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Angle rules and parallel lines

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Basic Angle Facts

  • Angles at a point sum to 360°: a + b + c + d = 360°.
  • Angles on a straight line sum to 180°: a + b + c = 180°.
  • Vertically opposite angles are equal: when two lines cross, the angles opposite each other are the same.
  • A right angle is 90°, often marked with a small square.
  • Complementary angles sum to 90°; supplementary angles sum to 180°.

Angles around a point

Angles around a point100°120°140°

Parallel Lines and Transversals

  • Parallel lines are marked with matching arrows and never meet.
  • A transversal intersects two or more lines at distinct points; those lines need not be parallel. The equal-angle rules below require parallel lines.
  • Corresponding angles are equal: they occupy matching positions on the parallel lines and the transversal.
  • Alternate angles are equal: they lie on opposite sides of the transversal and between the parallel lines (Z shape).
  • Co-interior angles (also called allied or interior angles) sum to 180°: they lie between the parallel lines on the same side of the transversal (C shape).
  • Do not confuse alternate (equal) with co-interior (sum to 180°).

Alternate interior angles

Alternate interior angles70°70°These marked angles are equal

Triangles

  • The interior angles of a triangle sum to 180°: a + b + c = 180°.
  • An equilateral triangle has all sides equal and all angles 60°.
  • An isosceles triangle has two equal sides and two equal base angles.
  • A right-angled triangle has one 90° angle; the other two angles sum to 90°.
  • The exterior angle of a triangle equals the sum of the two opposite interior angles.
  • Never assume a triangle is isosceles just from its appearance; check given markings or information.

Quadrilaterals

  • The interior angles of a quadrilateral sum to 360°.
  • A square has four equal sides and four right angles.
  • A rectangle has opposite sides equal and four right angles.
  • A parallelogram has opposite sides parallel and equal, and opposite angles equal.
  • A rhombus has all sides equal, opposite angles equal, and diagonals bisect at right angles.
  • A trapezium has one pair of parallel sides.
  • A kite has two pairs of adjacent equal sides and one pair of equal opposite angles.

Polygons: Interior Angles

  • For a simple polygon with n sides, the sum of interior angles is (n − 2) × 180°.
  • For a regular polygon (all sides and angles equal), each interior angle is \frac{(n-2) \times 180°}{n}.
  • A triangle (n=3) gives 180°; a quadrilateral (n=4) gives 360°; a pentagon (n=5) gives 540°.
  • To find the number of sides from an interior angle, rearrange the formula: n = \frac{360°}{180° - \text{interior angle}} (for regular polygons).
  • Be careful not to use the interior angle formula when you need the exterior angle.

Regular hexagon

Regular hexagon4 units120°

Polygons: Exterior Angles

  • For a convex polygon, one exterior turn at each vertex sums to 360°. For concave simple polygons use signed turning angles.
  • For a regular polygon, each exterior angle is \frac{360°}{n}, where n is the number of sides.
  • To find the number of sides given an exterior angle e, use n = \frac{360°}{e}.
  • The interior and exterior angles at a vertex are supplementary: they sum to 180°.
  • If a regular polygon has an exterior angle of 40°, it has 9 sides.

Reasoning and Justification

  • Always give a reason for each step when finding angles, using correct terminology.
  • Common reasons: 'angles on a straight line sum to 180°', 'angles at a point sum to 360°', 'vertically opposite angles are equal', 'corresponding angles are equal', 'alternate angles are equal', 'co-interior angles sum to 180°', 'interior angles of a triangle sum to 180°', 'interior angles of a quadrilateral sum to 360°', 'exterior angles sum to 360°'.
  • State the angle fact you used, not just the calculation.
  • If a diagram shows parallel lines, you may use the parallel line properties; otherwise, do not assume they are parallel.
  • When proving two angles equal, identify the relevant angle relationship and write a clear chain of reasoning.

Common Mistakes to Avoid

  • Assuming lines are parallel without markings or given information.
  • Confusing alternate (equal) with co-interior (sum to 180°).
  • Using the interior angle formula when the exterior angle is needed.
  • Assuming a triangle is isosceles from appearance only.
  • Omitting reasons in angle calculations – you lose marks even if the final answer is correct.
  • Forgetting that exterior turns around a convex polygon sum to 360°.

Corresponding angles

  • Corresponding angles occupy the same relative position at the two intersections.
  • When the two lines are parallel, corresponding angles are equal.

Corresponding angles

Corresponding angles70°70°These marked angles are equal

Co-interior angles

  • Co-interior angles lie between the parallel lines on the same side of the transversal.
  • For parallel lines their sum is 180°. They are equal only in the special case of two right angles.

Co-interior angles

Co-interior angles110°70°Same-side interior angles sum to 180°

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 63 câu hỏi. Đăng ký để xem tất cả.

  1. 1.What is the sum of angles at a point?

    Easy
    • A180°
    • B90°
    • C360°
    • D270°

  2. 2.What is the sum of angles on a straight line?

    Easy
    • A180°
    • B360°
    • C90°
    • D270°

  3. 3.Vertically opposite angles are always equal.

    Easy

    True or false?

  4. 4.Two angles are alternate angles on parallel lines. One is 70°. What is the other?

    Easy
    • A70°
    • B110°
    • C180°
    • D90°

  5. 5.Two angles are co-interior angles on parallel lines. One is 70°. What is the other?

    Easy
    • A110°
    • B70°
    • C180°
    • D90°

  6. 6.Which of the following statements are true for parallel lines cut by a transversal? (select all that apply) Select the relationships that hold in general, rather than only in a special right-angle case.

    Medium
    • ACorresponding angles are equal.
    • BAlternate angles sum to 180°.
    • CCo-interior angles sum to 180°.
    • DVertically opposite angles are equal.
    • ECorresponding angles sum to 180°.

  7. 7.Two parallel lines are cut by a transversal. Match each angle description with its value.

    Easy
    • Alternate interior angle to 70°
    • Corresponding angle to 50°
    • Co-interior angle to 40°
    • 70°
    • 50°
    • 140°

  8. 8.Arrange the steps to find the sum of interior angles of a hexagon.

    Medium
    • Use the formula: sum = (n - 2) × 180°
    • Substitute n = 6
    • Calculate (6 - 2) × 180° = 720°

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