# Geometry - Angles at a Point

**Question 1**

Find the value of $x$ in the following diagrams

a)

a
$x=30$

Question ID: 30020010010

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $60$° then for them to add up to $90$° the other angle must be $30$°

So $x=30$

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b)

a
$x=15$

Question ID: 30020010020

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $75$° then for them to add up to $90$° the other angle must be $15$°

So $x=15$

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c)

a
$x=45$

Question ID: 30020010030

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $45$° then for them to add up to $90$° the other angle must be $45$°

So $x=45$

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d)

a
$x=70$

Question ID: 30020010040

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $30$° then for them to add up to $90$° the other angle must be $70$°

So $x=70$

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e)

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$x=40$

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $50$° then for them to add up to $90$° the other angle must be $40$°

So $x=40$

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f)

a
$x=82$

Question ID: 30020010060

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $8$° then for them to add up to $90$° the other angle must be $82$°

So $x=82$

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g)

a
$x=38$

Question ID: 30020010070

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $52$° then for them to add up to $90$° the other angle must be $38$°

So $x=38$

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h)

a
$x=63$

Question ID: 30020010080

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The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $27$° then for them to add up to $90$° the other angle must be $63$°

So $x=63$

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**Question 2**

Find the value of $y$ in the following diagrams

a)

a
$y=120$

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b)

a
$y=134$

Question ID: 30020020020

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c)

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$y=90$

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d)

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$y=16$

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$y=100$

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$y=105$

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$y=69$

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$y=78$

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$y=37$

Question ID: 30020020090

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j)

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$y=90$

Question ID: 30020020100

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$y=107$

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**Question 3**

Find the value of $z$ in the following diagrams

a)

a
$z=120$

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b)

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$z=80$

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c)

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$z=60$

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d)

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$z=100$

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$z=25$

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f)

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$z=15$

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$z=31$

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$z=26$

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$z=43$

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$z=67$

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$z=28$

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l)

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$z=37$

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**Question 4**

Find the value of $q$ in the following diagrams

a)

a
$q=300$

Question ID: 30020040010

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b)

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$q=135$

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c)

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$q=270$

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$q=255$

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$q=155$

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$q=40$

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$q=186$

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**Question 5**

Find the value of $r$ in the following diagrams

a)

a
$r=60$

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b)

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$r=100$

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$r=163$

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d)

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$r=27$

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$r=90$

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$r=137$

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**Question 6**

Find the value of the acute angle $\angle AOB$ in the following diagrams, giving reasons

a)

a
$\angle AOB=30$°

Question ID: 30020060010

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b)

a
$\angle AOB=50$°

Question ID: 30020060020

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c)

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$\angle AOB=55$°

Question ID: 30020060030

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$\angle AOB=78$°

Question ID: 30020060040

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$\angle AOB=180$°

Question ID: 30020060050

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$\angle AOB=49$°

Question ID: 30020060060

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$\angle AOB=74$°

Question ID: 30020060070

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$\angle AOB=24$°

Question ID: 30020060080

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**Question 7**

Find the value of the pronumerals in the following diagrams, giving reasons

a)

a
$x=57$, $y=123$

Question ID: 30020070010

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b)

a
$a=36$, $b=144$, $c=69$

Question ID: 30020070020

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c)

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$m=46$

Question ID: 30020070030

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d)

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$h=44$, $j=136$, $k=46$

Question ID: 30020070040

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**Question 1**

Find the value of $x$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

**Question 2**

Find the value of $y$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

**Question 3**

Find the value of $z$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

**Question 4**

Find the value of $q$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

**Question 5**

Find the value of $r$ in the following diagrams

a)

b)

c)

d)

e)

f)

**Question 6**

Find the value of the acute angle $\angle AOB$ in the following diagrams, giving reasons

a)

b)

c)

d)

e)

f)

g)

h)

**Question 7**

Find the value of the pronumerals in the following diagrams, giving reasons

a)

b)

c)

d)

__Answers__

**Question 1**

a) $x=30$

b) $x=15$

c) $x=45$

d) $x=70$

e) $x=40$

f) $x=82$

g) $x=38$

h) $x=63$

**Question 2**

a) $y=120$

b) $y=134$

c) $y=90$

d) $y=16$

e) $y=100$

f) $y=105$

g) $y=69$

h) $y=78$

i) $y=37$

j) $y=90$

k) $y=107$

**Question 3**

a) $z=120$

b) $z=80$

c) $z=60$

d) $z=100$

e) $z=25$

f) $z=15$

g) $z=31$

h) $z=26$

i) $z=43$

j) $z=67$

k) $z=28$

l) $z=37$

**Question 4**

a) $q=300$

b) $q=135$

c) $q=270$

d) $q=255$

e) $q=155$

f) $q=40$

g) $q=186$

**Question 5**

a) $r=60$

b) $r=100$

c) $r=163$

d) $r=27$

e) $r=90$

f) $r=137$

**Question 6**

a) $\angle AOB=30$°

b) $\angle AOB=50$°

c) $\angle AOB=55$°

d) $\angle AOB=78$°

e) $\angle AOB=180$°

f) $\angle AOB=49$°

g) $\angle AOB=74$°

h) $\angle AOB=24$°

**Question 7**

a) $x=57$, $y=123$

b) $a=36$, $b=144$, $c=69$

c) $m=46$

d) $h=44$, $j=136$, $k=46$

**Question 1**

Find the value of $x$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

**Question 2**

Find the value of $y$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

**Question 3**

Find the value of $z$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

**Question 4**

Find the value of $q$ in the following diagrams

a)

b)

c)

d)

e)

f)

g)

**Question 5**

Find the value of $r$ in the following diagrams

a)

b)

c)

d)

e)

f)

**Question 6**

Find the value of the acute angle $\angle AOB$ in the following diagrams, giving reasons

a)

b)

c)

d)

e)

f)

g)

h)

**Question 7**

Find the value of the pronumerals in the following diagrams, giving reasons

a)

b)

c)

d)

__Answers__

**Question 1**

a) $x=30$

b) $x=15$

c) $x=45$

d) $x=70$

e) $x=40$

f) $x=82$

g) $x=38$

h) $x=63$

**Question 2**

a) $y=120$

b) $y=134$

c) $y=90$

d) $y=16$

e) $y=100$

f) $y=105$

g) $y=69$

h) $y=78$

i) $y=37$

j) $y=90$

k) $y=107$

**Question 3**

a) $z=120$

b) $z=80$

c) $z=60$

d) $z=100$

e) $z=25$

f) $z=15$

g) $z=31$

h) $z=26$

i) $z=43$

j) $z=67$

k) $z=28$

l) $z=37$

**Question 4**

a) $q=300$

b) $q=135$

c) $q=270$

d) $q=255$

e) $q=155$

f) $q=40$

g) $q=186$

**Question 5**

a) $r=60$

b) $r=100$

c) $r=163$

d) $r=27$

e) $r=90$

f) $r=137$

**Question 6**

a) $\angle AOB=30$°

b) $\angle AOB=50$°

c) $\angle AOB=55$°

d) $\angle AOB=78$°

e) $\angle AOB=180$°

f) $\angle AOB=49$°

g) $\angle AOB=74$°

h) $\angle AOB=24$°

**Question 7**

a) $x=57$, $y=123$

b) $a=36$, $b=144$, $c=69$

c) $m=46$

d) $h=44$, $j=136$, $k=46$

__Solutions__

**Question 1**

a) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $60$° then for them to add up to $90$° the other angle must be $30$°

So $x=30$

b) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $75$° then for them to add up to $90$° the other angle must be $15$°

So $x=15$

c) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $45$° then for them to add up to $90$° the other angle must be $45$°

So $x=45$

d) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $30$° then for them to add up to $90$° the other angle must be $70$°

So $x=70$

e) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $50$° then for them to add up to $90$° the other angle must be $40$°

So $x=40$

f) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $8$° then for them to add up to $90$° the other angle must be $82$°

So $x=82$

g) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $52$° then for them to add up to $90$° the other angle must be $38$°

So $x=38$

h) The square means that the whole angle is a right angle.

A right angle is $90$°

As one of the angles is $27$° then for them to add up to $90$° the other angle must be $63$°

So $x=63$

**Question 2**

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

**Question 3**

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

**Question 4**

a)

b)

c)

d)

e)

f)

g)

**Question 5**

a)

b)

c)

d)

e)

f)

**Question 6**

a)

b)

c)

d)

e)

f)

g)

h)

**Question 7**

a)

b)

c)

d)