krit.club logo

Shape and Space - Angle Relationships (Complementary, Supplementary, Vertically Opposite)

Grade 6IB

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

🔑Concepts

•

Angles and Vertices: An angle is the amount of turn between two rays that meet at a common endpoint called the vertex; imagine the corner of a table or the point where two hands of a clock meet.

•

Complementary Angles: Two angles are complementary if their measures sum to exactly 90∘90^{\circ}. Visually, these angles fit together to form a perfect 'L' shape or a right angle, which is often marked with a small square in the corner.

•

Supplementary Angles: Two angles are supplementary if their measures sum to 180∘180^{\circ}. When placed side-by-side, they form a straight line, resembling a flat horizon divided by a single ray.

•

Vertically Opposite Angles: These are pairs of equal angles formed by two straight lines intersecting. They sit across from each other at the vertex, creating an 'X' shape where the top angle equals the bottom, and the left equals the right.

•

Adjacent Angles: These are 'next-door' angles that share a common vertex and a common side but do not overlap; think of them as two adjacent slices of a circular pie.

•

Angles at a Point: The sum of all angles around a single point is always 360∘360^{\circ}. This represents a complete rotation, visually forming a full circle.

•

Right Angle and Straight Line: A right angle measures exactly 90∘90^{\circ} (a quarter turn), while a straight line measures 180∘180^{\circ} (a half turn).

📐Formulae

a+b=90∘a + b = 90^{\circ} (Complementary Angles)

a+b=180∘a + b = 180^{\circ} (Supplementary Angles)

Angle 1=Angle 2\text{Angle } 1 = \text{Angle } 2 (Vertically Opposite)

∑Angles at a Point=360∘\sum \text{Angles at a Point} = 360^{\circ}

💡Examples

Problem 1:

Find the value of xx if xx and 54∘54^{\circ} are complementary angles.

Solution:

  1. Since the angles are complementary, their sum is 90∘90^{\circ}. \ 2. Write the equation: x+54∘=90∘x + 54^{\circ} = 90^{\circ}. \ 3. Subtract 54∘54^{\circ} from both sides: x=90∘−54∘x = 90^{\circ} - 54^{\circ}. \ 4. Calculate: x=36∘x = 36^{\circ}.

Explanation:

We use the property that complementary angles must add up to a right angle (90∘90^{\circ}).

Problem 2:

Two lines intersect to form an 'X'. If one angle is 125∘125^{\circ}, find the measure of its vertically opposite angle and the measure of an adjacent supplementary angle.

Solution:

  1. Vertically opposite angles are equal, so the opposite angle is 125∘125^{\circ}. \ 2. Adjacent angles on a straight line are supplementary, so their sum is 180∘180^{\circ}. \ 3. Let the adjacent angle be yy: y+125∘=180∘y + 125^{\circ} = 180^{\circ}. \ 4. Solve for yy: y=180∘−125∘=55∘y = 180^{\circ} - 125^{\circ} = 55^{\circ}.

Explanation:

This problem applies two rules: opposite angles in an intersection are equal, and angles on a straight line total 180∘180^{\circ}.