Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Geometry → Lines and Angles ↓


Pairs of Angles


In geometry, angles are a fundamental concept that helps us understand the shapes and properties of objects. Specifically, a pair of angles is a set of two angles that share a certain relationship with each other. In this lesson, we will discuss in depth the variety of pairs of angles, how to identify them, and how they relate to each other in various geometric contexts.

Types of angle pairs

There are several common types of angle pairs you'll encounter in geometry. These include complementary angles, supplementary angles, adjacent angles, linear pairs, and vertically opposite angles. Let's look at each of these in detail.

Supplementary angles

Complementary angles are two angles that measure 90 degrees. Complementary angles are often found and they can either be adjacent or not. Here is a simple example:

A B C D

In the above example, angle ∠BAD and angle ∠DAC are complementary. Their sum is 90 degrees.

If ∠BAD = 30°, then ∠DAC = 60°.

A common real-world example of complementary angles is the hands of a clock at a quarter of the hour (i.e., 3:15). If you look at the hour and minute hands, they form complementary angles with every passing quarter-hour.

Obtuse angle

Complementary angles are two angles that measure 180 degrees. These angles often form a straight line when they are adjacent. Let's take a look:

X Y Jade Why

In this example, angle ∠XQY and angle ∠YQZ are complementary. The sum of their measures is 180 degrees.

If ∠XQY = 120°, then ∠YQZ = 60°.

Complementary angles are often used to create straight edges or alignments in construction, such as in architecture or carpentry.

Adjacent angles

Adjacent angles are two angles that share a common side and a common vertex but do not overlap. In polygons they are often seen in conjunction with each other. Below is an example:

1 2 3

In this figure, angle 1 and angle 2 are adjacent because they are side by side and share a ray and a vertex.

Angle 1 = ∠CAB
Angle 2 = ∠BAD

Linear pair of angles

A linear pair is a pair of adjacent angles formed when two lines intersect each other. The sum of the angles in a linear pair is always 180 degrees, thus they are complementary. They look like this:

P Why

Here, ∠PQY and ∠PQX form a linear pair as these two angles lie on opposite sides of a common ray PQ.

If ∠PQY = 70°, then ∠PQX = 110°.

Vertical opposite angles

When two lines cross each other, the angles opposite each other are called vertical opposite angles. These angles are always equal. It looks like this:

A B C D

In this figure, angle ∠AOC and angle ∠BOD are vertically opposite angles and hence equal.

If ∠AOC = 45°, then ∠BOD = 45°.

Vertically opposite angles are frequently encountered in the study of parallel lines cut by a transversal, as well as in various geometric proofs and constructions.

Applications of angle pairs

The concepts of angle pairs are widely used in real-world applications. Whether in construction, art, engineering design or even daily activities, understanding angle pairs can help in accurately measuring and constructing angles.

  • Architecture and construction: In these fields, angle pairs such as complementary angles are important in ensuring stability and symmetry in structures.
  • Design and art: Artists use complementary colors and angles when creating aesthetic patterns and effects in the visual arts.
  • Navigation and mapping: Using angles, sailors can determine direction and accurately plot routes on maps.

Practice problems

Let's solidify our understanding with some practice problems:

  1. Two angles are complementary. The measure of one angle is 35°. What is the measure of the other angle?
  2. Two angles are supplementary and the measure of one is 145°. What is the measure of the other angle?
  3. If two adjacent angles form a linear pair and one of the angles is twice the other, then find the measure of each angle.
  4. Two lines intersect each other, forming vertically opposite angles. One angle is 72°. Find the measure of the other three angles.

Solving such problems not only improves understanding but also improves the skill of applying the concepts of angle pairs in different scenarios. Practicing with different types of problems will also help in seeing consistent geometrical relationships.

Conclusion

Understanding the various pairs of angles and their properties is essential in geometry. This forms the basis of many more advanced concepts, and it is important for students to fully understand these fundamental ideas. Whether you are solving practical problems in architecture or designing a new artwork, the principles of angle pairs help us draw consistent and logical conclusions.

As we continue to discover and learn more about geometry, having a solid understanding of these basic concepts will prove incredibly valuable. Geometry teaches us how to see the world from a different perspective and provides the tools to analyze and create everything around us.


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Types of Angles
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Pairs of Angles

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