What type of angle is shown?
Worked Solution & Answer
Compare the angle with a right angle (90°) and a straight line (180°).
Reflex
Reflex
Answer: Reflex
Angles — Notes & Worked Examples Show / Hide
Explanation:
Memorise the five core angle facts and their exact wording for “give a reason” questions: (1) Angles on a straight line add up to 180°; (2) Angles around a point add up to 360°; (3) Vertically opposite angles are equal; (4) Angles in a triangle add up to 180°; and (5) Angles in a quadrilateral add up to 360°.
In an isosceles triangle, the two equal sides are marked with tick marks, and the base angles opposite those equal sides are equal.
Example 1: Find the size of angle x on the straight line below.
• Angles on a straight line add up to 180°:
• 48° + 65° = 113° → x = 180° − 113° = 67°
Example 2: An isosceles triangle has a top angle of 46° between its two equal sides. Find the base angle y.
• Remaining angle sum: 180° − 46° = 134°
• Base angles are equal: y = 134° ÷ 2 = 67°
Estimate the size of angle x
Worked Solution & Answer
45° B1
Answer: 45° (accept ± 10°)
PQR is a straight line.
(i) Work out the value of x. [2 marks]
x =
(ii) Give a reason for your answer. [1 mark]
Worked Solution & Answer
(i)
x = 180 – 85 – 30 = 65 M1 A1
Answer (i): x = 65
(ii)
Angles on a straight line add up to 180°
Answer (ii): Angles on a straight line add up to 180°
OA, OB and OC are three straight lines.
(i) Work out the size of the angle marked x. [2 marks]
(ii) Give a reason for your answer. [1 mark]
Worked Solution & Answer
(i)
360 – 220 – 90 = 50° M1 A1
Answer (i): 50
(ii)
Angles at a point add up to 360°
Answer (ii): Angles at a point add up to 360°
ABCD is a quadrilateral.
(a) Work out the size of angle x. [1 mark]
(b) Give a reason for your answer. [1 mark]
(b) The diagram is wrong.
Explain why. [1 mark]
Explain why. [1 mark]
Worked Solution & Answer
(a)
360 – (120 + 120 + 80) = 40° B1
Answer (a): 40
(b)
Angles in a quadrilateral add up to 360°
Answer (b): Angles in a quadrilateral add up to 360°
(b)
80 + 60 + 50 = 190° C1
Answer (b): Angles in a triangle add up to 180°, but 80 + 60 + 50 = 190°
(a)(i) Work out angle x on straight line ABC. (1) (ii) Give a reason. (1) (b) Right angle + 280° diagram is wrong. Explain why. (1)
(a) Work out the size of the angle marked x. [1 mark]
(b) Give a reason for your answer. [1 mark]
(b) Explain why. [1 mark]
Worked Solution & Answer
(a)
(Angles on a straight line add up to 180°)
180 – 150 = 30° B1
180 – 150 = 30° B1
Answer (a): 30
(b)
Angles on a straight line add up to 180°
Answer (b): Angles on a straight line add up to 180°
(b)
280 + 90 = 370°, which is not 360° C1
Answer (b): Angles around a point add up to 360°, but 280 + 90 = 370°
In rectangle ABCD
triangle ABE is equilateral
triangle CDE is isosceles, with CE = DE
triangle ABE is equilateral
triangle CDE is isosceles, with CE = DE
Work out the size of angle x.
Answer
degrees
Worked Solution & Answer
In equilateral triangle ABE, each angle is 60°, so ∠ AEB = 60° M1
In isosceles triangle CDE, ∠ EDC = 65°, so ∠ ECD = 65°
Therefore, ∠ DEC = 180° - (65° + 65°) = 50° M1
By symmetry, ∠ AED = ∠ BEC = x
The angles around point E sum to 360°: M1
∠ AEB + ∠ BEC + ∠ DEC + ∠ AED = 360°
60° + x + 50° + x = 360°
2x + 110° = 360° ⇒ 2x = 250° ⇒ x = 125°. A1
In isosceles triangle CDE, ∠ EDC = 65°, so ∠ ECD = 65°
Therefore, ∠ DEC = 180° - (65° + 65°) = 50° M1
By symmetry, ∠ AED = ∠ BEC = x
The angles around point E sum to 360°: M1
∠ AEB + ∠ BEC + ∠ DEC + ∠ AED = 360°
60° + x + 50° + x = 360°
2x + 110° = 360° ⇒ 2x = 250° ⇒ x = 125°. A1
Answer: Answer 125 degrees
ABC is a straight line, BCD is a triangle (30°, 35°, x°), ABDE is a quadrilateral (95°, y°, y°). (a)(i) Work out x. (1) (ii) Give a reason. (1) (b) Work out y. (2)
ABC is a straight line.
BCD is a triangle.
ABDE is a quadrilateral.
BCD is a triangle.
ABDE is a quadrilateral.
(a) Work out the value of x. [1 mark]
(b) Give a reason for your answer. [1 mark]
(b) Work out the value of y. [2 marks]
Worked Solution & Answer
(a)
x = 180 – (30 + 35) = 115 B1
Answer (a): 115
(b)
Angles in a triangle add up to 180°
Answer (b): Angles in a triangle add up to 180°
(b)
∠ABD = 180 – 115 = 65° (angles on a straight line) M1
in quadrilateral ABDE: 2y + 95 + 65 = 360 ⇒ 2y = 200 ⇒ y = 100 A1
in quadrilateral ABDE: 2y + 95 + 65 = 360 ⇒ 2y = 200 ⇒ y = 100 A1
Answer (b): 100
The diagram shows triangle ABD
ABC, BDE and ADF are straight lines.
angle CBD = 132° angle ADB = 58°
ABC, BDE and ADF are straight lines.
angle CBD = 132° angle ADB = 58°
(a) Write down the value of x [1 mark]
x =
(b) Give a reason for your answer. [1 mark]
(b) Work out the value of y [2 marks]
y =
Worked Solution & Answer
(a)
Vertically opposite to 58° B1
Answer (a): x = 58
(b)
Vertically opposite angles are equal
Answer (b): Vertically opposite angles are equal
(b)
∠ABD = 180° – 132° = 48° M1
y = 180° – (58° + 48°) = 74° (or exterior angle 132° – 58° = 74°) A1
y = 180° – (58° + 48°) = 74° (or exterior angle 132° – 58° = 74°) A1
Answer (b): y = 74
(a) BCD is a straight line.
Triangle ABC is equilateral.
CE = DE
16 (b) Amba is working out the size of an interior angle of a regular octagon.
Her method is: Interior angle = 360 ÷ 8
Is her method correct?
Tick a box
[ ] Yes [ ] No
Give a reason for your answer.
Triangle ABC is equilateral.
CE = DE
16 (b) Amba is working out the size of an interior angle of a regular octagon.
Her method is: Interior angle = 360 ÷ 8
Is her method correct?
Tick a box
[ ] Yes [ ] No
Give a reason for your answer.
(a) BCD is a straight line.
Triangle ABC is equilateral.
CE = DE [4 marks]
Triangle ABC is equilateral.
CE = DE [4 marks]
Work out the size of angle x.
Answer
degrees
(b) Amba is working out the size of an interior angle of a regular octagon.
Her method is: Interior angle = 360 ÷ 8
Is her method correct?
Tick a box
[ ] Yes [ ] No
Give a reason for your answer. [1 mark]
Her method is: Interior angle = 360 ÷ 8
Is her method correct?
Tick a box
[ ] Yes [ ] No
Give a reason for your answer. [1 mark]
Worked Solution & Answer
(a)
Triangle ABC is equilateral, so ∠ BCA = 60° M1
In isosceles triangle CDE, ∠ ECD = 180° - 28°2 = 76° M1
Angles on a straight line BCD sum to 180°: M1
60° + x + 76° = 180°
x = 180° - 136° = 44° A1
In isosceles triangle CDE, ∠ ECD = 180° - 28°2 = 76° M1
Angles on a straight line BCD sum to 180°: M1
60° + x + 76° = 180°
x = 180° - 136° = 44° A1
Answer (a): Answer 44 degrees
(b)
360 ÷ 8 = 45° is the exterior angle, not the interior angle
The interior angle is 180° - 45° = 135°. C1
The interior angle is 180° - 45° = 135°. C1
Answer (b): No