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Shapes & angles

Describe shapes using their properties and reason about angles. Use stated relationships rather than the appearance of a drawing to find an unknown angle.

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Geometric shapes and angle markings illustrate spatial relationships.

What you’ll learn

  • Describe a shape by relevant sides, corners, and relationships.
  • Use degrees to describe an angle as a turn.
  • Find an unknown angle from a stated angle relationship.

Properties define the shape

A shape keeps its properties when it is moved or rotated. A square remains a square when drawn on a corner. Use side lengths, parallel sides, right angles, and other stated properties to classify it.

Angles describe turns and relationships

A full turn is 360°, and a straight angle is 180°. In ordinary plane geometry, a triangle’s interior angles total 180°. These relationships let you reason about an unknown value.

A mistake to avoid

A diagram may not be drawn to scale. Do not assume two angles or lengths are equal because they look alike; use labels, markings, or facts given in the question.

Worked example

Find the third angle of a triangle

If two interior angles are 50° and 60°, the third is 180° − 50° − 60° = 70°. Check that 50° + 60° + 70° = 180°.

Learning paths that include this skill

Choose the route that fits your goal. The same skill can be useful in more than one subject or stage of learning.

Primary / elementary foundationsPrimary mathBuild a connected foundation in number, calculation, fractions, measurement, and data. Use the skills in order or return to the first idea that needs attention.Counting & comparing · Place value · Addition & subtraction · Multiplication · +6 moreSecondary geometry and trigonometryGeometry & trigonometryConnect measurement and geometric reasoning to triangle ratios and the unit circle. Keep the meaning of each angle, length, and coordinate visible.Shapes & angles · Area & volume · Linear equations · Geometric reasoning & proofs · +4 moreGoal-led refresherAdult math refresherRefresh the mathematics behind a real task, from a percentage or measurement to an equation or graph. Use familiar goals and revisit only the skills you need.Place value · Addition & subtraction · Multiplication · Long division · +12 more

Keep exploring

Area & volumeDistinguish length, area, and volume before using a formula. Explain rectangular area and cuboid volume through equal units, then check the result’s dimensions.Geometric reasoning & proofsBuild a short geometric argument from given facts and valid relationships. Distinguish what the diagram suggests from what the reasoning establishes.Right-triangle ratiosChoose the reference angle and identify the right sides before using sine, cosine, or tangent. Connect each ratio to similar right triangles.

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