Year 7 Mathematics: Angles, Geometry & Pythagoras' Theorem Review

Point

A location in space that has no length or width. Points are represented using a dot labelled with a capital letter.

Interval

A line joining two points. Intervals are named using a bar and the two end point labels.

Ray

A line that starts at a point and continues to infinity. Rays are named using an arrow, which indicates the direction of the ray.

Line

A line that continues to infinity in both directions. Lines are named using a double-headed arrow.

Perpendicular lines

Two lines that meet to form a right angle. Right angles are labelled with a small box where the lines meet. The symbol ⊥ means 'is perpendicular to'.

Parallel lines

Two lines that have the same slope and are always the same distance apart so that they never meet. Parallel lines are labelled with matching arrowheads. The symbol ∥ means 'is parallel to'.

Complementary angles

Two angles that add up to 90 degrees.

Supplementary angles

Two angles that add up to 180 degrees.

Vertically opposite angles

Angles that are opposite each other when two lines cross. They are equal in measure.

Angles at a point

The sum of angles around a point is 360 degrees.

Corresponding angles

Angles that are in the same position at each intersection where a straight line crosses two others.

Alternate angles

Angles that are on opposite sides of the transversal but inside the two lines.

Co-interior angles

Angles that are on the same side of the transversal and inside the two lines.

Transversal

A line that crosses at least two other lines.

Slope

The steepness of a line, calculated as the rise over the run.

Right angle

An angle that measures 90 degrees.

Acute angle

An angle that measures less than 90 degrees.

Obtuse angle

An angle that measures more than 90 degrees but less than 180 degrees.

Straight angle

An angle that measures exactly 180 degrees.

Reflex angle

An angle that measures more than 180 degrees but less than 360 degrees.

Angle measurement

The size of an angle measured in degrees.

Geometrical properties

Characteristics that define the relationships between angles and shapes.

Learning Intentions

In this activity we will investigate solving problems involving angles using reasoning.

Success Criteria

I can - Calculate and use angles at a point.

Classifying triangles by side lengths

Triangles can be classified based on the lengths of their sides.

Classifying triangles by size of the angles

Triangles can be classified based on the sizes of their angles.

Example 1

Calculate the size of ∠R.

Example 2

Find the value of 𝑎 in these isosceles triangles.

Example 3

Find the value of 𝑎.

Example 4

Calculate the value of the unknown marked angle in each diagram. These diagrams are not to scale.

Example 5

Write an equation for each triangle and solve it to find the value of the pronumeral.

Question 1

Give the common name of a triangle with these properties: a. One right angle.

Question 2

Use the angles sum of a triangle to help find the unknown angle in these triangles.

Question 3

Find the value of 𝑎.

Question 4

Write an equation for each triangle and solve it to find the value of the pronumeral and the unknown angles.

Question 5

Use all your knowledge of angles to find the value of each unknown marked in the diagrams.

Quadrilateral

A polygon with four sides.

Convex Quadrilateral

A quadrilateral with all vertices pointing outwards.

Non-convex Quadrilateral

A quadrilateral with one vertex pointing inwards.

Angle Sum of Quadrilaterals

The angle sum of a quadrilateral is 2 × 180° = 360°.

Perimeter of a Rectangle

The total distance around the rectangle, calculated as 2 × (length + width).

Rhombus

A quadrilateral with all sides of equal length.

Kite

A quadrilateral with two pairs of equal sides.

Diagonal Intersections in Squares, Rhombuses, and Kites

The diagonals intersect at right angles.

Special Quadrilaterals with Equal Length Sides

Quadrilaterals that have all sides of equal length include squares and rhombuses.

Special Quadrilaterals with Parallel Sides

Quadrilaterals that have two pairs of parallel sides include rectangles, parallelograms, and squares.

Diagonals Meeting at Right Angles

Quadrilaterals that have diagonals meeting at right angles include kites and squares.

Diagonals of Equal Length

Quadrilaterals that have diagonals of equal length include rectangles and squares.

Quadrilaterals with Four Right Angles

Quadrilaterals that have four right angles include rectangles and squares.

Two Pairs of Equal Opposite Angles

Quadrilaterals that have two pairs of equal opposite angles include parallelograms, rectangles, and rhombuses.

Diagonals that Bisect Each Other

Quadrilaterals that have diagonals that bisect each other include parallelograms, rectangles, rhombuses, and squares.

Identical Markings on Angles

Angles with identical markings are the same size.

Parallel Sides Markings

Sides with the same number of arrows are parallel.

Equal Length Markings

Identical number of short strokes indicate equal lengths.

Finding Missing Angles

You can use the angle sum to find a missing angle without using a protractor.

Identifying Quadrilaterals

Identifying and classifying squares, rectangles, rhombuses, parallelograms, kites, and trapeziums.

Finding Unknown Angles

Finding unknown angles using the sum of angles in a quadrilateral.

Classifying Quadrilaterals

Classifying quadrilaterals according to their properties.

Pythagoras' theorem

A mathematical principle used to determine whether a triangle is right-angled.

Finding the Hypotenuse

Applying Pythagoras' theorem to find the length of the hypotenuse in a right-angled triangle.

Finding the Shorter Side

Applying Pythagoras' theorem to find the length of the shorter side in a right-angled triangle.

Problem Solving

Using Pythagoras' theorem to solve practical problems.

Pythagorean Triads

Justifying whether a set of 3 integers forms a Pythagorean triad.

Equilateral triangle

A triangle with all three sides of equal length. All angles are 60^[\smallsetminus]circ .

Isosceles triangle

A triangle with two sides of equal length. The angles opposite the equal sides are also equal.

Scalene triangle

A triangle with all three sides of different lengths. All angles are also different.

Acute-angled triangle

A triangle where all three angles are acute (less than 90^[\smallsetminus]circ).

Obtuse-angled triangle

A triangle with one obtuse angle (greater than 90^[\smallsetminus]circ).

Right-angled triangle

A triangle with one right angle (90^[\smallsetminus]circ). The side opposite the right angle is called the hypotenuse.

Area of a triangle

Calculated as A = \frac{1}{2} \times \text{base} \times \text{height}.

Area of a rectangle

Calculated as A = \text{length} \times \text{width}.

Area of a square

Calculated as A = \text{side} \times \text{side} = \text{side}^2.

Perimeter of a triangle

The sum of the lengths of all three sides.

Perimeter of a square

Calculated as P = 4 \times \text{side}.