Properties of Quadrilaterals & Their Proofs (Grade 10) — Worksheet & Memo

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Grade 10 · Euclidean Geometry

Properties of Quadrilaterals & Their Proofs (Grade 10)

MARKING GUIDELINE

Marks

Duration

45 minutes

Questions

Name:

Class:

Date:

Mark

/ 30

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Which property does a RECTANGLE have that a general parallelogram does NOT?
(1)
A)its diagonals are equal✓
B)its opposite sides are parallel
C)its diagonals bisect each other
D)its opposite angles are equal
Answer: A — Equal diagonals is the extra property a rectangle gains; the other three belong to every parallelogram.
- B — every parallelogram has both pairs of opposite sides parallel
- C — every parallelogram's diagonals bisect each other
- D — every parallelogram has equal opposite angles
1.2
Which property does a RHOMBUS have that a rectangle does NOT?
(1)
A)its diagonals cross at right angles✓
B)its diagonals are equal
C)its opposite angles are equal
D)its opposite sides are parallel
Answer: A — Four equal sides force the diagonals of a rhombus to be perpendicular; a rectangle's are not.
- B — EQUAL diagonals is the rectangle's property, not the rhombus's
- C — both shapes have equal opposite angles — both are parallelograms
- D — both shapes have opposite sides parallel
1.3
A quadrilateral with exactly ONE pair of parallel sides is a …
(1)
A)trapezium✓
B)parallelogram
C)kite
D)rhombus
Answer: A — A trapezium has one pair of parallel sides; a parallelogram has two.
- B — a parallelogram has TWO pairs of parallel sides
- C — a kite has no parallel sides — it has two pairs of adjacent equal sides
- D — a rhombus has two pairs of parallel sides
1.4
In parallelogram ABCD, ∠A = 70°. Then ∠B = …
(1)
A)110°✓
B)70°
C)20°
D)290°
Answer: A — ∠A and ∠B are co-interior between the parallel sides AD and BC, so they are supplementary.
- B — that is ∠C, the OPPOSITE angle, which is the equal one
- C — subtracted from 90° instead of from 180°
- D — subtracted from 360° instead of from 180°
1.5
Which ONE of these proves that a quadrilateral is a parallelogram?
(1)
A)both pairs of opposite sides are equal✓
B)one pair of opposite sides is equal
C)the diagonals are equal
D)one pair of opposite angles is equal
Answer: A — Both pairs of opposite sides equal is one of the standard tests for a parallelogram.
- B — one pair must be equal AND parallel; equal on its own is not enough
- C — equal diagonals give a RECTANGLE, and only once it is already a parallelogram
- D — BOTH pairs of opposite angles must be equal
1.6
The diagonals of a KITE …
(1)
A)cross at 90°, and the long one bisects the other✓
B)bisect each other
C)are equal
D)are both axes of symmetry
Answer: A — A kite has one axis of symmetry — the long diagonal — and it cuts the other diagonal in half at right angles.
- B — only ONE of the two diagonals is bisected
- C — equal diagonals belong to the rectangle
- D — only the LONG diagonal is an axis of symmetry
1.7
The interior angles of any quadrilateral add up to …
(1)
A)360°✓
B)180°
C)540°
D)720°
Answer: A — A quadrilateral splits into two triangles, and 2 × 180° = 360°.
- B — that is the sum for a TRIANGLE
- C — that is the sum for a pentagon
- D — that is the sum for a hexagon
1.8
In rhombus PQRS the diagonals meet at T. Then △PTQ is …
(1)
A)right-angled✓
B)equilateral
C)obtuse-angled
D)scalene with no special angle
Answer: A — The diagonals of a rhombus cross at 90°, so ∠PTQ = 90°.
- B — PT, TQ and PQ are not all equal
- C — ∠PTQ is exactly 90°, which is not obtuse
- D — ∠PTQ is exactly 90°, which IS a special angle
1.9
A square is …
(1)
A)both a rectangle and a rhombus✓
B)a rectangle but not a rhombus
C)a rhombus but not a rectangle
D)neither a rectangle nor a rhombus
Answer: A — A square has four right angles, which makes it a rectangle, and four equal sides, which makes it a rhombus.
- B — four equal sides makes it a rhombus as well
- C — four right angles makes it a rectangle as well
- D — it satisfies the definition of both
1.10
In parallelogram ABCD, AB = 3x − 4 and DC = x + 8. Then x = …
(1)
A)6✓
B)3
C)4
D)12
Answer: A — Opposite sides of a parallelogram are equal, so 3x − 4 = x + 8 and 2x = 12.
- B — divided the 12 by 4 instead of by 2
- C — dropped the −4 and solved 3x = x + 8
- D — reached 2x = 12 but never divided by the 2

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[9 MARKS]

ABCD is a parallelogram. It is given that Â = (3x + 10)° and Ĉ = (5x − 30)°.

2.1
Give a reason why Â = Ĉ.
(2)
The OPPOSITE angles of a parallelogram are equal (reason 2)
2.2
Calculate the value of x.
(3)
3x + 10 = 5x − 30 (1)
40 = 2x (1)
x = 20 (1)
2.3
Hence calculate the size of Â and of B̂, with reasons.
(4)
Â = 3(20) + 10 = 70° (1)
Â + B̂ = 180° (1) — co-interior angles, AD ∥ BC (reason 1)
B̂ = 110° (1)

Question 3

[11 MARKS]

Answer the questions below on the properties of quadrilaterals.

3.1
State THREE properties that a rhombus has but a general parallelogram does not.
(3)
All four sides are equal (1)
The diagonals cross at right angles (1)
The diagonals bisect the interior angles (1)
3.2
A quadrilateral has one pair of opposite sides both parallel AND equal. Prove that it must be a parallelogram.
(3)
The equal, parallel sides make the two triangles formed by a diagonal congruent (1)
So the other pair of sides is equal too (1)
A quadrilateral with both pairs of opposite sides equal is a parallelogram (1)
3.3
A quadrilateral has diagonals that bisect each other at right angles but are NOT equal in length. Name the quadrilateral and give a reason.
(3)
A rhombus (1) — bisecting diagonals make it a parallelogram (1), and perpendicular diagonals make it a rhombus; equal diagonals would have made it a square (1)
3.4
Explain why every square is a rectangle, but not every rectangle is a square.
(2)
A square has four right angles, which is the definition of a rectangle (1); a rectangle need not have four equal sides (1)

TOTAL: 30 marks

This question paper consists of 3 questions.