What is an Included Angle? Guide & Examples

In geometry, triangles possess specific characteristics that define their shape and size, and understanding these properties often involves analyzing their angles and sides. Included angles, found within triangles and other polygons, play a crucial role in proving congruence using theorems such as Side-Angle-Side (SAS). Specifically, Euclid's Elements extensively uses the properties of included angles to explain geometric proofs. The included angle is essentially the angle formed by two specific sides of a polygon; therefore, understanding what is an included angle is paramount for students studying geometrical proofs. Tools like protractors from companies such as Staedtler can aid in accurately measuring these angles, which are pivotal for architects and engineers alike when designing structures and ensuring structural integrity.

Welcome to the fascinating world of included angles!

This concept serves as a cornerstone in geometry, enabling us to unlock solutions to a wide range of problems.

Understanding included angles is not just about memorizing definitions.

It’s about grasping a fundamental relationship that governs the properties of shapes and their interactions.

Defining the Included Angle

So, what exactly is an included angle?

Simply put, an included angle is the angle formed by two sides of a polygon that share a common vertex.

Think of it as the angle “sandwiched” between two adjacent sides.

This seemingly simple definition holds immense power in geometrical analysis.

It allows us to accurately describe and classify geometric figures.

The Significance in Congruence and Similarity

Included angles play a pivotal role in proving the congruence and similarity of geometric shapes.

Congruence, meaning identical in shape and size, and similarity, meaning identical in shape but potentially different in size, rely heavily on relationships between angles and sides.

Specifically, theorems like Side-Angle-Side (SAS) and Angle-Side-Angle (ASA) congruence hinge on the included angle.

These theorems provide a powerful framework for proving that two triangles are exactly the same or that two triangles are proportional.

Therefore, mastering included angles is a crucial step towards mastering geometric proofs.

Real-World Relevance: Beyond the Textbook

The applications of included angles extend far beyond the realm of textbooks and theoretical exercises.

In architecture, included angles are critical for designing stable and aesthetically pleasing structures.

They influence everything from roof slopes to the arrangement of support beams.

Engineers rely on the principles of included angles to calculate forces, ensure structural integrity, and optimize designs for bridges, buildings, and machines.

Even in everyday life, an understanding of included angles can help us to better understand the spatial relationships around us.

From setting up the perfect camera angle, to maximizing space in your home, included angles are always at work.

By understanding included angles, we gain a deeper appreciation for the geometry that shapes our world.

Core Concepts: Building Blocks of Included Angle Understanding

Welcome to the foundational layer of included angle mastery!

Before diving into the intricacies of included angles, it's crucial to solidify our understanding of the underlying geometric principles.

This section serves as a meticulous review, ensuring we possess the essential tools to dissect and comprehend the behavior of included angles within various shapes.

Let's embark on this crucial journey, building a rock-solid geometric foundation!

Understanding Angles: The Cornerstone

At the heart of included angles lies the fundamental concept of the angle itself.

An angle, in its simplest form, is the measure of the amount of turn between two lines or rays that diverge from a common point.

Understanding angle measurement and classification is vital for working with included angles.

Angle Measurement: Degrees and Radians

Angles are commonly measured in two primary units: degrees and radians.

A degree (°) is defined as 1/360th of a full rotation.

Radians, on the other hand, relate the angle to the radius of a circle.

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Familiarity with both degree and radian measures is essential for various geometrical and trigonometric applications.

Types of Angles: A Quick Review

Angles can be categorized based on their measure.

Here’s a brief refresher:

• Acute angles measure less than 90°.
• Right angles measure exactly 90°.
• Obtuse angles measure greater than 90° but less than 180°.
• Straight angles measure exactly 180°.

Recognizing these angle types is a foundational skill for identifying and analyzing included angles.

The Vertex: The Angle's Origin

The vertex is the cornerstone from which an angle emerges.

It is the point where two rays or line segments meet to form the angle.

In the context of included angles, the vertex is particularly significant, as it is the shared point between two adjacent sides of a polygon.

Accurately identifying the vertex is paramount to correctly locating and understanding the included angle.

Without the vertex, there is no angle!

Sides of an Angle: Defining the Boundaries

The sides of an angle are the two rays or line segments that emanate from the vertex.

These sides define the boundaries of the angle and determine its measure.

Understanding the properties of these sides is crucial for determining the specific type and size of the angle.

Whether they are rays extending infinitely or line segments with defined endpoints, these sides are essential in defining the angle's form.

Adjacent Sides: The Key to Inclusion

Adjacent sides are sides that share a common vertex.

Within a polygon, it is the angle formed between adjacent sides that we define as the included angle.

Understanding the concept of adjacency is therefore vital to identifying included angles within geometric shapes.

Consider a square: each side is adjacent to two other sides, forming included angles of 90 degrees at each vertex.

It's this relationship between adjacent sides that brings the included angle into being.

Included Angles in Common Geometric Shapes: Triangles and Parallelograms

Now that we've established a firm understanding of the core concepts, it's time to see included angles in action! Let’s take a journey through two fundamental geometric shapes: triangles and parallelograms.

By dissecting these familiar figures, we’ll solidify our grasp on how included angles manifest and influence their properties.

This section provides concrete examples that bridge the gap between abstract theory and practical application.

Get ready to see included angles come alive!

Included Angles in Triangles: A Foundation of Geometry

Triangles, the simplest of polygons, provide an excellent starting point for understanding included angles.

The beauty of a triangle lies in the fact that every angle is an included angle!

This stems directly from the definition: an included angle is formed by two sides sharing a common vertex, and in a triangle, every angle is formed by two sides that share a vertex.

Visualizing Included Angles in Triangles

Imagine a triangle ABC. Angle A is formed by sides AB and AC.

Angle B is formed by sides BA and BC.

And angle C is formed by sides CA and CB.

Therefore, angles A, B, and C are all included angles.

This may seem obvious, but recognizing this fundamental characteristic is essential for more complex geometric reasoning.

Understanding triangles and their properties is the cornerstone to other geometric topics.

The Sum of Included Angles in a Triangle

A key property of triangles is that the sum of their included angles always equals 180°.

This holds true for all types of triangles – acute, right, obtuse, equilateral, isosceles, and scalene.

This seemingly simple rule is a powerful tool for solving a wide range of geometric problems.

For instance, if you know two angles of a triangle, you can easily determine the third, utilizing the 180° sum property.

Included Angles in Parallelograms: Unveiling Relationships

Moving beyond triangles, let's explore included angles within parallelograms, a more complex quadrilateral.

A parallelogram is a four-sided figure with two pairs of parallel sides.

Included angles play a crucial role in defining and understanding the properties of parallelograms.

Opposite Angles and Included Angles

One of the defining features of a parallelogram is that its opposite angles are congruent (equal in measure).

These opposite angles are, of course, included angles formed by the adjacent sides of the parallelogram.

For example, in parallelogram ABCD, angle A (formed by sides AB and AD) is congruent to angle C (formed by sides CB and CD).

Similarly, angle B (formed by sides BA and BC) is congruent to angle D (formed by sides DA and DC).

This equality of opposite included angles is a direct consequence of the parallel sides and the transversal lines that form the parallelogram.

Adjacent Angles and Included Angles

Another key relationship in parallelograms involves adjacent angles.

Adjacent angles in a parallelogram are supplementary, meaning they add up to 180°.

Since these are included angles, we understand the special connection between included angles and the overall form of the parallelogram.

For example, in parallelogram ABCD, angle A + angle B = 180°, angle B + angle C = 180°, and so on.

The Significance of Included Angles in Parallelograms

Understanding the relationships between included angles in parallelograms allows us to solve for unknown angles and side lengths.

This understanding is crucial for geometric proofs, constructions, and practical applications in fields like architecture and engineering.

By analyzing the included angles, we unlock the secrets held within the parallelogram's form.

Included Angles and Congruence Theorems: SAS and ASA

Included angles aren't just theoretical concepts; they're the cornerstones of powerful geometric tools.

Among these tools, two congruence theorems, Side-Angle-Side (SAS) and Angle-Side-Angle (ASA), stand out.

Understanding the role of included angles within these theorems unlocks the ability to prove that two triangles are exactly the same – congruent.

Let's delve into how these theorems utilize the included angle to establish congruence.

SAS (Side-Angle-Side) Congruence: The Included Angle in Action

The Side-Angle-Side (SAS) congruence theorem states:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Notice the key phrase: included angle. This is not just any angle.

It's the angle formed by the two sides we're comparing.

The Importance of "Included" in SAS

The "included" aspect of SAS is crucial for establishing congruence.

If the angle were not included (i.e., not formed by the two sides being compared), the triangles would not necessarily be congruent.

Imagine two triangles with two pairs of sides that are the same length.

If the angles opposite one of the sides are equal, they may not be congruent because of the placement of the angle.

Visualizing SAS

Imagine triangle ABC and triangle XYZ.

If AB is congruent to XY, AC is congruent to XZ, and angle A (formed by sides AB and AC) is congruent to angle X (formed by sides XY and XZ), then, according to SAS, triangle ABC is congruent to triangle XYZ.

Without angle A being the included angle, the congruence cannot be proven by SAS.

Applying SAS in Proofs

In geometric proofs, SAS allows us to confidently declare that two triangles are identical if we can demonstrate the congruence of two sides and the included angle.

This theorem is frequently used in construction, engineering, and design, where ensuring exact replication of shapes is essential.

ASA (Angle-Side-Angle) Congruence: The Included Side's Role

The Angle-Side-Angle (ASA) congruence theorem states:

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Here, the focus shifts to the included side.

The included side is the side that lies between the two angles that we are comparing.

The Importance of "Included" in ASA

Just as with SAS, the "included" aspect of ASA is vital.

It mandates that the side we are comparing must be between the two angles we are comparing.

If the side were not included (i.e., not between the two angles), the triangles wouldn't necessarily be congruent.

Visualizing ASA

Consider triangles DEF and PQR.

If angle D is congruent to angle P, angle F is congruent to angle R, and side DF (between angles D and F) is congruent to side PR (between angles P and R), then triangle DEF is congruent to triangle PQR according to ASA.

ASA and Included Angles

While ASA focuses on the included side, it's inherently linked to included angles.

The included side defines the two included angles adjacent to it.

These two angles contribute to fulfilling the requirements of the ASA theorem.

Applying ASA in Proofs

ASA, similar to SAS, is used extensively in geometry to prove triangle congruence.

It is extremely useful when angle measures and a side length are known, particularly in situations where direct side length comparison is difficult.

Engineers often rely on ASA to guarantee structural integrity when dealing with triangular frameworks.

By understanding SAS and ASA, we equip ourselves with powerful tools for analyzing and proving geometric relationships.

These theorems highlight the profound connection between included angles (or sides) and the fundamental concept of congruence.

Mastering these principles unlocks a deeper understanding of the elegance and precision within geometry.

Measuring and Constructing Included Angles: A Practical Guide

Understanding included angles moves beyond theory when you can accurately measure and construct them.

This skill bridges the gap between abstract concepts and tangible geometric applications.

This section provides step-by-step guidance on using a protractor, the essential tool for mastering this skill.

Let's dive in and learn how to bring these geometric concepts to life.

Mastering the Protractor: Your Angle-Measuring Companion

The protractor is indispensable for precise angle measurement and construction.

Its semi-circular (or circular) scale, marked in degrees, allows us to quantify angles with accuracy.

Anatomy of a Protractor: Key Features

Familiarizing yourself with the protractor's components is the first step.

The baseline (the straight edge of the protractor) serves as the reference line.

The center point (located at the midpoint of the baseline) is aligned with the vertex of the angle you're measuring.

The degree scale (ranging from 0° to 180° or 0° to 360°) provides the numerical value of the angle.

Step-by-Step Guide to Measuring an Included Angle

Follow these steps to accurately measure an included angle:

1. Placement: Place the protractor's center point directly on the vertex of the angle.
2. Alignment: Align the protractor's baseline with one of the sides of the angle, ensuring it reads 0°.
3. Reading the Angle: Observe where the other side of the angle intersects the degree scale. This reading indicates the measure of the angle in degrees.

Remember, depending on the protractor, you may have two sets of numbers going in opposite directions.

Choose the set that starts at 0° on the side you aligned with the baseline.

Avoiding Common Errors: Precision is Key

Even with a clear process, errors can occur.

Here's how to minimize mistakes:

• Misalignment: Ensure the center point and baseline are perfectly aligned. Even a slight error can skew the reading.
• Reading the Wrong Scale: Always double-check that you're reading the correct degree scale (the one starting from 0° on your aligned side).
• Parallax Error: View the protractor directly from above to avoid parallax errors (errors caused by viewing the scale from an angle).

Constructing Included Angles: Bringing Angles to Life

Constructing included angles with a specific degree measure is a crucial skill in geometry and design.

Using a protractor, you can create angles accurately and reliably.

Step-by-Step Guide to Constructing an Included Angle

Follow these steps to construct an angle of a desired measure:

1. Draw a Baseline: Begin by drawing a straight line segment. This will be one side of your angle.
2. Mark the Vertex: Mark a point on the line segment. This point will be the vertex of your angle.
3. Position the Protractor: Place the center point of the protractor on the vertex you marked, and align the baseline of the protractor with the line segment you drew.
4. Mark the Degree: Find the desired degree measure on the protractor's scale and make a small mark on the paper at that point.
5. Draw the Second Side: Remove the protractor and draw a straight line segment connecting the vertex to the mark you made on the paper.

You have now successfully constructed an angle with the desired measure.

Diagrams and Illustrations: The Visual Language of Included Angles

Visual aids are essential for understanding included angles.

Diagrams help clarify concepts and relationships within geometric shapes.

Clear and well-labeled diagrams assist in identifying included angles within triangles, quadrilaterals, and other polygons.

Examples of Diagrams Showcasing Included Angles

Here are some examples of diagrams you might encounter:

• Triangle: A triangle with each angle clearly labeled as an included angle formed by its two adjacent sides.
• Square: A square with one angle highlighted, showing it's formed by the two adjacent sides of the square.
• Parallelogram: A parallelogram illustrating how the included angles are formed by its adjacent sides, highlighting opposite angles and their relationships.

When studying geometry, pay close attention to diagrams and use them to visualize the concepts being explained.

The ability to both measure and construct included angles transforms your understanding of geometry from passive observation to active engagement.

With practice and attention to detail, you'll gain confidence in your ability to work with angles accurately, laying a solid foundation for more advanced geometric concepts.

Practical Examples: Identifying Included Angles in Various Shapes

Let's solidify your understanding of included angles by exploring them in various geometric shapes.

Identifying these angles correctly is crucial for success in geometry, particularly when applying congruence theorems or analyzing shape properties.

The following examples will guide you through identifying included angles, reinforcing your knowledge with visual aids and clear explanations.

Triangles: A Fundamental Case

In a triangle, every angle is an included angle.

This is because each angle is formed by the two sides that meet at its vertex.

Example: Scalene Triangle ABC

Consider scalene triangle ABC, where all three sides have different lengths.

Angle A is included between sides AB and AC, angle B is included between sides BA and BC, and angle C is included between sides CA and CB.

Visualize this: Imagine standing at vertex A. The two sides that create the corner where you're standing (AB and AC) are the sides that include angle A.

Diagrammatically, angle A, angle B, and angle C will be clearly marked to highlight their position as included angles formed by their corresponding adjacent sides.

Quadrilaterals: Stepping Up the Complexity

Quadrilaterals, with their four sides and four angles, provide slightly more complex examples of included angles.

Remember, the included angle is always formed by two adjacent sides.

Example: Rectangle DEFG

In rectangle DEFG, each angle is a right angle (90 degrees).

Angle D is included between sides DE and DG.

Similarly, angle E is included between sides EF and DE, angle F is included between sides FG and EF, and angle G is included between sides GD and GF.

Each of these angles is crucial for understanding the properties of a rectangle, such as opposite sides being equal and parallel.

Example: Parallelogram HIJK

A parallelogram has opposite sides parallel and equal, but its angles are not necessarily right angles.

Angle H is included between sides HI and HK, angle I is included between sides IJ and IH, angle J is included between sides JK and JI, and angle K is included between sides KH and KJ.

Understanding these included angles helps to prove properties such as opposite angles being congruent.

Polygons: Expanding the Concept

The concept of included angles extends beyond triangles and quadrilaterals to all polygons.

Whether it's a pentagon, hexagon, or any n-sided figure, the included angle remains the angle formed by two adjacent sides meeting at a vertex.

Example: Regular Pentagon LMNOP

In a regular pentagon LMNOP, all sides and angles are equal.

Angle L is included between sides LM and LO, angle M is included between sides MN and ML, angle N is included between sides NO and NM, angle O is included between sides OP and ON, and angle P is included between sides PL and PO.

Identifying these included angles is essential in calculating the sum of interior angles and understanding the pentagon's symmetry.

Why is Identifying Included Angles Important?

Identifying included angles is not just a theoretical exercise.

It is crucial for several reasons:

• Applying Congruence Theorems: The Side-Angle-Side (SAS) congruence theorem explicitly requires identifying the included angle between two pairs of congruent sides to prove that two triangles are congruent.

• Analyzing Geometric Properties: Understanding included angles helps in deducing various properties of shapes, such as angle relationships in parallelograms or the sum of interior angles in polygons.

• Solving Geometric Problems: Many geometric problems require you to work with the relationships between sides and angles.

Knowing which angle is included between specific sides is essential for setting up the problem correctly and finding the solution.

By consistently practicing the identification of included angles in different shapes, you will significantly enhance your geometric problem-solving skills.

The more you work with included angles, the more intuitive their identification becomes, making complex geometric tasks easier to tackle!

Advanced Concepts: Included Angles in Congruence and Similarity Proofs

The journey into the world of included angles doesn't stop with basic identification. It extends to the core of geometric reasoning: proving congruence and similarity.

Understanding how included angles function in these proofs unlocks a deeper appreciation for the elegance and precision of geometry.

This section delves into the advanced applications of included angles, showcasing their pivotal role in establishing geometric truths.

Included Angles and Congruence Theorems: A Powerful Partnership

Congruence theorems provide a systematic way to prove that two geometric figures are identical in shape and size. Included angles are often indispensable components of these theorems.

Side-Angle-Side (SAS) Congruence Theorem: The Included Angle's Central Role

The SAS theorem states that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.

Notice the crucial word: included. It's not just any angle; it's the angle nestled between the two specified sides.

This precise placement is what allows us to definitively conclude congruence. If a non-included angle were used, the triangles might still differ, invalidating the proof.

Imagine constructing a triangle. If you know the lengths of two sides and the angle between them, there's only one possible triangle you can create.

This is the essence of the SAS theorem and highlights the power of the included angle.

Angle-Side-Angle (ASA) Congruence Theorem: The Included Side's Influence

While not explicitly named an "included angle," the ASA theorem relies on a similar concept of inclusion.

It states that if two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent.

Here, the included side is the side that lies between the two specified angles. The relationship is reciprocal.

Each of the angles is, by definition, the included angle for the included side.

This theorem emphasizes that knowing two angles and the side connecting them uniquely determines a triangle.

The included side provides the critical link between the angles, ensuring congruence.

Included Angles and Similarity Theorems: Scaling Proportions

Similarity, unlike congruence, allows for scaling. Two figures are similar if they have the same shape but different sizes.

Included angles play a vital role in establishing similarity, especially through the Side-Angle-Side (SAS) Similarity Theorem.

Side-Angle-Side (SAS) Similarity Theorem: Proportionality and the Included Angle

The SAS Similarity Theorem asserts that if two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles are congruent, then the two triangles are similar.

This theorem illustrates how the included angle acts as a bridge between proportionality and similarity.

The proportional sides establish a scaling factor, while the congruent included angle ensures that the triangles maintain the same shape.

Without the congruent included angle, the triangles could have different shapes, even with proportional sides.

Examples of Proofs Involving Included Angles: Putting Theory into Practice

Let's examine some proof scenarios to solidify your understanding.

Imagine you are given two triangles, ABC and DEF, where AB ≅ DE, AC ≅ DF, and ∠A ≅ ∠D. You want to prove that ΔABC ≅ ΔDEF.

This proof directly applies the SAS congruence theorem. Since ∠A is the included angle between sides AB and AC, and ∠D is the included angle between sides DE and DF, the conditions for SAS are met.

Therefore, you can confidently conclude that ΔABC ≅ ΔDEF.

Now, consider two triangles, GHI and JKL, where GH/JK = GI/JL and ∠G ≅ ∠J. Here, we aim to prove ΔGHI ~ ΔJKL.

This is a clear application of the SAS similarity theorem. Because sides GH and GI are proportional to sides JK and JL respectively, and the included angles ∠G and ∠J are congruent, you can conclude that ΔGHI ~ ΔJKL.

These examples, though simplified, represent the fundamental logic used in more complex geometric proofs involving included angles.

By mastering the application of these theorems, you'll gain the ability to tackle intricate geometric challenges.

Tools and Technologies: Visualizing Included Angles with Software

Geometry, at its heart, is a visual discipline. While textbooks and traditional methods provide a foundation, dynamic geometric software elevates understanding to a whole new level.

These tools empower you to not just see included angles, but to interact with them, manipulating shapes and observing the resulting changes in real-time.

This section will explore how readily available geometric software can transform your grasp of included angles from abstract concepts to tangible realities.

Unleashing the Power of Dynamic Geometry Software

Software like GeoGebra and Desmos are more than just digital drawing boards. They are sophisticated environments specifically designed for geometric exploration.

They allow you to create geometric figures with precise measurements and then dynamically alter them, observing how angles, side lengths, and relationships shift in response.

This immediate visual feedback is invaluable for solidifying your intuition about included angles and their behavior within different shapes.

GeoGebra: A Comprehensive Toolkit for Geometric Discovery

GeoGebra is a free, open-source software packed with features for exploring geometry, algebra, calculus, and more.

Its dynamic interface makes it ideal for visualizing included angles and their role in geometric proofs.

Step-by-Step Tutorial: Constructing and Exploring Included Angles in GeoGebra

1. Creating a Triangle: Begin by selecting the "Polygon" tool. Click three distinct points on the GeoGebra canvas to form a triangle.
2. Measuring Included Angles: Use the "Angle" tool. Click on three points in order (e.g., B, A, C) to measure the angle ∠BAC. This angle is included between sides AB and AC.
3. Dynamic Manipulation: Now, click and drag any of the triangle's vertices. Observe how the measured angle changes in real-time as you alter the side lengths. Notice how only changing a non-adjacent side does not change the included angle!
4. Exploring SAS Congruence: Construct a second triangle. Set the lengths of two sides and the included angle of the first triangle to be equal to the corresponding elements in the second triangle (using the "Distance or Length" and "Angle with Given Size" tools). Verify the triangles are congruent.

By actively building and manipulating these figures, you gain a deep, intuitive understanding of how included angles function in geometric constructions.

Desmos Geometry: Simplicity and Elegance in Visualizing Angles

Desmos is renowned for its user-friendly interface and accessibility. Its Geometry tool is equally intuitive, offering a simplified yet powerful environment for exploring geometric concepts.

Step-by-Step Tutorial: Visualizing Included Angles in Parallelograms with Desmos

1. Drawing a Parallelogram: Use the "Polygon" tool to create a quadrilateral. Then, use the properties of a parallelogram to ensure that opposite sides are parallel. Desmos' grid and line tools make this simple.
2. Measuring Included Angles: Utilize the "Angle" tool to measure the angles formed at each vertex of the parallelogram. Observe the relationships between adjacent and opposite angles.
3. Exploring Relationships: Drag the vertices of the parallelogram. Observe how opposite angles remain congruent, and adjacent angles remain supplementary, regardless of the shape's distortion. The included angle of any side remains constant to the side opposite it.

Desmos's clear visual representations and ease of use make it an excellent choice for beginners and experienced learners alike.

Key Benefits of Using Geometric Software

• Visual Reinforcement: These tools offer immediate visual feedback, turning abstract concepts into concrete images.
• Interactive Exploration: Dynamic manipulation fosters deeper understanding and intuition.
• Error Reduction: Software provides precise measurements, minimizing errors associated with manual constructions.
• Increased Engagement: The interactive nature of the software makes learning more engaging and enjoyable.

By incorporating geometric software into your study routine, you can unlock a new level of understanding and appreciation for included angles and the broader world of geometry.

Embrace these powerful tools and watch your geometric skills soar!

So, there you have it! Hopefully, this guide clears up any confusion about what an included angle really is and how to spot one in a figure. Now you can confidently tackle those geometry problems and impress your friends with your included angle knowledge!