KS3 Maths · Key Stage 3 · Unit 10: Congruence and Trigonometry

Congruent triangles, proof and right-angled trigonometry

Year 9About 85 minutesCalculator allowed89 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/ks3-congruence-and-trigonometry/

NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Two angles of a triangle are $5 3^{\circ}$ and $7 1^{\circ}$ . Work out the size of the third angle.
[1 mark]
Answer

A2
A right-angled triangle has shorter sides of 20 cm and 21 cm. Work out the length of its longest side.
[2 marks]
Answer
A3
Solve $\frac{x}{8} = 0.65$ .
[1 mark]
Answer $x = \dots\dots$
A4
Solve $\frac{12}{x} = 0.75$ .
[1 mark]
Answer $x = \dots\dots$
A5
Round 17.849 to 1 decimal place.
[1 mark]
Answer

Examples, then your turn

Core

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Triangles $A B C$ and $D E F$ have $A B = D E = 7$ cm, $B C = E F = 9$ cm, and angle $A B C$ = angle $D E F = 5 0^{\circ}$ . Which condition shows that the triangles are congruent? Give a reason.

Hint Check where the angle sits. Angle

B

is between the sides

A B

and

B C

, so this is two sides and the angle between them.

B1
In the diagram, $P Q = X Y = 6.5$ cm, angle $P$ = angle $X = 4 0^{\circ}$ and angle $Q$ = angle $Y = 7 2^{\circ}$ . Which condition shows that the triangles are congruent? Give a reason.
[2 marks]
Not drawn accurately
Answercondition $\dots\dots\dots$

Example 2

A right-angled triangle has a hypotenuse of 14 cm and an angle of $3 5^{\circ}$ . Work out the length of the side opposite the $3 5^{\circ}$ angle. Give your answer to 1 decimal place.

Hint Label the sides from the

3 5^{\circ}

angle: opposite, adjacent, hypotenuse. Opposite and hypotenuse means sine, so the side is

14 sin 3 5^{\circ}

B2
Work out the length $x$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Example 3

In triangle $A B C$ , angle $B = 9 0^{\circ}$ , angle $A = 2 8^{\circ}$ and $B C = 6$ cm. Work out the length of $A C$ . Give your answer to 1 decimal place.

Hint

B C

is opposite the

2 8^{\circ}

angle and

A C

is the hypotenuse, so use sine. The unknown is on the bottom of the fraction, so you divide.

B3
Work out the length $x$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Example 4

A right-angled triangle has a hypotenuse of 9 cm. The side opposite the angle $θ$ is 5 cm. Work out the size of $θ$ . Give your answer to 1 decimal place.

Hint Opposite and hypotenuse means sine:

sin θ = \frac{5}{9}

. To get the angle back from the ratio, use the inverse,

sin^{- 1}

B4
Work out the size of the angle $θ$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Practice

Core

Each question asks for something different. Show your working.

C1
Triangles $A B C$ and $D E F$ each have a right angle. $A C = D F = 41$ cm and $A B = D E = 9$ cm.
Not drawn accurately
(a)
Which condition shows that the two triangles are congruent?
[1 mark]
Answer
(b)
Work out the length of $E F$ .
[2 marks]
Answer
(c)
Work out the area of triangle $D E F$ .
[1 mark]
Answer
Total for C1: 4 marks

C2
Triangle $G H I$ has $G H = 11$ cm, $H I = 8$ cm and angle $H G I = 4 0^{\circ}$ . Triangle $J K L$ has $J K = 11$ cm, $K L = 8$ cm and angle $K J L = 4 0^{\circ}$ .
Kim says, "The two triangles must be congruent." Is Kim right? Give a reason.
[2 marks]
AnswerYes or No: $\dots\dots$
C3
Triangles $A B C$ and $P QR$ are congruent. $A$ matches $P$ , $B$ matches $Q$ and $C$ matches $R$ . Angle $A B C = 7 4^{\circ}$ and angle $P R Q = 3 9^{\circ}$ .
Work out the size of angle $QP R$ .
[2 marks]
Answer
C4
The diagram shows a right-angled triangle with sides of 12 cm, 35 cm and 37 cm. Write each of these as a fraction.
Not drawn accurately
(a)
$sin θ$
[1 mark]
Answer
(b)
$cos θ$
[1 mark]
Answer
(c)
$tan θ$
[1 mark]
Answer
Total for C4: 3 marks
C5
A ladder 7.5 m long leans against a vertical wall. The top of the ladder is 6.8 m above the ground.
Work out the angle $θ$ between the ladder and the ground. Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer
C6
Jo stands 35 m from the foot of a tree on level ground. The angle of elevation of the top of the tree from the ground where she stands is $2 3^{\circ}$ .
Work out the height $h$ of the tree. Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer
C7
An isosceles triangle has two sides of 11 cm and a base of 14 cm. The dashed line is its line of symmetry.
Work out the size of the angle $x$ . Give your answer to 1 decimal place.
[3 marks]
Not drawn accurately
Answer
C8
$A B C D$ is a kite with $A B = C B$ and $A D = C D$ . The line $B D$ is drawn.
(a)
Prove that triangle $A B D$ is congruent to triangle $C B D$ .
[2 marks]
(b)
Angle $B A D = 10 8^{\circ}$ . Write down the size of angle $B C D$ .
[1 mark]
Answer
Total for C8: 3 marks

Test-style questions

Stretch

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Sam works out the length $x$ . He writes, " $x = 12 sin 5 0^{\circ} = 9.2$ cm."
Explain the mistake Sam has made. Work out the correct value of $x$ , to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

D2
$A B C D$ is a parallelogram. The diagonal $A C$ is drawn.
Prove that triangle $A B C$ is congruent to triangle $C D A$ .
[3 marks]
D3
Leah flies a kite. The string is 40 m long and is pulled straight. It makes an angle of $5 2^{\circ}$ with the horizontal. Leah holds the end of the string 1.3 m above the ground.
Work out the height of the kite above the ground. Give your answer to 1 decimal place.
[3 marks]
Answer
D4
In the diagram, angle $A B C = 9 0^{\circ}$ and angle $A C D = 9 0^{\circ}$ . $A B = 9$ cm, angle $B A C = 4 0^{\circ}$ and angle $C A D = 2 5^{\circ}$ .
Work out the length of $A D$ . Give your answer to 1 decimal place. You must show your working.
[4 marks]
Not drawn accurately
Answer
D5
A zip wire runs in a straight line from the top $T$ of a tower 18 m tall to a point $G$ on level ground 64 m from the foot of the tower.
Not drawn accurately
(a)
Work out the angle between the zip wire and the ground. Give your answer to 1 decimal place.
[2 marks]
Answer
(b)
Work out the length of the zip wire. Give your answer to 1 decimal place.
[2 marks]
Answer
(c)
A safety rule says the angle between a zip wire and the ground must be no more than $1 2^{\circ}$ . The tower stays the same. How much further from the tower must $G$ be moved? Give your answer to 1 decimal place.
[2 marks]
Answer
Total for D5: 6 marks
D6
Triangles $A B C$ and $D E F$ are congruent. $A$ matches $D$ , $B$ matches $E$ and $C$ matches $F$ . $A B = (2 y + 3)$ cm and $D E = (4 y - 9)$ cm. $B C = 11$ cm and $A C = 13$ cm.
(a)
Work out the value of $y$ .
[2 marks]
Answer $y = \dots\dots$
(b)
Work out the perimeter of triangle $D E F$ .
[2 marks]
Answer
Total for D6: 4 marks
D7
An isosceles triangle has two sides of 15 cm. The angle between them is $4 8^{\circ}$ .
Work out the area of the triangle. Give your answer to 1 decimal place. You must show your working.
[4 marks]
Not drawn accurately
Answer
D8
A boat sails 14 km due north from a harbour, then 9 km due east.
Work out the angle between due north and the straight line from the harbour to the boat. Give your answer to 1 decimal place.
[3 marks]
Answer
D9
In a right-angled triangle, the hypotenuse is 20 cm. One of the other angles is $x$ , and $tan x = 0.75$ .
Work out the perimeter of the triangle. You must show your working.
[3 marks]
Answer

Challenge

UKMT level

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A right-angled triangle has a perimeter of 60 cm. Its smallest angle is $θ$ , and $tan θ = \frac{5}{12}$ . Find the area of the triangle.
[4 marks]
Hint 1 · What to try
The two shorter sides are in the ratio $5 : 12$ . Call them $5 k$ and $12 k$ .
Hint 2 · The first line
The hypotenuse is $(5 k)^{2} + (12 k)^{2} = 13 k$ , so the perimeter is $30 k$ .
Hint 3 · The full method
$30 k = 60$ , so $k = 2$ . The shorter sides are $10$ cm and $24$ cm, and the area is $\frac{1}{2} \times 10 \times 24 = 120$ cm².
Answer

E2
Triangle $P QR$ has $P Q = P R$ . $M$ is the midpoint of $QR$ . $QR = 30$ cm and the area of triangle $P QR$ is 300 cm². Find the perimeter of triangle $P QR$ .
[4 marks]
Hint 1 · What to try
Triangles $P QM$ and $P R M$ are congruent (SSS), so the angles at $M$ are equal. They add to $18 0^{\circ}$ .
Hint 2 · The first line
So $P M$ is perpendicular to $QR$ and is the height of the triangle: $\frac{1}{2} \times 30 \times P M = 300$ .
Hint 3 · The full method
$P M = 20$ cm and $QM = 15$ cm, so $P Q = 1 5^{2} + 2 0^{2} = 25$ cm. The perimeter is $25 + 25 + 30 = 80$ cm.
Answer
E3
A ladder leans against a vertical wall and makes an angle of $6 2^{\circ}$ with the level ground. The foot of the ladder is moved 1.5 m further from the wall. The ladder now makes an angle of $4 8^{\circ}$ with the ground. Find the length of the ladder, to 1 decimal place.
[4 marks]
Hint 1 · What to try
Call the length of the ladder $L$ . Write the distance from the wall to the foot in each position.
Hint 2 · The first line
The distances are $L cos 6 2^{\circ}$ and $L cos 4 8^{\circ}$ , and they differ by $1.5$ m.
Hint 3 · The full method
$L (cos 4 8^{\circ} - cos 6 2^{\circ}) = 1.5$ , so $L = \frac{1.5}{0.66913 - 0.46947} = 7.5$ m.
Answer
E4
A right-angled triangle has a hypotenuse of 53 cm and an area of 630 cm². Find the size of its smallest angle, to 1 decimal place.
[4 marks]
Hint 1 · What to try
Call the two shorter sides $a$ and $b$ . The area gives $ab = 1260$ , and Pythagoras gives $a^{2} + b^{2} = 2809$ .
Hint 2 · The first line
$(a + b)^{2} = a^{2} + b^{2} + 2 ab = 2809 + 2520 = 5329$ , and $(a - b)^{2} = 2809 - 2520 = 289$ .
Hint 3 · The full method
$a + b = 73$ and $a - b = 17$ , so the sides are $45$ cm and $28$ cm. The smallest angle is opposite the $28$ cm side: $tan^{- 1} (\frac{28}{45}) = 31. 9^{\circ}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove that two triangles are congruent using SSS, SAS, ASA or RHS, and explain why two sides and a non-included angle are not enough.
I can use sin, cos and tan to find a missing side or angle in a right-angled triangle.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers