Euclidean Geometry Riders: Proving It, With Reasons (Grade 10)

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Grade 10 · Euclidean Geometry

MEMORANDUM

Name:

Class:

Date:

Mark

/ 30

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Write down the reason you would quote for the statement “x = 65°” if the two angles are on opposite sides of a transversal, between two parallel lines AB and CD.
(2)
alt ∠s; AB ∥ CD✓✓ (2)
2
Why is a correct angle written with no reason usually worth only half the marks?
(2)
The marking guideline awards a mark for the statement and a separate mark for the reason; without the reason the second mark cannot be given. ✓✓(2)
3
In △PQR, ST ∥ QR with S on PQ and T on PR. P̂ = 44° and Q̂ = 76°. Determine, with reasons, the size of PŜT.
(3)
PŜT = 76° ✓ (corresp ∠s; ST ∥ QR) ✓✓(3)
4
Using the same triangle, determine, with a reason, the size of R̂.
(3)
R̂ = 180° − 44° − 76° ✓ = 60° ✓ (∠s of a △) ✓(3)
5
List the FOUR conditions for congruency and state which one applies only to right-angled triangles.
(4)
SSS, SAS, AAS and RHS ✓✓✓; RHS applies only to right-angled triangles ✓(4)
6
ABCD is a parallelogram with diagonal BD drawn. Prove that △ABD ≡ △CDB.
(4)
AB = CD ✓ (opp sides of a parm); AD = CB ✓ (opp sides of a parm); BD = DB ✓ (common); ∴ △ABD ≡ △CDB ✓ (SSS)(4)
7
State THREE different ways of proving that a quadrilateral is a parallelogram.
(3)
Any three of: both pairs of opp sides parallel ✓; both pairs of opp sides equal ✓; both pairs of opp ∠s equal ✓; one pair of opp sides equal and parallel; diagonals bisect each other(3)
8
In △ABC, D and E are the midpoints of AB and AC. If DE = 7,5 cm and AĈB = 58°, determine BC and AÊD, with reasons.
(4)
BC = 2 × 7,5 = 15 cm ✓✓ (midpt theorem); AÊD = 58° ✓ (corresp ∠s; DE ∥ BC) ✓(4)
9
Explain why you may NOT use the reason “opp sides of a parm” inside a proof whose conclusion is that the shape is a parallelogram.
(2)
It assumes the very thing you are asked to prove ✓. Properties of a parallelogram may only be used once the shape has been established as one ✓.(2)
10
PQRS is a quadrilateral with PQ ∥ SR and PQ = SR. Prove that PQRS is a parallelogram.
(3)
PQ ∥ SR ✓ (given); PQ = SR ✓ (given); ∴ PQRS is a parallelogram ✓ (one pair of opp sides equal and parallel)(3)

TOTAL: 30 marks