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Geometry - Classifying triangles and quadrilaterals

Grade 6IGCSE

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

🔑Concepts

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Triangles classified by sides: Equilateral (3 equal sides), Isosceles (2 equal sides), and Scalene (no equal sides).

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Triangles classified by angles: Acute (all angles < 90°), Right (one angle = 90°), and Obtuse (one angle > 90°).

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Quadrilaterals: Four-sided polygons including Square, Rectangle, Parallelogram, Rhombus, Trapezium, and Kite.

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Properties of Parallelograms: Opposite sides are parallel and equal; opposite angles are equal.

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Properties of Rhombus: All four sides are equal; diagonals bisect each other at 90°.

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Properties of Trapezium: At least one pair of opposite sides are parallel.

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Angle Sum Property: The sum of interior angles in any triangle is 180°.

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Angle Sum Property: The sum of interior angles in any quadrilateral is 360°.

📐Formulae

Sum of angles in a triangle: a+b+c=180∘a + b + c = 180^{\circ}

Sum of angles in a quadrilateral: a+b+c+d=360∘a + b + c + d = 360^{\circ}

Perimeter of a triangle: P=side1+side2+side3P = side_1 + side_2 + side_3

Perimeter of a rectangle: P=2(l+w)P = 2(l + w)

💡Examples

Problem 1:

A triangle has angles measuring 40∘40^{\circ} and 70∘70^{\circ}. Find the third angle and classify the triangle by its sides and angles.

Solution:

Third angle = 70∘70^{\circ}; Classed as an Acute Isosceles triangle.

Explanation:

To find the third angle: 180∘−(40∘+70∘)=70∘180^{\circ} - (40^{\circ} + 70^{\circ}) = 70^{\circ}. Since all angles are less than 90∘90^{\circ}, it is acute. Since two angles are equal (70∘70^{\circ} and 70∘70^{\circ}), two sides must be equal, making it isosceles.

Problem 2:

A quadrilateral has four equal sides, but its internal angles are not 90∘90^{\circ}. Identify the shape.

Solution:

Rhombus

Explanation:

A square and a rhombus both have four equal sides. However, a square must have four 90∘90^{\circ} angles. If the angles are not right angles, the shape is a rhombus.

Problem 3:

In a quadrilateral ABCDABCD, ∠A=100∘\angle A = 100^{\circ}, ∠B=80∘\angle B = 80^{\circ}, and ∠C=100∘\angle C = 100^{\circ}. Find ∠D\angle D and identify if this could be a parallelogram.

Solution:

∠D=80∘\angle D = 80^{\circ}; Yes, it is a parallelogram.

Explanation:

Sum of angles = 360∘360^{\circ}. So, ∠D=360∘−(100∘+80∘+100∘)=80∘\angle D = 360^{\circ} - (100^{\circ} + 80^{\circ} + 100^{\circ}) = 80^{\circ}. Since opposite angles are equal (100°=100° and 80°=80°), it satisfies the property of a parallelogram.