Practice Maths

Topic Review — Quadrilaterals and Geometric Reasoning

Mixed Practice — L54 – L56

This review covers all lessons in the Quadrilaterals and Geometric Reasoning topic. Try each question before checking your answers. Give a reason for every step in angle-finding questions.

Review Questions

  1. Properties of quadrilaterals. Fluency

    1. Name the quadrilateral: all four sides equal, all angles 90°.
    2. Name the quadrilateral: all four sides equal, opposite angles equal, no right angles.
    3. A parallelogram has one angle of 68°. Find the other three angles and give reasons.
    4. In a rhombus, the diagonals are 16 cm and 12 cm. Find the side length of the rhombus (use Pythagoras).
    5. ABCD is a rectangle. AC is a diagonal of length 26 cm. Find the length of each half-diagonal from the centre.
    6. Identify all the properties that a square and a rhombus share.

  2. Angles in polygons. Fluency

    1. Find the interior angle sum of a nonagon (9 sides).
    2. Find each interior angle and each exterior angle of a regular hexagon.
    3. A regular polygon has exterior angles of 36°. How many sides does it have?
    4. A pentagon has four angles: 95°, 110°, 120°, 105°. Find the fifth angle.
    5. A regular polygon has an interior angle of 156°. How many sides does it have?
    6. Find the interior angle sum of a 12-sided polygon.

  3. Geometric reasoning with parallel lines and triangles. Understanding

    1. Two parallel lines are cut by a transversal. One angle is 74°. Find: (i) the alternate angle (ii) the co-interior angle (iii) the corresponding angle. Give a reason for each.
    2. Triangle ABC is isosceles with AB = AC. ∠BAC = 52°. Find ∠ABC and ∠ACB with reasons.
    3. Two lines intersect. One angle is 118°. Find all four angles at the intersection with reasons.
    4. ABCD is a parallelogram with ∠DAB = (4x − 10)° and ∠ABC = (2x + 40)°. Find x and all four angles with reasons.

  4. Geometric proofs and structured arguments. Understanding

    1. Write a structured geometric argument to show that the exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
    2. PQRS is a rectangle. A diagonal PR is drawn. ∠QPR = 35°. Find ∠PRS. Give a reason for each step.
    3. ABCD is a rhombus. Show that its diagonals bisect the angles at each vertex.
    4. Prove or disprove: “If a quadrilateral has all four sides equal, it must have all four angles equal.” Provide a geometric argument or counterexample.

  5. Mixed geometric problem solving. Problem Solving

    1. A designer is tiling a floor using regular hexagons and equilateral triangles that share edges. Show that these two shapes can fit together without gaps by demonstrating the angles at a shared meeting point sum to 360°.
    2. In quadrilateral ABCD, AB ∥ DC. ∠DAB = 55° and ∠ABC = 80°. Find ∠BCD and ∠CDA. Name this quadrilateral with a reason.
    3. A diagonal of a rectangle creates a triangle. If the rectangle has length 15 cm and width 8 cm, find: (i) the length of the diagonal (ii) both acute angles of the triangle formed (to 1 decimal place, using known ratios or trigonometry if studied).
    4. A student says: “I can draw a polygon where the interior angle sum is 630°.” Is this possible? If yes, how many sides does the polygon have? If no, explain why not.
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