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Calculating angles

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visibility 248update 6 months agobookmarkshare

🎯 In this topic you will

Use the properties of parallel lines to calculate angles
Use the properties of triangles and quadrilaterals to calculate angles
Apply several different angle properties together

📘 Worked example

Calculate the values of $x$ and $y$.

Answer:

The sum of angles on a straight line is $180^\circ$ and so $x = 180^\circ - 118^\circ = 62^\circ$.

Angle $C$ of the quadrilateral is $360^\circ - (62^\circ 80^\circ 134^\circ) = 84^\circ$.

Angles $DCF$ and $ADC$ are alternate angles so angle $DCF = x^\circ = 62^\circ$.

Then $y = 180 - (84 62) = 34$.

First, use the straight-line rule to find $x$ by subtracting $118^\circ$ from $180^\circ$.

Next, find angle $C$ in the quadrilateral by subtracting the sum of the other three angles from $360^\circ$.

Recognise that $DCF$ and $ADC$ are alternate angles, so $DCF = x = 62^\circ$.

Finally, use the triangle angle sum to find $y$ by subtracting $84^\circ$ and $62^\circ$ from $180^\circ$.

🧠 PROBLEM-SOLVING Strategy

Angles on lines, in triangles & quadrilaterals

Use these steps to find unknown angles when parallel lines, triangles, quadrilaterals, or circles appear.

Mark all given angles and facts (parallel marks, equal sides) directly on the diagram. Convert any exterior angles to interior ones using $180^\circ$ on a straight line.
Use line/point facts first: straight line sums to $180^\circ$; around a point sums to $360^\circ$; vertically opposite angles are equal.
If there are parallel lines, apply corresponding/alternate/co-interior angle rules. Write equalities like $\angle \text{corresponding} = \angle \text{matching}$ or $\angle \text{co-int} \angle \text{co-int}=180^\circ$.
Handle triangles next: sum to $180^\circ$; in isosceles triangles, base angles are equal; an exterior angle equals the sum of the two opposite interior angles.
For quadrilaterals, use the interior-angle sum $360^\circ$. For special types: kite (two pairs adjacent equal sides ⇒ one pair of equal opposite angles), parallelogram (opposite angles equal, adjacent sum $180^\circ$), rectangle/square (right angles).
In circle questions with centre $O$: radii make isosceles triangles (e.g., $OA=OB$), central angle is twice the subtended angle at the circumference: $\angle AOB = 2\angle ACB$.
Write short equations for each step (e.g., $x=180^\circ-115^\circ$, $y=360^\circ-(a b c)$) and solve systematically.
Check by substituting back and verifying required sums ($180^\circ$ or $360^\circ$) are satisfied.

Fact	Sum / Relation
Straight line	$180^\circ$
Around a point	$360^\circ$
Triangle	$180^\circ$
Quadrilateral	$360^\circ$
Co-interior on parallel lines	$\text{sum}=180^\circ$
Central vs inscribed (circle)	$\angle \text{centre} = 2\times \angle \text{circumference}$

❓ EXERCISES

1. Work out the angles of this triangle.

👀 Show answer

Exterior angles at the base are $120^\circ$ and $155^\circ$, so interior base angles are $180^\circ-120^\circ=60^\circ$ and $180^\circ-155^\circ=25^\circ$. The third angle is $180^\circ-(60^\circ 25^\circ)=95^\circ$. Answer: $60^\circ$, $25^\circ$, $95^\circ$.

2. In this diagram, $AB=AC=AD$

a. Calculate $x$ and $y$.

b. Work out angle $C$ of quadrilateral $ABCD$.

c. Show that the sum of the four angles of the quadrilateral is $360^\circ$.

👀 Show answer

a. From the straight-line rule, $x=180^\circ-118^\circ=62^\circ$. Using angle sums and alternate angles in the diagram gives $y=34^\circ$.

b. $\angle C=360^\circ-(62^\circ 80^\circ 134^\circ)=84^\circ$.

c. The interior angles of any quadrilateral sum to $360^\circ$, so the four angles add to $360^\circ$.

3. Calculate angles $a$, $b$, $c$ and $d$.

👀 Show answer

Not enough information provided to determine unique values for $a$, $b$, $c$, and $d$ without the precise diagram measurements and marked relationships.

4. The angles of a quadrilateral are $x^\circ$, $(x 10)^\circ$, $(x 20)^\circ$ and $(x 30)^\circ$. Work out the value of $x$.

👀 Show answer

Sum of angles in a quadrilateral is $360^\circ$: $x (x 10) (x 20) (x 30)=360 \;\Rightarrow\; 4x 60=360 \;\Rightarrow\; 4x=300 \;\Rightarrow\; x=75^\circ$.

❓ EXERCISES

5. a. What type of quadrilateral is $ABCD$? Give a reason for your answer.
b. Work out the angles of $ABCD$.

👀 Show answer

The screenshot does not state all required parallel/equal-side facts explicitly, so the exact type and all interior angles of $ABCD$ cannot be determined with certainty from the image alone. Not enough information provided.

6. $ABCD$ is a kite. Work out the angles of $ABCD$.

👀 Show answer

Without the exact angle positions and side labels from the diagram (e.g., which angles are interior vs exterior), the four interior angles of the kite cannot be uniquely determined. Not enough information provided.

7. $O$ is the centre of the circle. $OA$, $OB$ and $OC$ are radii. Angle $AOB=72^\circ$. Work out
a. angle $OAB$ b. angle $OCB$ c. angle $ABC$.
Give reasons for your answers.

👀 Show answer

a. In $\triangle AOB$, $OA=OB$ so it is isosceles. Base angles are equal and sum with the vertex angle to $180^\circ$: $\displaystyle \angle OAB=\angle OBA=\frac{180^\circ-72^\circ}{2}=54^\circ$.

b. Determining $\angle OCB$ requires knowing $\angle BOC$ (or the arc $BC$), which is not specified. Not enough information provided.

c. $\angle ABC$ is an inscribed angle subtending arc $AC$ and equals $\tfrac12\angle AOC$, but $\angle AOC$ is not given. Not enough information provided.

8. Work out the values of $x$ and $y$. Justify your answers.

👀 Show answer

The exact positions of $70^\circ$, $115^\circ$, $x^\circ$ and $y^\circ$ in relation to the indicated parallel lines are not fully readable, so a unique pair $(x,y)$ cannot be justified. Not enough information provided.

9. Work out the value of $x$.

👀 Show answer

The diagram’s exact relationships (e.g., which lines are parallel) are not fully specified in the text, so $x$ cannot be determined uniquely. Not enough information provided.

❓ EXERCISES

10. ACE is a straight line.

a. Calculate angle BAD.

b. Show that the sum of the angles of ABCD is $360^\circ$.

👀 Show answer

a. From the text alone we know only that $A$, $C$, and $E$ are collinear. The diagram shows angles labelled near $B$, $C$, and $D$, but it does not specify which of the angles at $C$ (the ones marked $110^\circ$ and $123^\circ$ about the line $CE$) is the interior angle of the quadrilateral $ABCD$. Without that explicit identification, the value of $\angle BAD$ cannot be determined uniquely. Not enough information provided.

b. Draw diagonal $AC$ to split quadrilateral $ABCD$ into $\triangle ABC$ and $\triangle ADC$. The interior angles of each triangle sum to $180^\circ$, so the four interior angles of $ABCD$ sum to $180^\circ 180^\circ = 360^\circ$. Hence, the sum of the angles of $ABCD$ is $360^\circ$.

📘 What we've learned

Angles on a straight line sum to $180^\circ$.
Angles around a point sum to $360^\circ$.
Vertically opposite angles are equal.
The sum of interior angles in a triangle is $180^\circ$, and in a quadrilateral is $360^\circ$.
In an isosceles triangle, the base angles are equal.
For parallel lines, corresponding and alternate angles are equal, and co-interior angles sum to $180^\circ$.
Exterior angle of a triangle equals the sum of the two opposite interior angles.
In circle geometry, the central angle is twice the inscribed angle subtending the same arc.
We applied these properties together to solve multi-step angle problems.

Calculating angles

🎯 In this topic you will

🧠 PROBLEM-SOLVING Strategy

Angles on lines, in triangles & quadrilaterals

❓ EXERCISES

❓ EXERCISES

❓ EXERCISES

📘 What we've learned

Related Past Papers

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