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Basic Geometrical Ideas - Angles, Vertices, and Sides

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An angle is formed when two rays originate from a common starting point. This common point is called the vertex, and the two rays are known as the arms or sides of the angle. In the notation ∠ABC\angle ABC, the middle letter BB represents the vertex.

Diagram showing angle ABC with vertex B and arms AB and BC.
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A polygon is a closed figure made up entirely of line segments. The line segments forming the polygon are called its sides. The point where any two sides meet is called a vertex.

A quadrilateral showing sides and vertices.
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The interior of an angle is the region between the arms that extends infinitely. Any point located inside this space is said to be in the interior, while points outside the arms are in the exterior. Points on the rays themselves are said to be 'on the angle'.

Diagram showing point P in the interior and point Q in the exterior of an angle.
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Adjacent sides of a polygon are any two sides that share a common vertex. For example, in a triangle ABCABC, the sides AB‾\overline{AB} and BC‾\overline{BC} are adjacent because they meet at vertex BB.

Triangle ABC illustrating adjacent sides meeting at vertices.

📐Formulae

Number of Sides in any Polygon=Number of Vertices in that Polygon\text{Number of Sides in any Polygon} = \text{Number of Vertices in that Polygon}

Angle Notation:∠(Point on Arm 1)(Vertex)(Point on Arm 2)\text{Angle Notation}: \angle \text{(Point on Arm 1)(Vertex)(Point on Arm 2)}

Line Segment AB=AB‾\text{Line Segment } AB = \overline{AB}

Ray starting at P through Q=PQ→\text{Ray starting at } P \text{ through } Q = \overrightarrow{PQ}

💡Examples

Problem 1:

Given an angle named ∠DEF\angle DEF, identify the vertex and the arms of the angle.

Solution:

Step 1: Identify the middle letter in the angle notation ∠DEF\angle DEF. The middle letter is EE, so the Vertex is EE. Step 2: The arms are the rays that start from the vertex EE and pass through the other two points DD and FF. Therefore, the Arms are Ray ED→\overrightarrow{ED} and Ray EF→\overrightarrow{EF}.

Explanation:

In geometric notation for angles, the vertex is always placed in the center of the three-letter name. The arms originate from this vertex point.

Problem 2:

A triangle is named PQRPQR. List all the sides and all the vertices of this triangle.

Solution:

Step 1: Identify the vertices. The vertices are the individual points that define the corners of the triangle: P,Q, and RP, Q, \text{ and } R. Step 2: Identify the sides. The sides are the line segments connecting these points: PQ‾\overline{PQ}, QR‾\overline{QR}, and RP‾\overline{RP}.

Explanation:

A triangle is a polygon with three sides and three vertices. Each side is a line segment connecting two consecutive vertices.

Problem 3:

Observe the quadrilateral KLMNKLMN. Name all the pairs of opposite sides and all the pairs of opposite angles.

A square-shaped quadrilateral KLMN with vertices labeled.

Solution:

Opposite sides: (KL‾,MN‾)(\overline{KL}, \overline{MN}) and (LM‾,NK‾)(\overline{LM}, \overline{NK}). Opposite angles: (∠K,∠M)(\angle K, \angle M) and (∠L,∠N)(\angle L, \angle N).

Explanation:

Opposite sides in a quadrilateral are the sides that do not meet at a common vertex. Opposite angles are the angles at vertices that are not connected by a single side.

Problem 4:

In the given figure, identify three angles that have point OO as their common vertex.

Three rays OA, OB, and OC originating from a common point O.

Solution:

The three angles are ∠AOB\angle AOB, ∠BOC\angle BOC, and ∠AOC\angle AOC.

Explanation:

An angle is formed by any two rays meeting at a point. Here, rays OA→\overrightarrow{OA}, OB→\overrightarrow{OB}, and OC→\overrightarrow{OC} all meet at OO. We can pair them as (OA,OB)(OA, OB), (OB,OC)(OB, OC), and (OA,OC)(OA, OC) to form three distinct angles.