Angles in Triangles and Quadrilaterals: Full Guide — Worksheet & Memo

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Grade 8–9 · Geometry of 2D Shapes

Angles in Triangles and Quadrilaterals: Full Guide

MARKING GUIDELINE

Marks

Duration

1 hour

Questions

Name:

Class:

Date:

Mark

/ 37

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
The angles of a triangle add up to …
(1)
A)180°✓
B)360°
C)540°
D)90°
Answer: A — True for every triangle.
- B — that is a quadrilateral
- C — that is a pentagon
- D — that is a right angle
1.2
The angles of a quadrilateral add up to …
(1)
A)540°
B)360°✓
C)270°
D)180°
Answer: B — A quadrilateral splits into TWO triangles.
- A — that is a pentagon
- C — that is three right angles, not four
- D — that is one triangle
1.3
Two angles of a triangle are 55° and 65°. The third is …
(1)
A)60°✓
B)240°
C)120°
D)70°
Answer: A — 180° − 55° − 65° = 60°.
- B — used 360°
- C — that is the sum of the two given angles
- D — arithmetic slip
1.4
A triangle with all three sides equal is called …
(1)
A)scalene
B)right-angled
C)isosceles
D)equilateral✓
Answer: D — Equilateral means equal sides, and all its angles are 60°.
- A — scalene has no equal sides
- B — that describes an angle, not the sides
- C — isosceles has exactly two equal sides
1.5
The base angles of an ISOSCELES triangle are …
(1)
A)always 60°
B)equal to each other✓
C)always right angles
D)always different
Answer: B — Equal sides are opposite equal angles.
- A — that is an EQUILATERAL triangle
- C — only in a right-angled isosceles triangle is one angle 90°
- D — they are equal
1.6
The EXTERIOR angle of a triangle equals …
(1)
A)the largest interior angle
B)the adjacent interior angle
C)180°
D)the sum of the two opposite interior angles✓
Answer: D — Exterior angle = sum of the two interior opposite angles.
- A — not necessarily
- B — it is SUPPLEMENTARY to the adjacent angle
- C — an exterior angle is less than 180°
1.7
Which quadrilateral has diagonals that bisect each other AT RIGHT ANGLES but are NOT equal?
(1)
A)square
B)rectangle
C)trapezium
D)rhombus✓
Answer: D — A rhombus has perpendicular bisecting diagonals of different lengths.
- A — a square's are equal AND perpendicular
- B — a rectangle's diagonals are equal and not perpendicular
- C — a trapezium's do not bisect each other
1.8
Opposite angles of a PARALLELOGRAM are …
(1)
A)complementary
B)supplementary
C)equal✓
D)right angles
Answer: C — Opposite angles are equal; CO-INTERIOR angles are supplementary.
- A — they add to 180°, not 90°
- B — that is true of CO-INTERIOR angles
- D — only in a rectangle
1.9
Three angles of a quadrilateral are 80°, 95° and 110°. The fourth is …
(1)
A)85°
B)95°
C)75°✓
D)285°
Answer: C — 360° − 80° − 95° − 110° = 75°.
- A — added the three given angles as 275° instead of 285°
- B — copied one of the given angles
- D — that is the SUM of the three given angles, not the fourth
1.10
Every square is a rectangle because …
(1)
A)it has four right angles✓
B)its diagonals are equal
C)it is a quadrilateral
D)all its sides are equal
Answer: A — A rectangle is defined by its four right angles, which a square has.
- B — equal diagonals follow from the right angles
- C — not every quadrilateral is a rectangle
- D — equal sides make it a rhombus, not a rectangle

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

[10 MARKS]

In △ABC below the three interior angles are given as x, 2x and 3x. The diagram is not drawn to scale.

2.1
Calculate the value of x.
(3)
x + 2x + 3x = 180° (sum of the angles of a triangle) (1)
6x = 180° (1)
x = 30° (1)
2.2
Hence write down the size of each angle, and state what kind of triangle ABC is. Give a reason.
(4)
Â = 30°, B̂ = 60°, Ĉ = 90° (2)
It is a right-angled triangle (1)
because one of its angles is 90° (1)
2.3
In a different triangle the three angles are equal. Determine the size of each angle and name the triangle.
(3)
3x = 180°, so x = 60° (2)
It is an equilateral triangle (1)

Question 3

[7 MARKS]

In the diagram below, one side of the triangle has been produced to form the exterior angle y. The diagram is not drawn to scale.

3.1
Determine, with a reason, the size of y.
(3)
y = 55° + 63° (exterior angle = sum of the opposite interior angles) (1)
= 118° (2)
3.2
Hence determine the size of the third interior angle of the triangle, in two different ways.
(4)
Method 1: 180° − 118° = 62° (angles on a straight line) (2)
Method 2: 180° − 55° − 63° = 62° (angle sum of a triangle) (2)

Question 4

[10 MARKS]

Answer the following questions about quadrilaterals.

4.1
The four angles of a quadrilateral are x, (x + 20)°, 2x and (2x + 40)°. Calculate the value of x.
(3)
x + (x + 20) + 2x + (2x + 40) = 360° (sum of the angles of a quadrilateral) (1)
6x + 60° = 360° (1)
x = 50° (1)
4.2
Hence write down the sizes of the four angles and check your answer.
(3)
50° ; 70° ; 100° ; 140° (2)
50 + 70 + 100 + 140 = 360° ✓ (1)
4.3
PQRS is a parallelogram with P̂ = 70°. Determine, with reasons, the sizes of Q̂ and R̂.
(4)
Q̂ = 180° − 70° = 110° (co-interior angles, PS ∥ QR) (2)
R̂ = 70° (1)
(opposite angles of a parallelogram are equal) (1)

TOTAL: 37 marks

This question paper consists of 4 questions.