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Geometry and Measure - Shapes and geometric reasoning

Grade 7Cambridge (IGCSE)

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

🔑Concepts

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Angles on a straight line always sum to 180∘180^\circ. This property allows for the calculation of an unknown angle when other angles meeting at the same point on a line are known.

A straight line with a ray creating two adjacent angles a and b that sum to 180 degrees.
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When two parallel lines are cut by a transversal, alternate angles (forming a 'Z' shape) are equal, and corresponding angles (forming an 'F' shape) are equal.

Parallel lines showing alternate interior angles x and y.
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Vertically opposite angles are formed when two straight lines intersect. These angles are always equal to each other.

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Regular polygons have all sides of equal length and all interior angles of equal size. The sum of interior angles depends on the number of sides nn.

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The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

📐Formulae

Sum of interior angles of an nn-sided polygon = (n−2)×180∘(n - 2) \times 180^\circ

Sum of exterior angles of any convex polygon = 360∘360^\circ

Individual exterior angle of a regular nn-sided polygon = 360∘n\frac{360^\circ}{n}

Individual interior angle of a regular nn-sided polygon = 180∘−exterior angle180^\circ - \text{exterior angle}

Area of a Triangle = 12×base×height\frac{1}{2} \times \text{base} \times \text{height}

Area of a Parallelogram = base×height\text{base} \times \text{height}

💡Examples

Problem 1:

In an isosceles triangle, the vertex angle is 40∘40^\circ. Find the size of the two base angles.

Solution:

70∘70^\circ each

Explanation:

The sum of angles in a triangle is 180∘180^\circ. Subtract the vertex angle: 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ. Since it is an isosceles triangle, the two base angles are equal. Therefore, 140∘÷2=70∘140^\circ \div 2 = 70^\circ.

Problem 2:

Calculate the size of one interior angle of a regular hexagon.

Solution:

120∘120^\circ

Explanation:

A hexagon has n=6n=6 sides. First, find the exterior angle: 360∘÷6=60∘360^\circ \div 6 = 60^\circ. Since the interior and exterior angles lie on a straight line, the interior angle is 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ.

Problem 3:

Two parallel lines are intersected by a transversal. If an alternate angle is 55∘55^\circ, what is the size of its corresponding co-interior angle?

Solution:

125∘125^\circ

Explanation:

If the alternate angle is 55∘55^\circ, the angle adjacent to it on the straight line is 180∘−55∘=125∘180^\circ - 55^\circ = 125^\circ. This adjacent angle is equal to the co-interior partner because co-interior angles must sum to 180∘180^\circ (180∘−55∘=125∘180^\circ - 55^\circ = 125^\circ).

Problem 4:

In the following diagram, lines L1L1 and L2L2 are parallel. Calculate the value of angle xx.

Two parallel lines L1 and L2 intersected by a transversal. One angle is 115 degrees, and the unknown angle x is alternate to it.

Solution:

  1. Identify that the angle 115∘115^\circ and the angle adjacent to xx are corresponding angles if the transversal is considered.
  2. Alternatively, the angle 115∘115^\circ and the angle co-interior to xx sum to 180∘180^\circ.
  3. Let the co-interior angle be yy: y=180∘−115∘=65∘y = 180^\circ - 115^\circ = 65^\circ.
  4. Since xx and yy are on a straight line: x=180∘−65∘=115∘x = 180^\circ - 65^\circ = 115^\circ.
  5. More simply, xx and the 115∘115^\circ angle are alternate exterior angles, or xx is corresponding to the vertically opposite angle of 115∘115^\circ. Therefore, x=115∘x = 115^\circ.

Explanation:

Using the properties of parallel lines, we can determine that xx is equivalent to the given angle because they are in corresponding positions.

Problem 5:

Calculate the perimeter of the composite shape shown below, which consists of a rectangle and a right-angled triangle.

A composite shape made of a rectangle and a right-angled triangle attached to its side.

Solution:

  1. Identify the missing side lengths. The bottom side of the rectangle is 1212 cm.
  2. The vertical side of the rectangle is 55 cm, which is the same as the base of the triangle shown.
  3. The hypotenuse of the triangle is given as 1313 cm.
  4. Total Perimeter = sum of all outer boundaries.
  5. P=12+5+12+13=42P = 12 + 5 + 12 + 13 = 42 cm.

Explanation:

To find the perimeter, we sum only the lengths of the external boundaries of the composite shape.