angles of a pentagon.
The sum is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ.
The formula for the sum of interior angles of any convex polygon is (n−2)×180∘(n - 2) \times 180^\circ, where nn is the number of sides.
2
Set up an algebraic equation representing the sum of the five interior angles.
3x+180∘=540∘3x + 180^\circ = 540^\circ
Three angles are equal (let them be xx), and the remaining two angles are supplementary, meaning their combined sum is exactly 180∘180^\circ.
3
Solve the equation for xx.
3x=360∘⇒x=120∘3x = 360^\circ \Rightarrow x = 120^\circ
Subtracting 180∘180^\circ from both sides isolates the 3x3x term. Dividing both sides by 3 gives the value of a single equal angle.

Anahtar Kavram

The sum of interior angles of a polygon and the geometric property of supplementary angles.
Soru 13Soru →

In a convex pentagon, two of the interior angles are equal in measure. The other three interior angles are also equal to each other, and each of these three is 40∘40^\circ larger than the measure of one of the first two angles. What is the measure of the largest exterior angle of this pentagon?

Cevabı ve açıklamayı göster

Cevap: 96∘96^\circ

Cevap

The largest exterior angle is 96∘96^\circ.
The correct answer is 96∘96^\circ because setting up the equation for a pentagon's sum of interior angles, 2x+3(x+40)=5402x + 3(x + 40) = 540, gives x=84∘x = 84^\circ as the smallest interior angle. The exterior angle is supplementary to the interior angle, so 180∘−84∘=96∘180^\circ - 84^\circ = 96^\circ.

Adım Adım Çözüm

1
Determine the sum of the interior angles of a pentagon.
Using the formula (n−2)×180∘(n - 2) \times 180^\circ, for n=5n = 5, the sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
You must know the total sum of the interior angles before you can solve for any individual unknown angle.
2
Set up an algebraic equation to represent the interior angles.
Let the two smaller equal angles be xx. The three larger equal angles are (x+40∘)(x + 40^\circ). The equation is 2x+3(x+40∘)=540∘2x + 3(x + 40^\circ) = 540^\circ.
Translating the word problem into algebra allows us to solve for the specific angle measures.
3
Solve the equation for xx.
Distribute the 3 to get 2x+3x+120∘=540∘2x + 3x + 120^\circ = 540^\circ. Combine like terms: 5x+120∘=540∘5x + 120^\circ = 540^\circ. Subtract 120 from both sides: 5x=420∘5x = 420^\circ. Divide by 5: x=84∘x = 84^\circ.
Solving for xx provides the measure of the smallest interior angles.
4
Determine the largest exterior angle.
An exterior angle is supplementary to its adjacent interior angle. The largest exterior angle corresponds to the smallest interior angle. The largest exterior angle is 180∘−84∘=96∘180^\circ - 84^\circ = 96^\circ.
Because exterior and interior angles on a straight line add up to 180 degrees, the smallest interior angle creates the largest exterior angle.

Anahtar Kavram

Quadrilaterals and Polygons
Soru 14Soru →

Match each geometric description to the correct polygon or quadrilateral it uniquely describes.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

A regular polygon in which the sum of the interior angles is exactly five times the sum of its exterior angles.
A convex polygon in which the total number of diagonals is exactly 2.52.5 times the number of its vertices.
A quadrilateral whose diagonals are equal in length and bisect each other, but are not mutually perpendicular.
A regular polygon in which the measure of each interior angle is exactly 100∘100^\circ greater than its adjacent exterior angle.

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The geometric descriptions correspond perfectly to the Dodecagon, Octagon, Rectangle, and Nonagon based on their algebraic angle properties and specific diagonal traits.
Each provided description translates to a specific algebraic or geometric property that uniquely identifies a single polygon. Setting up and solving the respective equations for nn (number of sides) or applying logical diagnostic properties of quadrilaterals produces the correct matching pairs.

Adım Adım Çözüm

1
Analyze the polygon where the interior angle sum is five times the exterior angle sum.
180(n−2)=5(360)⇒180(n−2)=1800⇒n−2=10⇒n=12180(n-2) = 5(360) \Rightarrow 180(n-2) = 1800 \Rightarrow n-2 = 10 \Rightarrow n=12.
Applying the universal constant for exterior angle sums (360∘360^\circ) allows solving algebraically for the 1212-sided dodecagon.
2
Solve for the polygon where the number of diagonals equals 2.52.5 times the number of vertices (nn).
n(n−3)2=2.5n⇒n−3=5⇒n=8\frac{n(n-3)}{2} = 2.5n \Rightarrow n-3 = 5 \Rightarrow n=8.
Equating the formula for the total number of diagonals to 2.5n2.5n allows us to simplify and identify the 88-sided octagon.
3
Deduce the quadrilateral with congruent, bisecting, non-perpendicular diagonals.
The shape is uniquely identified as a rectangle (excluding a square).
Bisecting diagonals prove it is a parallelogram, congruent diagonals prove it is a rectangle, and non-perpendicularity proves it is not a square.
4
Calculate the sides of the polygon where the interior angle is 100∘100^\circ larger than the exterior angle.
I+E=180∘I + E = 180^\circ and I−E=100∘I - E = 100^\circ. Solving the system gives E=40∘E = 40^\circ. Since n=360En = \frac{360}{E}, n=36040=9n = \frac{360}{40} = 9.
Recognizing that interior and exterior angles form a linear pair lets us set up a system of equations. Finding the exterior angle easily leads to the 99-sided nonagon.

Anahtar Kavram

Advanced Polygon Analysis and Quadrilateral Properties
Quadrilaterals and Polygons Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) | Examkin