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 → Triangles ↓


Angle Sum Property


The angle sum property is an important concept in the study of triangles in geometry. This property states that the sum of all interior angles in a triangle is always 180 degrees. Regardless of the type of triangle or its orientation, this property will be true. Let's look at this concept in more detail with an explanation, examples, and some simple visuals.

Understanding the triangle

A triangle is a three-sided polygon with three vertices and three edges. The angles inside a triangle are called its interior angles. The angle sum property deals specifically with these interior angles.

Explanation of angle sum property

According to the angle sum property, if you have a triangle ABC, the sum of the angles (angles A, B, and C) at the corners of this triangle will always be equal to 180 degrees. Mathematically, this is expressed as:

angle A + angle B + angle C = 180°

Visual example

Consider triangle ABC. It can be represented like this:

A B C ∠A ∠B ∠C

Here, when you measure the interior angles ∠A, ∠B, and ∠C and add them together, their sum will be 180 degrees according to the angle sum property.

Why is the angle sum property true?

This property is true because of the geometric structure of the triangle. One way to see this is to draw a line through the opposite vertex parallel to one side of the triangle and use alternate interior angles. Let's look at this using a visual approach.

Parallel line explanation

Consider this scenario:

A B C ∠A ∠B ∠C

Here, when a line is drawn through point A parallel to BC, the properties of parallel lines and angles ensure that the sum of the angles around point A equals 180 degrees. Therefore, the sum of the interior angles of a triangle is also 180 degrees through the supplementary angles and angle-transversal relationships.

Examples of angle sum property

Example 1: Find the third angle

If two angles of a triangle are 50 degrees and 60 degrees, then find the third angle.

First Angle = 50° Second Angle = 60° Let the Third Angle be x. According to the angle sum property, 50° + 60° + x = 180° => 110° + x = 180° => x = 180° - 110° => x = 70°

The third angle is 70 degrees.

Example 2: Equilateral triangle

All angles in an equilateral triangle are equal. What is each angle of an equilateral triangle?

Let each angle of the equilateral triangle be x. According to the angle sum property, x + x + x = 180° => 3x = 180° => x = 180° / 3 => x = 60°

Each angle of an equilateral triangle is 60 degrees.

Example 3: Right-angled triangle

In a right-angled triangle one angle is 90 degrees. If one of the other angles is 30 degrees, find the remaining angle.

First Angle = 90° (Right angle) Second Angle = 30° Let the Third Angle be x. According to the angle sum property, 90° + 30° + x = 180° => 120° + x = 180° => x = 180° - 120° => x = 60°

The remaining angle is 60 degrees.

Different types of triangles and angle sum property

  • Equilateral triangle: All three sides and angles are equal. Each angle is 60 degrees.
  • Isosceles triangle: Two sides are equal, and two angles are equal. This property still applies today: the sum of all angles is 180 degrees.
  • Scalene triangle: All sides and angles are different. Still, the angle sum property is true.
  • Right angle: One angle is 90 degrees. The sum of the other two angles is 90 degrees.

Conclusion

The angle sum property is a fundamental aspect of triangles in geometry. By understanding this property, we not only gain insight into solving problems, but also understand the inherent symmetries and constraints of triangular shapes. Through constant experimentation and practice, mastering this property becomes a straightforward part of mathematics, especially in geometry.


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Angle Sum Property

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