Starter: Can you use your knowledge of angles to find the missing ones?

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10/7/2023
Properties of Shapes – Angles in Parallel lines
Learning
Intention

To understand and apply angle facts on parallel lines

Success Criteria

• To be able to recall and apply basic angle facts (straight line,

around a point, vertically opposite, angles in a triangle and angles in
a quadrilateral)

• To be able to identify the names of angles on parallel lines.

• To be able to solve one step problems finding angles in parallel lines

• To integrate knowledge of angles on parallel lines to solve problems

involving multiple angle facts

Keywords:
Angle, acute, obtuse, reflex, right angle, parallel, degree,
protractor, triangle, isosceles, scalene, equilateral
Mathematical Learning Journey
RECAP: Parts of a Circle

!
?
(Minor)
Arc ?
Sector

?
Chord ?
Radius

(Minor)
?
Segmen
t ?
Diameter

?
Tangent
?
Circumference
RECAP: Parts of a Circle

!
(Minor)
Arc Sector

Chord Radius

(Minor)
Segmen
Diameter
t

Tangent
Circumference
What are Circle Theorems
Circle Theorems are laws that apply to both angles and lengths when circles are
involved. We’ll deal with them in groups.

#1 Non-Circle Theorems

These are not circle theorems, but are useful in questions involving circle theorems.

?
13
0
50

Angles in a quadrilateral
The radius is of constant length
add up to 360. When you have multiple radii, put a mark
on each of them to remind yourself
they’re the same length.
 #2 Circle Theorems Involving Right Angles
Remember the wording in the black boxes, because you’re often
required to justify in words a particular angle in an exam.

“Angle in semicircle is
90.”
Note that the hypotenuse of
the triangle MUST be the
diameter.
#3 Circle Theorems Involving Other Angles

!
a
a a

2a

“Angles in same segment “Angle at centre is twice the

and standing on the same angle at the circumference.”
chord are equal.”
The angle subtended by an arc at the centre is twice
the angle subtended at the circumference

An arc is just a
connected
section of the
circumference
#3 Circle Theorems Involving Other Angles

!
x

180-x
x

Opposite angles of cyclic

quadrilateral add up to 180.
Is t=40°?
Which Circle Theorem?
Identify which circle theorems you could use to solve Angle in semicircle is 90
each question. Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

quadrilateral add to
180
Angles in same segment
are equal
O
Angle at centre is twice
?
160 angle at circumference

Lengths of the tangents

100 from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Which Circle Theorem?
Identify which circle theorems you could use to solve Angle in semicircle is 90
each question. Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

70 quadrilateral add to
180
Angles in same segment
are equal
70
?
60 Angle at centre is twice
angle at circumference

Lengths of the tangents

from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Which Circle Theorem?
Identify which circle theorems you could use to solve Angle in semicircle is 90
each question. Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

quadrilateral add to
180
Angles in same segment
?
70 are equal

Angle at centre is twice
angle at circumference

Lengths of the tangents

from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Which Circle Theorem?
Identify which circle theorems you could use to solve Angle in semicircle is 90
each question. Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

quadrilateral add to
180
Angles in same segment
?
32
are equal

Angle at centre is twice
angle at circumference

Lengths of the tangents

from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Helper sheet.
Today we are only
looking at the top 4.
Task 1
 105
101 95
89

58
49

76 122
1.ai) 180 – 2 × 25 = 130°
M1 for 180 – 2 × 25
A1 cao
(ii) Mark for mentioning isosceles and equal (or base) angles or equal sides and equal
(or base) angles
(b) 180 – 95 = 85°
2.ai) 2x70= 140°
(ii) The angle subtended by an arc at the centre is twice the angle subtended at the
circumference

(bi) 180 – 70 = 110°

(bii) Opposite angles in a cyclic quadrilateral add to 180°.
3. (a)60°
(b) ADB = 60 – 25= 35°

(c)Ben is correct; angle DAB = 65 + 25 = 90 and since angle in a

semi-circle is 90°, BD must be a diameter
Plenary
J=60

I = 60

K = 60

L=120

H= 30
Plenary

Find angle i. Give a

reason for your
answer.

I = 39
Starter
Starter
Starter
10/7/2023
Properties of Shapes – Angles in Parallel lines
Learning
Intention

To understand and apply angle facts on parallel lines

Success Criteria

• To be able to recall and apply basic angle facts (straight line,

around a point, vertically opposite, angles in a triangle and angles in
a quadrilateral)

• To be able to identify the names of angles on parallel lines.

• To be able to solve one step problems finding angles in parallel lines

• To integrate knowledge of angles on parallel lines to solve problems

involving multiple angle facts

Keywords:
Angle, acute, obtuse, reflex, right angle, parallel, degree,
protractor, triangle, isosceles, scalene, equilateral
Mathematical Learning Journey
Circle Theorems Involving Right Angles

“A line from the centre to the

“Angle between radius
midpoint of a chord is
and tangent is 90”.
perpendicular (at 90) to the
chord”.
Circle Theorems Involving Lengths

There’s
only one
you need
to know...

Lengths of the tangents from a

point to the circle are equal.
Which Circle Theorem?
Identify which circle theorems you could use to Angle in semicircle is 90
solve each question.
Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

quadrilateral add to
180
Angles in same segment
are equal
?
115
Angle at centre is twice
angle at circumference

Lengths of the tangents

from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Which Circle Theorem?
Identify which circle theorems you could use to solve Angle in semicircle is 90
each question. Reveal
Angle between tangent
and radius is 90

Opposite angles of cyclic

quadrilateral add to
?
31 180
 Angles in same segment
are equal

Angle at centre is twice

angle at circumference

Lengths of the tangents

from a point to the circle
are equal

Two angles in isosceles

triangle the same
Angles of quadrilateral
add to 360
Tangent to a circle
Task

Question 12: find x

Exam style question/s
Exam style question/s
 PQ and PT are tangents to a circle with centre
O. Find the unknown angles giving reasons.

yo
Q
xo O
angle w 90o (tan/rad)
98o
=
angle x 90o (tan/rad)
=
angle y= 49o (angle at centre)
angle z = 360o – 278 = 82o (quadrilateral)
zo wo
P T
 PQ and PT are tangents to a circle with centre
O. Find the unknown angles giving reasons.

Q
yo
O
angle w 90o (tan/rad)
xo =
angle x 180 – 140 = 40o (angles sum tri)
=
angle y= 50o (isos triangle)

80o wo 50o
P T
Plenary question
Alternate Segment Theorem
This one is probably the hardest to remember

!
Click to Start
This is called the animation
alternate segment
because it’s the segment
on the other side of the
chord.


...is equal to the angle in the
alternate segment

tangent


The angle between the

tangent and a chord...
Check Your Understanding

z = ?58
Check Your Understanding
Source: IGCSE Jan 2014 (R)

Angle ABC = Angle AOC = Angle CAE =

112 ? 136 ? 68 ?
Give a reason: Give a reason: Give a reason:
Supplementary angles of Angle at centre is double Alternate Segment
?
cyclic quadrilateral add ?
angle at circumference. Theorem. ?
up to 180.
Exam style question/s
Exam style question/s
Answers to more difficult questions
Source: IGCSE May 2013

?39

?
64

?
77

Determine angle ADB.

Answers to more difficult questions
(Towards the end of your sheet)

? ?
32
11 2
42?
1 Angle at centre is twice
6
3 angle at circumference

Two angles in isosceles

triangle the same

Alternate Segment
Theorem