How to Identify and Classify Triangles in Geometry
What are Triangles in Geometry?
Shapes and patterns have always fascinated us since the dawn of time. The rigidness in some and swirls in others have often puzzled our minds and nature housing such beautiful designs is beyond fascinating. Triangles are one of those shapes that possess a variety of features and applications in our world. A triangle is defined as a closed two-dimensional figure with 3 sides, 3 angles, and 3 vertices in Geometry. They are a type of polygon whose sum of all three angles usually equals 180°.
Types of Triangles
Since a triangle is a two-dimensional shape, the types of triangles are classified based on their sides and angle measurement.
A triangle is classified into three types based on the length of its sides,
Scalene Triangle – The length of all sides differs or is not equal.
Isosceles Triangle – The properties of an isosceles triangle involve a triangle having two sides equal in length and the third is not equal.
Equilateral Triangle – The properties of an equilateral triangle involve a triangle having the length of all three sides equal.
A triangle is classified based on the angles,
Acute Angle Triangle – The angles of a triangle are less than 90°
Obtuse Angle Triangle – One angle of a triangle is greater than 90°
Right Angle Triangle – One angle of a triangle is equal to 90°
Properties of a Triangle
Triangles follow certain properties and rules to achieve a particular state and to solve complex geometric problems. Some of the properties of triangles are listed below,
The presence of three sides, three angles, and three vertices.
All the interior angles equal to 180°.
The third side of a triangle is less than the sum of the other two sides.
The half product of the base and the height gives the area of the triangle.
The sum of all the three sides of a triangle provides the perimeter of the triangle.
Properties of Median in a Triangle
The following properties are established to find the median of a triangle,
A triangle has 3 medians, one from each vertex.
All medians meet at a single point.
The Centroid of the triangle is the point where the 3 medians meet.
The median of a triangle divides it into two smaller triangles.
What is a Congruent Triangle?
When all three sides and three angles of a given triangle are equal it is referred to as a congruent triangle. It is based on the shape and size of the triangle. The object and its mirror image are generally referred to as congruence. Two images are congruent if they superimpose each other. In geometric figures, a similar length of line segments are congruent and so is it’s the angle of measurement.
Conditions for a Congruent Triangle are established as the property of sides of a triangle.
SSS (Side-Side-Side)
All three sides are equivalent concerning the second triangle.
SAS (Side-Angle-Side)
Any two sides and an angle are equivalent concerning the second triangle.
ASA (Angle-Side-Angle)
Any two angles and a side are equivalent concerning property f a second triangle.
AAS (Angle-Angle-Side)
A non-included side and two angles are equal to corresponding angles and sides of another triangle.
RHS (Right Angle-Hypotenuse-Side)
The hypotenuse with a side of a right-angled triangle is equivalent to the second triangle’s hypotenuse and right-angled side.
Example of a Congruent Triangle
In the figure, ΔABC and ΔPQR are congruent triangles. Therefore,
Vertices: A and P, B and Q, C and R vertices are equal
Sides: AB=PQ, QR= BC, and AC=PR;
Angles: ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R
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State The Angle Sum Property of a Triangle
The Angle Sum Property of a given triangle is defined as the sum of all the interior angles of a triangle is equal to 180°.
Devise a theorem to calculate the angle sum property of a triangle.
Proof:
Let’s take a ΔABC , prove the property of the triangle by drawing a line PQ,
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Since PQ is a straight line,
∠PAB + ∠BAC + ∠QAC = 180⁰
PQ ll BC and AB, AC are transversals
Hence, ∠QAC = ∠ACB and ∠PAB = ∠CBA are a pair of alternate angles
Substituting ∠QAC and ∠PAB , we get
∠ACB + ∠BAC + ∠CBA = 180⁰
Hence, the sum of all interior angles is equal to 180⁰.
FAQs on What Are Triangles? Explanation, Types & Properties
1. What is a triangle in geometry?
A triangle is a polygon with three sides, three vertices (corners), and three interior angles. It is one of the most basic shapes in geometry, forming a closed, two-dimensional figure. The sum of its three interior angles always equals 180 degrees. For example, a shape with corners A, B, and C, and sides AB, BC, and CA is a triangle.
2. How are triangles classified based on their sides and angles?
Triangles are classified in two main ways: by the length of their sides and by the measure of their angles.
- Classification by Sides:
- Equilateral Triangle: All three sides are of equal length, and all three angles are equal (60° each).
- Isosceles Triangle: Two sides are of equal length, and the angles opposite these sides are also equal.
- Scalene Triangle: All three sides have different lengths, and all three angles have different measures.
- Classification by Angles:
- Acute Triangle: All three interior angles are acute (less than 90°).
- Right Triangle: One of the interior angles is a right angle (exactly 90°).
- Obtuse Triangle: One of the interior angles is an obtuse angle (greater than 90°).
3. What is the most fundamental property of the angles in a triangle?
The most fundamental property is the Angle Sum Property of a Triangle. This property states that the sum of the measures of the three interior angles of any triangle is always 180 degrees. This holds true for all types of triangles, whether they are scalene, isosceles, equilateral, acute, right, or obtuse.
4. What are the basic formulas to find the perimeter and area of a triangle?
The two basic formulas for a triangle are for its perimeter and area.
- Perimeter (P): The perimeter is the total length of the boundary of the triangle. It is calculated by adding the lengths of all three sides (a, b, and c):
P = a + b + c - Area (A): The area is the space enclosed by the triangle. The most common formula uses the base (b) and height (h):
A = ½ × base × height
For triangles where the height is not known but all three side lengths are, an advanced method called Heron's formula can be used.
5. Why can a triangle not have two right angles?
A triangle cannot have two right angles because the sum of its three interior angles must be exactly 180 degrees. If a triangle had two right angles (90° + 90°), the sum of just those two angles would already be 180°. This would leave 0° for the third angle, which is impossible as a triangle must have three angles and three vertices to form a closed shape. Therefore, a triangle can have at most one right angle.
6. What is the Triangle Inequality Theorem and why does it matter?
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides a, b, and c, this means:
- a + b > c
- a + c > b
- b + c > a
7. Where can we see examples of triangles being used in the real world?
Triangles are fundamental to design and engineering because of their inherent strength and stability. Some real-world examples include:
- Architecture and Construction: Roof trusses, bridges (like the Truss bridge), and support beams use triangles to distribute weight and withstand pressure.
- Navigation and Surveying: GPS technology and land surveyors use a method called triangulation to determine precise locations and distances.
- Art and Design: Artists use triangles for composition, creating a sense of balance, perspective, and guiding the viewer's eye.
- Everyday Objects: You can find triangles in bicycle frames, sandwich slices, pyramids, and traffic warning signs.
