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Shape and Space - Lines and Angles (Acute, Obtuse, Right, Straight, Reflex)

Grade 6IB

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

🔑Concepts

An angle is formed when two rays or line segments meet at a common endpoint called a vertex. The size of the angle is measured in degrees (^{\circ}) using a tool called a protractor. Visually, you can think of an angle as the amount of 'turn' between two lines.

Acute Angles are those that measure more than 00^{\circ} but less than 9090^{\circ}. Visually, these angles appear 'sharp' or narrow, similar to the tip of a pencil or a partially opened pair of scissors.

Right Angles measure exactly 9090^{\circ}. These represent a perfect quarter-turn and are visualized as the 'L' shape found in the corners of a square, a book, or a room. In geometry diagrams, a right angle is uniquely marked with a small square at the vertex instead of a curved arc.

Obtuse Angles measure more than 9090^{\circ} but less than 180180^{\circ}. Visually, these angles look 'blunt' or wide, like the shape of a reclining beach chair or the hands of a clock showing 4:00.

Straight Angles measure exactly 180180^{\circ}. A straight angle forms a perfectly flat line. Visually, it looks like a single straight line with a vertex point in the middle, representing a half-turn of a circle.

Reflex Angles are angles that measure more than 180180^{\circ} but less than 360360^{\circ}. Visually, they represent the 'outside' bend of an angle, similar to the angle formed on the back of your elbow when you bend your arm.

Full Rotation or Angles at a Point measure exactly 360360^{\circ}. This represents a complete circle. Visually, if you start at one line and turn all the way back to the start, you have completed a 360360^{\circ} turn.

Adjacent angles on a straight line always add up to 180180^{\circ}. Visually, if you split a straight line into two or more parts with lines coming out of a single point, the sum of those 'inner' angles will always equal the straight angle (180180^{\circ}).

📐Formulae

Acute Angle: 0<θ<90\text{Acute Angle: } 0^{\circ} < \theta < 90^{\circ}

Right Angle: θ=90\text{Right Angle: } \theta = 90^{\circ}

Obtuse Angle: 90<θ<180\text{Obtuse Angle: } 90^{\circ} < \theta < 180^{\circ}

Straight Angle: θ=180\text{Straight Angle: } \theta = 180^{\circ}

Reflex Angle: 180<θ<360\text{Reflex Angle: } 180^{\circ} < \theta < 360^{\circ}

Angles on a Straight Line: a+b=180\text{Angles on a Straight Line: } a + b = 180^{\circ}

Angles around a Point: a+b+c+...=360\text{Angles around a Point: } a + b + c + ... = 360^{\circ}

Complementary Angles: a+b=90\text{Complementary Angles: } a + b = 90^{\circ}

💡Examples

Problem 1:

Two angles lie on a straight line. If one angle measures 6565^{\circ}, calculate the value of the missing angle xx.

Solution:

  1. Identify the relationship: Angles on a straight line sum to 180180^{\circ}.
  2. Set up the equation: x+65=180x + 65^{\circ} = 180^{\circ}.
  3. Subtract 6565^{\circ} from both sides: x=18065x = 180^{\circ} - 65^{\circ}.
  4. Calculate the result: x=115x = 115^{\circ}.

Explanation:

Since the angles are supplementary (forming a straight line), we subtract the known angle from 180180^{\circ} to find the unknown. The resulting angle is obtuse.

Problem 2:

An interior angle of a triangle is measured as 110110^{\circ}. Calculate the reflex angle yy that exists on the outside of this vertex.

Solution:

  1. Identify the relationship: A full rotation around a point is 360360^{\circ}.
  2. Set up the equation: y+110=360y + 110^{\circ} = 360^{\circ}.
  3. Subtract 110110^{\circ} from both sides: y=360110y = 360^{\circ} - 110^{\circ}.
  4. Calculate the result: y=250y = 250^{\circ}.

Explanation:

To find a reflex angle when the interior angle is known, subtract the interior angle from a full circle (360360^{\circ}). The result 250250^{\circ} is a reflex angle because it is between 180180^{\circ} and 360360^{\circ}.