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Cool Angle Rules: Vertically Opposite Angles and Circle Theorems

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Harsimran

15/04/2023

Biology

geography, biology and maths

Cool Angle Rules: Vertically Opposite Angles and Circle Theorems

Geometry helps us understand shapes and angles through important rules and properties that make solving problems easier.

Vertically opposite angles properties and rules are fundamental concepts where two lines intersect to create four angles. When lines cross, the angles opposite each other are always equal. This means if one angle is 45 degrees, its vertically opposite angle must also be 45 degrees. Understanding this property helps in solving complex geometric problems and proofs.

Circle theorems related to angles and tangents teach us several key principles about circles. A tangent line touches a circle at exactly one point and forms a 90-degree angle with the radius at that point. When two tangents are drawn from a point outside the circle, they are equal in length. Inside a circle, angles at the center are twice the size of angles at the circumference when both angles are subtended by the same arc. The angle in a semicircle is always 90 degrees, and angles in the same segment are equal. These circle properties are essential for understanding advanced geometry and real-world applications like engineering and architecture.

Angle rules and properties in polygons and quadrilaterals provide a framework for analyzing shapes. In any polygon, the sum of interior angles can be calculated using the formula (n-2) × 180°, where n is the number of sides. Regular polygons have equal sides and equal angles. In quadrilaterals, the sum of interior angles is always 360 degrees. Special quadrilaterals like rectangles have all right angles (90 degrees), while parallelograms have opposite angles equal to each other. In a rhombus, diagonals intersect at right angles. These properties help us understand shape relationships and solve geometric problems systematically. Understanding these angle properties is crucial for fields like construction, design, and mathematical reasoning.

...

15/04/2023

97

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Understanding Angle Properties and Rules in Geometry

When studying geometry, understanding vertically opposite angles properties and rules is fundamental. Vertically opposite angles are formed when two straight lines intersect, creating two pairs of equal angles. These angles are always equal in measure, regardless of the lines' orientation. This principle is crucial for solving complex geometric problems and understanding more advanced concepts.

In parallel lines, corresponding angles and alternate angles follow specific patterns. When two parallel lines are cut by a transversal, corresponding angles are equal, and alternate angles maintain the same measure. This relationship creates a foundation for understanding more complex geometric structures and proofs.

The angle properties in polygons follow systematic rules. In any polygon, interior angles sum to n2n-2 × 180°, where n represents the number of sides. For example, in a triangle, the angles sum to 180°, while in a quadrilateral, they total 360°. Exterior angles of a polygon, when summed, always equal 360°, providing a useful tool for geometric calculations.

Definition: Vertically opposite angles are pairs of angles formed when two lines intersect, with each pair being equal in measure.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Circle Theorems and Their Applications

Circle theorems related to angles and tangents provide essential tools for geometric analysis. One fundamental theorem states that the angle at the center is twice the angle at the circumference when both angles intercept the same arc. This relationship is crucial for calculating arc measures and understanding circular geometry.

Another important theorem involves angles in the same segment of a circle, which are always equal. This principle applies to semicircles as well, where angles inscribed in a semicircle are always 90 degrees. These properties make circle theorems particularly useful in architectural design and engineering applications.

The relationship between tangents and radii follows specific rules. When a radius meets a tangent at the point of contact, they form a right angle. Additionally, two tangents drawn from an external point to a circle are equal in length, creating symmetrical properties useful in construction and design.

Highlight: The angle at the center of a circle is always twice the angle at the circumference when both angles intercept the same arc.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Polygon Angles and Their Properties

Angle rules and properties in polygons and quadrilaterals form the backbone of geometric analysis. In regular polygons, all interior angles are equal, and their measure can be calculated using the formula n2n-2 × 180° ÷ n, where n is the number of sides. This relationship helps in understanding both regular and irregular polygons.

Interior angles of quadrilaterals always sum to 360 degrees, while exterior angles of any polygon sum to 360 degrees. These properties remain constant regardless of the polygon's shape or size, making them reliable tools for geometric calculations and proofs.

The relationship between interior and exterior angles in polygons provides insights into shape properties and transformations. For each vertex, the exterior and interior angles are supplementary, meaning they sum to 180 degrees. This relationship helps in solving complex geometric problems and understanding polygon properties.

Example: In a regular pentagon 5sides5 sides, each interior angle measures 108 degrees, calculated using 525-2 × 180° ÷ 5.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Advanced Geometric Applications and Relationships

Understanding geometric relationships extends beyond basic angle properties to practical applications. In cyclic quadrilaterals, opposite angles are supplementary, meaning they sum to 180 degrees. This property is particularly useful in construction and engineering design.

The relationship between tangents and circles creates important geometric properties used in various fields. When two tangents are drawn to a circle from an external point, they form equal angles with the line joining the point to the center. This principle is essential in designing curved structures and understanding circular motion.

These geometric principles find applications in real-world scenarios, from architecture to engineering. Understanding how angles relate to each other in various geometric shapes helps in designing stable structures, creating efficient layouts, and solving spatial problems in practical situations.

Vocabulary: Cyclic quadrilateral - a quadrilateral whose vertices all lie on a circle's circumference.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Understanding DNA and Reproduction

DNA, or deoxyribonucleic acid, is the fundamental building block of life that carries genetic information. This complex molecule determines every inherited characteristic in living organisms through its unique double helix structure and specific genetic coding.

The genome represents the complete set of genetic material within an organism. DNA functions as a polymer made up of two intertwined strands that form the iconic double helix shape. Within this structure, specific sequences of nucleotides form genes that code for particular proteins. These proteins then carry out essential functions in cells, such as producing blood cells or controlling inherited traits.

Sexual reproduction involves the fusion of male and female gametes spermandeggcellssperm and egg cells through a process called meiosis. This combination of genetic material from two parents creates variation in offspring, as genes are mixed and recombined. The resulting children inherit a unique combination of genes from both parents, leading to diversity within species.

Definition: DNA DeoxyribonucleicacidDeoxyribonucleic acid is the chemical compound that contains all genetic information and instructions for building and maintaining an organism.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Rates of Chemical Reactions and Collision Theory

Understanding reaction rates requires examining several key factors that influence how quickly chemicals interact. Temperature, concentration, surface area, and catalysts all play crucial roles in determining reaction speeds.

The collision theory explains how reactions occur at the molecular level. According to this theory, particles must collide with sufficient energy activationenergyactivation energy and proper orientation for a successful reaction. Increasing temperature causes particles to move faster and collide more frequently, thereby speeding up reactions. Similarly, higher concentrations mean more particles in the same volume, resulting in more frequent collisions.

Surface area significantly impacts reaction rates - larger surface areas allow for more particle collisions. This explains why powdered substances react faster than larger chunks. Catalysts provide an alternative reaction pathway with lower activation energy, increasing reaction rates without being consumed in the process.

Example: When adding diluted acid to calcium carbonate, increasing the temperature will make the reaction produce gas more quickly due to faster-moving particles.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

India's Economic Development and Global Position

India's strategic location in Southern Asia has significantly contributed to its rapid economic growth and development. As a member of the World Trade Organization with the world's second-largest population, India has become a major player in global trade and commerce.

The country's industrial structure has evolved dramatically, with the tertiary and quaternary sectors now employing a larger portion of the workforce compared to traditional primary industries. Cities like Hyderabad have emerged as global business hubs, with government-constructed business districts attracting international corporations like Unilever.

India's economic development has brought both positive and negative impacts. While access to clean water has improved significantly and wages have increased, environmental challenges have emerged. The country's growing energy consumption, particularly from fossil fuels, has raised concerns about greenhouse gas emissions and environmental sustainability.

Highlight: India's transformation from an agriculture-based economy to a service-oriented one has created millions of jobs but also presents environmental challenges.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Kenya's National Parks and Tourism Development

Kenya's tourism sector, anchored by its 24 national parks, has become a crucial component of the country's economy. The industry employs approximately 11 million people and contributes significantly to Kenya's GDP, though numbers have fluctuated over time.

The country's appeal lies in its diverse wildlife, unspoiled scenery, and rich cultural heritage, particularly the Maasai communities. To boost tourism, Kenya has implemented various measures, including reducing visa fees by 50% in 2009 and lowering airport landing fees to make the destination more accessible.

However, tourism development has brought both benefits and challenges. While it has increased the Human Development Index HDIHDI and created employment opportunities, issues like environmental damage and cultural displacement have emerged. The Maasai communities, in particular, have faced pressure as tourism development sometimes forces them from their traditional lands.

Highlight: Kenya's tourism industry provides significant economic benefits but must balance development with environmental and cultural preservation.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

View

Understanding the Coordination and Control Systems in Organisms

The coordination and control systems in living organisms are essential for maintaining stable internal conditions and responding to changes. These systems work together through complex networks of communication and regulation.

Definition: Homeostasis is the maintenance of optimal internal conditions within an organism through various regulatory mechanisms and feedback systems.

The nervous system provides rapid, precise responses to stimuli through a network of specialized cells. When a stimulus is detected by receptors, the information travels along sensory neurons to the central nervous system. Here, the information is processed and decisions are made. Motor neurons then carry commands to effectors musclesorglandsmuscles or glands which produce appropriate responses. This entire process happens within milliseconds, allowing for quick reactions to environmental changes.

The endocrine system works alongside the nervous system but operates through chemical messengers called hormones. Key endocrine glands include the pancreas producinginsulinproducing insulin, thyroid producingthyroxineproducing thyroxine, ovaries producingestrogenproducing estrogen, testes producingtestosteroneproducing testosterone, and the pituitary gland. Unlike the rapid nervous system responses, hormonal responses are slower but longer-lasting and affect multiple target tissues throughout the body.

Example: When blood glucose levels rise, the pancreas detects this change and releases insulin. This hormone signals cells throughout the body to take up glucose, thereby reducing blood glucose levels back to normal.

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Biology

97

15 Apr 2023

15 pages

Cool Angle Rules: Vertically Opposite Angles and Circle Theorems

user profile picture

Harsimran

@harsimran_nagra

Geometry helps us understand shapes and angles through important rules and properties that make solving problems easier.

Vertically opposite angles properties and rulesare fundamental concepts where two lines intersect to create four angles. When lines cross, the angles opposite... Show more

vertically opposite
angles
straight
line
equal
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lines
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180⁰
Lacorresponding
angles
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angles in a
triangle
Angl

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Understanding Angle Properties and Rules in Geometry

When studying geometry, understanding vertically opposite angles properties and rules is fundamental. Vertically opposite angles are formed when two straight lines intersect, creating two pairs of equal angles. These angles are always equal in measure, regardless of the lines' orientation. This principle is crucial for solving complex geometric problems and understanding more advanced concepts.

In parallel lines, corresponding angles and alternate angles follow specific patterns. When two parallel lines are cut by a transversal, corresponding angles are equal, and alternate angles maintain the same measure. This relationship creates a foundation for understanding more complex geometric structures and proofs.

The angle properties in polygons follow systematic rules. In any polygon, interior angles sum to n2n-2 × 180°, where n represents the number of sides. For example, in a triangle, the angles sum to 180°, while in a quadrilateral, they total 360°. Exterior angles of a polygon, when summed, always equal 360°, providing a useful tool for geometric calculations.

Definition: Vertically opposite angles are pairs of angles formed when two lines intersect, with each pair being equal in measure.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Circle Theorems and Their Applications

Circle theorems related to angles and tangents provide essential tools for geometric analysis. One fundamental theorem states that the angle at the center is twice the angle at the circumference when both angles intercept the same arc. This relationship is crucial for calculating arc measures and understanding circular geometry.

Another important theorem involves angles in the same segment of a circle, which are always equal. This principle applies to semicircles as well, where angles inscribed in a semicircle are always 90 degrees. These properties make circle theorems particularly useful in architectural design and engineering applications.

The relationship between tangents and radii follows specific rules. When a radius meets a tangent at the point of contact, they form a right angle. Additionally, two tangents drawn from an external point to a circle are equal in length, creating symmetrical properties useful in construction and design.

Highlight: The angle at the center of a circle is always twice the angle at the circumference when both angles intercept the same arc.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Polygon Angles and Their Properties

Angle rules and properties in polygons and quadrilaterals form the backbone of geometric analysis. In regular polygons, all interior angles are equal, and their measure can be calculated using the formula n2n-2 × 180° ÷ n, where n is the number of sides. This relationship helps in understanding both regular and irregular polygons.

Interior angles of quadrilaterals always sum to 360 degrees, while exterior angles of any polygon sum to 360 degrees. These properties remain constant regardless of the polygon's shape or size, making them reliable tools for geometric calculations and proofs.

The relationship between interior and exterior angles in polygons provides insights into shape properties and transformations. For each vertex, the exterior and interior angles are supplementary, meaning they sum to 180 degrees. This relationship helps in solving complex geometric problems and understanding polygon properties.

Example: In a regular pentagon 5sides5 sides, each interior angle measures 108 degrees, calculated using 525-2 × 180° ÷ 5.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Advanced Geometric Applications and Relationships

Understanding geometric relationships extends beyond basic angle properties to practical applications. In cyclic quadrilaterals, opposite angles are supplementary, meaning they sum to 180 degrees. This property is particularly useful in construction and engineering design.

The relationship between tangents and circles creates important geometric properties used in various fields. When two tangents are drawn to a circle from an external point, they form equal angles with the line joining the point to the center. This principle is essential in designing curved structures and understanding circular motion.

These geometric principles find applications in real-world scenarios, from architecture to engineering. Understanding how angles relate to each other in various geometric shapes helps in designing stable structures, creating efficient layouts, and solving spatial problems in practical situations.

Vocabulary: Cyclic quadrilateral - a quadrilateral whose vertices all lie on a circle's circumference.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Understanding DNA and Reproduction

DNA, or deoxyribonucleic acid, is the fundamental building block of life that carries genetic information. This complex molecule determines every inherited characteristic in living organisms through its unique double helix structure and specific genetic coding.

The genome represents the complete set of genetic material within an organism. DNA functions as a polymer made up of two intertwined strands that form the iconic double helix shape. Within this structure, specific sequences of nucleotides form genes that code for particular proteins. These proteins then carry out essential functions in cells, such as producing blood cells or controlling inherited traits.

Sexual reproduction involves the fusion of male and female gametes spermandeggcellssperm and egg cells through a process called meiosis. This combination of genetic material from two parents creates variation in offspring, as genes are mixed and recombined. The resulting children inherit a unique combination of genes from both parents, leading to diversity within species.

Definition: DNA DeoxyribonucleicacidDeoxyribonucleic acid is the chemical compound that contains all genetic information and instructions for building and maintaining an organism.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Rates of Chemical Reactions and Collision Theory

Understanding reaction rates requires examining several key factors that influence how quickly chemicals interact. Temperature, concentration, surface area, and catalysts all play crucial roles in determining reaction speeds.

The collision theory explains how reactions occur at the molecular level. According to this theory, particles must collide with sufficient energy activationenergyactivation energy and proper orientation for a successful reaction. Increasing temperature causes particles to move faster and collide more frequently, thereby speeding up reactions. Similarly, higher concentrations mean more particles in the same volume, resulting in more frequent collisions.

Surface area significantly impacts reaction rates - larger surface areas allow for more particle collisions. This explains why powdered substances react faster than larger chunks. Catalysts provide an alternative reaction pathway with lower activation energy, increasing reaction rates without being consumed in the process.

Example: When adding diluted acid to calcium carbonate, increasing the temperature will make the reaction produce gas more quickly due to faster-moving particles.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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India's Economic Development and Global Position

India's strategic location in Southern Asia has significantly contributed to its rapid economic growth and development. As a member of the World Trade Organization with the world's second-largest population, India has become a major player in global trade and commerce.

The country's industrial structure has evolved dramatically, with the tertiary and quaternary sectors now employing a larger portion of the workforce compared to traditional primary industries. Cities like Hyderabad have emerged as global business hubs, with government-constructed business districts attracting international corporations like Unilever.

India's economic development has brought both positive and negative impacts. While access to clean water has improved significantly and wages have increased, environmental challenges have emerged. The country's growing energy consumption, particularly from fossil fuels, has raised concerns about greenhouse gas emissions and environmental sustainability.

Highlight: India's transformation from an agriculture-based economy to a service-oriented one has created millions of jobs but also presents environmental challenges.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Kenya's National Parks and Tourism Development

Kenya's tourism sector, anchored by its 24 national parks, has become a crucial component of the country's economy. The industry employs approximately 11 million people and contributes significantly to Kenya's GDP, though numbers have fluctuated over time.

The country's appeal lies in its diverse wildlife, unspoiled scenery, and rich cultural heritage, particularly the Maasai communities. To boost tourism, Kenya has implemented various measures, including reducing visa fees by 50% in 2009 and lowering airport landing fees to make the destination more accessible.

However, tourism development has brought both benefits and challenges. While it has increased the Human Development Index HDIHDI and created employment opportunities, issues like environmental damage and cultural displacement have emerged. The Maasai communities, in particular, have faced pressure as tourism development sometimes forces them from their traditional lands.

Highlight: Kenya's tourism industry provides significant economic benefits but must balance development with environmental and cultural preservation.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Understanding the Coordination and Control Systems in Organisms

The coordination and control systems in living organisms are essential for maintaining stable internal conditions and responding to changes. These systems work together through complex networks of communication and regulation.

Definition: Homeostasis is the maintenance of optimal internal conditions within an organism through various regulatory mechanisms and feedback systems.

The nervous system provides rapid, precise responses to stimuli through a network of specialized cells. When a stimulus is detected by receptors, the information travels along sensory neurons to the central nervous system. Here, the information is processed and decisions are made. Motor neurons then carry commands to effectors musclesorglandsmuscles or glands which produce appropriate responses. This entire process happens within milliseconds, allowing for quick reactions to environmental changes.

The endocrine system works alongside the nervous system but operates through chemical messengers called hormones. Key endocrine glands include the pancreas producinginsulinproducing insulin, thyroid producingthyroxineproducing thyroxine, ovaries producingestrogenproducing estrogen, testes producingtestosteroneproducing testosterone, and the pituitary gland. Unlike the rapid nervous system responses, hormonal responses are slower but longer-lasting and affect multiple target tissues throughout the body.

Example: When blood glucose levels rise, the pancreas detects this change and releases insulin. This hormone signals cells throughout the body to take up glucose, thereby reducing blood glucose levels back to normal.

vertically opposite
angles
straight
line
equal
Parrallel
lines
add to
180⁰
Lacorresponding
angles
TOURSHOW TEMPURA
angles in a
triangle
Angl

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Negative Feedback and Homeostatic Control

Negative feedback is the primary mechanism for maintaining homeostasis in living organisms. This sophisticated control system helps regulate various bodily functions including metabolism, heart rate, and body temperature.

Highlight: Negative feedback works by opposing changes to bring conditions back to optimal levels. When conditions deviate from the normal range, the body initiates responses to restore balance.

The process begins when receptors detect a deviation from optimal conditions. This information is sent to coordination centers eitherthenervoussystemorendocrinesystemeither the nervous system or endocrine system, which then trigger appropriate responses through effectors. If levels are too high, the response works to decrease them; if levels are too low, the response works to increase them. This process continues until optimal conditions are restored.

Understanding these control systems is crucial because they maintain essential life processes. When these systems malfunction, various disorders can develop. For example, diabetes occurs when the negative feedback system controlling blood glucose levels doesn't work properly, either due to insufficient insulin production or cells not responding to insulin correctly.

Vocabulary: Effectors are organs or tissues that carry out responses to restore balance, such as muscles, glands, or other specialized cells.

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Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user