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Geometry - Classifying triangles and quadrilaterals

Grade 6Cambridge (IGCSE)

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

🔑Concepts

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Triangles are classified by their sides: Equilateral (3 equal sides), Isosceles (2 equal sides), and Scalene (no equal sides). They are also classified by their internal angles: Acute (all angles <90∘< 90^{\circ}), Right-angled (one angle =90∘= 90^{\circ}), and Obtuse (one angle >90∘> 90^{\circ}).

Comparison of an equilateral triangle and a right-angled triangle.
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A Parallelogram is a quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. Consecutive angles are supplementary (sum=180∘sum = 180^{\circ}).

Parallelogram showing equal opposite angles a and b.
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A Trapezium (or Trapezoid) is a quadrilateral with at least one pair of parallel sides. An Isosceles Trapezium has non-parallel sides of equal length and equal base angles.

Diagram of an isosceles trapezium.
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A Kite is a quadrilateral with two pairs of adjacent equal sides. The diagonals intersect at 90∘90^{\circ}, and one diagonal bisects the other.

Kite with perpendicular diagonals.

📐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.

Problem 4:

In triangle PQRPQR, side PQ=5 cmPQ = 5\text{ cm}, QR=5 cmQR = 5\text{ cm}, and ∠Q=90∘\angle Q = 90^{\circ}. Calculate the values of the other two angles and classify the triangle by both its sides and its angles.

A right-angled isosceles triangle PQR with equal legs of 5cm.

Solution:

  1. Since PQ=QRPQ = QR, the triangle is isosceles.
  2. In an isosceles triangle, angles opposite equal sides are equal. Let ∠P=∠R=x\angle P = \angle R = x.
  3. Sum of angles: x+x+90∘=180∘x + x + 90^{\circ} = 180^{\circ}.
  4. 2x=90∘  ⟹  x=45∘2x = 90^{\circ} \implies x = 45^{\circ}.
  5. Classification: Isosceles Right-angled triangle.

Explanation:

We use the property that isosceles triangles have two equal base angles and that all angles in a triangle sum to 180∘180^{\circ}. Because one angle is 90∘90^{\circ}, it is a right-angled triangle.

Problem 5:

A quadrilateral JKLMJKLM has ∠J=110∘\angle J = 110^{\circ} and ∠K=70∘\angle K = 70^{\circ}. If side JKJK is parallel to MLML and JMJM is parallel to KLKL, find the measure of ∠L\angle L and identify the shape.

Parallelogram JKLM with angles J and K labeled.

Solution:

  1. Since opposite sides are parallel (JK∥MLJK \parallel ML and JM∥KLJM \parallel KL), the shape is a parallelogram.
  2. In a parallelogram, opposite angles are equal: ∠L=∠J\angle L = \angle J.
  3. Therefore, ∠L=110∘\angle L = 110^{\circ}.
  4. Also, check ∠M=∠K=70∘\angle M = \angle K = 70^{\circ}. Sum: 110+70+110+70=360∘110 + 70 + 110 + 70 = 360^{\circ}.

Explanation:

By definition, a quadrilateral with two pairs of parallel sides is a parallelogram. A key property of parallelograms is that opposite angles are congruent.