Practice Maths

Topic Review — Geometric Reasoning

Mixed Practice — L34 – L36

This review covers all lessons in the Geometric Reasoning topic. Try each question before checking your answers.

Review Questions

  1. Angle relationships fluency. Fluency

    1. Find the complement of 34°.
    2. Find the supplement of 97°.
    3. Two parallel lines are cut by a transversal. A co-interior angle is 74°. Find its co-interior partner.
    4. Two lines intersect. One angle is 48°. Find the vertically opposite angle and the two adjacent angles.
    5. In a triangle, two angles are 55° and 78°. Find the third angle.
    6. The exterior angle of a triangle is 142°. One non-adjacent interior angle is 89°. Find the other.

  2. Triangle and quadrilateral angles. Fluency

    1. An isosceles triangle has an apex angle of 28°. Find the base angles.
    2. A parallelogram has one angle of 53°. Find the other three angles.
    3. A rhombus has one angle of 118°. Find all four angles.
    4. In a trapezium, one pair of co-interior angles is 91° and y°. Find y.
    5. The four angles of a quadrilateral are 2x, 3x, 4x, and 51°. Find x and each angle.
    6. In an equilateral triangle, one angle is expressed as (3m − 3)°. Find m.

  3. Congruence and geometric proof. Understanding

    1. Name the congruence test: two triangles with two equal angles and the included side equal.
    2. ABCD is a parallelogram. Without measuring, explain why ▵ABC and ▵CDA are congruent, naming the test used.
    3. Prove that the base angles of an isosceles triangle are equal, using a formal proof with a line of symmetry (bisect the apex angle).
    4. In ▵XYZ, XY = XZ. M is the midpoint of YZ. Prove XM ⊥ YZ using congruent triangles.

  4. Quadrilateral properties and reasoning. Understanding

    1. A quadrilateral has diagonals that bisect each other. What type(s) could it be?
    2. PQRS is a rectangle. The diagonal PR = 5k − 3 and SQ = 3k + 9. Find k and the diagonal length.
    3. In a rhombus with side 13 cm and one diagonal of 10 cm, use Pythagoras to find the other diagonal.
    4. Explain the difference between a rhombus and a square. What additional property must a rhombus have to also be a square?

  5. Mixed geometric reasoning problem solving. Problem Solving

    1. Lines AB and CD are parallel. Point P is between them. ∠PAB = 72° and ∠PCD = 38°. Find ∠APD, showing all steps with reasons. (Hint: draw a line through P parallel to AB and CD.)
    2. In quadrilateral ABCD, AB = BC, AD = DC, and ∠B = 90°. Identify the type of quadrilateral, then find ∠A if ∠D = 130°.
    3. A triangular plot of land has angles in the ratio 1 : 2 : 3. A surveyor extends the shortest side, creating an exterior angle. Find the exterior angle at the vertex with the largest interior angle, then verify using the exterior angle theorem.
    4. ABCD is a square with side 8 cm. E is the midpoint of BC. Prove ▵ABE ≡ ▵DCE and find the length AE.
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