AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I · Lesson 3

Geometric proof

About 75 minutesNo calculator55 marks

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Date

Do Now

The first is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
In triangle $A B C$ , angle $A = (2 x + 10)^{\circ}$ and angle $B = (3 x - 5)^{\circ}$ . The exterior angle at $C$ is $10 5^{\circ}$ . Work out the value of $x$ .
[2 marks]
Answer
A2
Last lesson
$A B C D$ is a cyclic quadrilateral. Angle $A = (2 x + 16)^{\circ}$ and angle $C = (x + 38)^{\circ}$ . Work out the size of angle $A$ .
[2 marks]
Answer
A3
From chapter 4
Show that $(n + 3)^{2} - (n - 3)^{2}$ is a multiple of $12$ for every integer $n$ .
[2 marks]
A4
From chapter 5
Work out the gradient of a line perpendicular to the line through $(1, 4)$ and $(5, 12)$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it. Give a reason for every step.

Example 1

$P Q = P R$ . $QR S$ is a straight line. Angle $QP R = x$ . Prove that angle $P R S = 9 0^{\circ} + \frac{1}{2} x$ .

Not drawn accurately

Hint Start with the two equal angles of the isosceles triangle. Then use the straight line.

B1
$A B$ is parallel to $C D$ . $E F = E G$ . Angle $A E F = x$ and angle $F E G = y$ . Prove that $y = 18 0^{\circ} - 2 x$ .
[3 marks]
Not drawn accurately

Example 2

$T A$ is a tangent to the circle at $A$ . $A B = A C$ . Prove that $T A$ is parallel to $B C$ .

Not drawn accurately

Hint Use the alternate segment theorem for the angle between

T A

and

A B

. Then look for alternate angles.

B2
$T A$ and $T C$ are tangents to the circle at $A$ and $C$ . $B$ is on the major arc $A C$ . Angle $A B C = x$ . Prove that angle $A T C = 18 0^{\circ} - 2 x$ .
[3 marks]
Not drawn accurately

Example 3

$O$ is the centre. $C$ is on the major arc $A B$ . Angle $O A B = x$ . Prove that angle $A C B = 9 0^{\circ} - x$ .

Not drawn accurately

Hint Triangle

O A B

has two radii as sides. Find angle

A O B

, then use the angle at the centre.

B3
$O$ is the centre. $D$ is on the minor arc $A B$ . Angle $O A B = x$ . Prove that angle $A D B = 9 0^{\circ} + x$ .
[4 marks]
Not drawn accurately

Example 4

$A$ , $B$ , $C$ and $D$ are points on a circle. Angle $B A C = 2 x$ , angle $C A D = 3 x$ and angle $B C D = 5 x$ . Prove that $B D$ is a diameter of the circle.

Not drawn accurately

Hint Opposite angles of a cyclic quadrilateral add up to

18 0^{\circ}

. A chord is a diameter if the angle it makes at the circumference is

9 0^{\circ}

B4
$A$ , $B$ , $C$ and $D$ are points on a circle. Angle $C A D = x$ and angle $A B D = 9 0^{\circ} - x$ . Prove that $A C$ is a diameter of the circle.
[3 marks]
Not drawn accurately

Practice

Level 2

C1
$A B C D$ is a quadrilateral. $A B = A D$ . Angle $A B D = x$ and angle $B C D = 2 x$ . Prove that $A B C D$ is a cyclic quadrilateral.
[3 marks]
Not drawn accurately
C2
$O$ is the centre. $T A$ is a tangent to the circle at $A$ . Angle $T A B = x$ . Prove that angle $O B A = 9 0^{\circ} - x$ .
[3 marks]
Not drawn accurately

C3
$S$ is a point on the side $P R$ of triangle $P QR$ . $P S = S Q = QR$ . Angle $QP S = x$ . Prove that angle $P QR = 18 0^{\circ} - 3 x$ .
[3 marks]
Not drawn accurately

Exam-style questions

AQA exam style

D1
$T A$ is a tangent to the circle at $A$ . $B$ and $C$ are points on the circle. Ben says, "Angle $T A B$ = angle $A B C$ , by the alternate segment theorem."
Ben is wrong. Which angle in the diagram is equal to angle $T A B$ ? Give the reason.
[2 marks]
Not drawn accurately
Answerangle $\dots\dots$

D2
$A$ , $B$ and $C$ are points on a circle. $C B P$ is a straight line. Angle $A C B = x$ , angle $A B C = 3 x$ and angle $A P B = 2 x$ .
Not drawn accurately
(a)
Prove that $P A$ is a tangent to the circle.
[4 marks]
(b)
Angle $B A C = 4 4^{\circ}$ . Work out angle $A P B$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D2: 6 marks
D3
$O$ is the centre. $B$ is on the minor arc $A C$ and $D$ is on the major arc $A C$ . Angle $O A B = x$ and angle $O C B = y$ . Prove that angle $A D C = 18 0^{\circ} - x - y$ .
[4 marks]
Not drawn accurately
D4
$O$ is the centre. $T A$ and $T B$ are tangents to the circle at $A$ and $B$ . $A C$ is a diameter.
Not drawn accurately
(a)
Prove that angle $A T B = 2 \times$ angle $B A C$ .
[4 marks]
(b)
Angle $A T B = 6 4^{\circ}$ . Hence work out angle $A C B$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D4: 6 marks
D5
$A$ , $B$ , $C$ and $D$ are points on a circle. Angle $A C B = x$ and angle $D B C = x$ . Prove that $A D$ is parallel to $B C$ .
[3 marks]
Not drawn accurately

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$A B C D$ is a cyclic quadrilateral with $A B = A D$ . Prove that $C A$ bisects angle $B C D$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Draw the diagonal $B D$ . Which two angles does $A B = A D$ make equal?
Hint 2 · The first line
Angle $A B D$ = angle $A D B$ (isosceles triangle). Now find an angle at $C$ equal to each of them.
Hint 3 · The full method
Angle $A C D$ = angle $A B D$ and angle $A C B$ = angle $A D B$ (angles in the same segment), so angle $A C B$ = angle $A C D$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a geometric proof with a reason for every step.
I can use the circle theorems and their converses in a proof.
I can use a proved result to work out an angle.

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