Angles at Points and on Lines Revision Notes | GCSE AQA Higher Maths
Vertically opposite angles are the angles directly across from each other where two straight lines cross, and they are always equal. Free GCSE Maths notes.
Updated 21 Mar 2026 · Revise Right Team
Angles at Points and on Lines is part of Properties and constructions in GCSE AQA Higher Maths.
3 min
Reading time
42
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
Vertically opposite angles are the angles directly across from each other where two straight lines cross, and they are always equal. Free GCSE Maths notes.
Apply the property that angles at a point sum to 360° to find missing angles
Apply the property that angles on a straight line sum to 180° to find missing angles
Identify and use vertically opposite angles to find unknown angles
Construct clear mathematical reasoning when finding angles, using correct angle facts as justification
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Use this lesson to stay inside Properties and constructions while connecting the detail back to the wider Geometry and measures topic.
Angles at a point: Angles formed around a single point that together make a full turn. They always sum to 360°.
Angles on a straight line: Angles formed on one side of a straight line at a common point. They always sum to 180°.
Vertically opposite angles: Pairs of angles formed directly across from each other when two straight lines intersect. They are always equal.
Revision notes
Read the core explanations from Angles at Points and on Lines before testing yourself from memory.
1. Why Angles at Points and Lines Matter
Every geometry problem you will ever meet — from finding a missing angle in a triangle to proving two lines are parallel — rests on three angle facts. Master these three, and you have the foundation for almost all of GCSE geometry.
Think about a clock face. At 12 o'clock, all the hands point in the same direction. As they spread out, they carve up the full turn around the centre. That full turn is always $360^\circ$ — no matter how many hands there are or how they are arranged. That is the first of our three facts.
2. Angles at a Point
You can picture this by imagining standing at the centre of a roundabout and turning all the way around until you face the same direction again — that is $360^\circ$. Any set of angles that fills that complete turn must add up to $360^\circ$.
Worked example: Finding a Missing Angle at a Point
Three angles meet at a point. Two of them are $110^\circ$ and $120^\circ$. Find the third angle $x$.
$110^\circ + 120^\circ + x = 360^\circ$ — Angles at a point sum to 360°. Write this as an equation. [M1]
$110 + 120 = 230$ — Find the total of the two known angles.
$x = 130^\circ$ (angles at a point sum to $360^\circ$) — Always write the angle fact as your reason — examiners award marks for this.
Answer: $x = 130^\circ$
3. Angles on a Straight Line
A straight line is half a full turn. If you stand facing right and turn to face left, you have rotated $180^\circ$. Any angles sitting on one side of a straight line at a common point must share that half-turn — so they sum to $180^\circ$.
Worked example: Finding a Missing Angle on a Straight Line
Two angles sit on a straight line. One angle is $55^\circ$. Find the other angle $y$.
$55^\circ + y = 180^\circ$ — Angles on a straight line sum to 180°. [M1]
$y = 180^\circ - 55^\circ$ — Subtract the known angle from 180°.
$y = 125^\circ$ — Final answer. [A1]
$y = 125^\circ$ (angles on a straight line sum to $180^\circ$) — Writing the reason is expected in GCSE Higher exam questions that ask you to 'find' or 'work out' an angle.
Answer: $y = 125^\circ$
4. Vertically Opposite Angles
When two straight lines cross each other, they create four angles at the intersection point. The angles that sit directly across from each other — on opposite sides of the crossing — are called vertically opposite angles, and they are always equal.
Here is why. Label the four angles $p$, $q$, $r$, $s$ going around the crossing point. Angles $p$ and $q$ sit on a straight line, so $p + q = 180^\circ$. Angles $q$ and $r$ also sit on a straight line, so $q + r = 180^\circ$. Both expressions equal $180^\circ$, which means $p = r$. The same logic gives $q = s$. Vertically opposite angles are equal — always.
Worked example: Using Vertically Opposite Angles
Two straight lines intersect. One of the four angles is $47^\circ$. Find all four angles.
$47^\circ + \text{adjacent} = 180^\circ$
$\text{adjacent} = 180^\circ - 47^\circ = 133^\circ$ — Angles on a straight line sum to 180°. [M1]
Fourth angle $= 133^\circ$ (vertically opposite to $133^\circ$) — The fourth angle is vertically opposite to the 133° angle. [A1]
$47^\circ + 133^\circ + 47^\circ + 133^\circ = 360^\circ$ — All four angles at the intersection must sum to 360°. This confirms the answers are correct.
Answer: Two angles of $47^\circ$ and two angles of $133^\circ$
5. Putting All Three Facts Together
In harder exam questions, you will need to chain two or three of these facts in a single problem. The approach is always the same: identify which fact applies at each step, write the equation, and state your reason.
Worked example: Multi-Step Angle Problem
Two straight lines intersect. One angle at the intersection is $(2x + 15)^\circ$ and the angle on a straight line next to it is $(3x - 5)^\circ$. Find the value of $x$ and all four angles at the intersection.
$(2x + 15)^\circ + (3x - 5)^\circ = 180^\circ$ — The two labelled angles are adjacent on a straight line, so they sum to 180°. [M1]
Angles: $83^\circ$, $97^\circ$, $83^\circ$, $97^\circ$ — Vertically opposite angles are equal, so the four angles at the intersection are two pairs. [A1]
$83 + 97 + 83 + 97 = 360^\circ$ — All four angles at a point sum to 360°. Confirmed. ✓
Answer: $x = 34$; angles at the intersection are $83^\circ$, $97^\circ$, $83^\circ$, $97^\circ$
6. Formula Reference and Exam Technique
Angle facts at a point and on a straight line are basic, but they power a huge amount of later geometry. Treat them as conservation rules: angles on a straight line must total 180 degrees, and angles around a point must total 360 degrees. When several unknowns appear, write one equation for each fact you are using and keep your reasoning in order. Students often lose marks by skipping the equation and jumping straight to the answer. A short statement such as "angles on a straight line sum to 180 degrees" makes the logic explicit and earns method credit.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
Vertically opposite angles are the angles directly across from each other where two straight lines cross, and they are always equal. If one angle is 47°, the opposite angle is also 47° and the other two are both 180° - 47° = 133°.
What do angles on a straight line add up to?
Angles on a straight line add up to 180°, which is half a full turn. If one angle on a straight line is 55°, the other is 180° - 55° = 125°.
What do angles at a point add up to?
Angles at a point add up to 360°, because together they form one complete rotation. If two angles at a point are 110° and 120°, the third angle is 360° - 230° = 130°.
What are supplementary angles?
Supplementary angles are angles that add up to 180°, such as angles on a straight line at a common point. For example, 55° and 125° are supplementary because 55° + 125° = 180°.
Why are vertically opposite angles equal?
Label the four angles p, q, r and s around the crossing point. Since p + q = 180° and q + r = 180° (angles on a straight line), p = r, and the same logic gives q = s.
How do you solve straight-line angle problems with algebra?
Write an equation using the angle fact, then solve. If adjacent angles on a straight line are (2x + 15)° and (3x - 5)°, then 5x + 10 = 180, so x = 34 and the angles are 83° and 97°.
How do you find a missing angle at a point when the others are 110° and 120°?
The missing angle is 130°, because angles at a point sum to 360°. So 110° + 120° + x = 360°, giving x = 360° - 230° = 130°.
Two lines cross and one angle is 47°. What are the other angles?
When two lines cross and one angle is 47°, the angles are 47°, 133°, 47° and 133°. The opposite angle is 47° because vertically opposite angles are equal, and 180° - 47° = 133° on the straight line.
Why do angles at a point add up to 360°?
Angles at a point add up to 360° because together they form one complete rotation. It is like standing at the centre of a roundabout and turning all the way round until facing the same direction again.
Why do angles on a straight line add up to 180°?
Angles on a straight line add up to 180° because a straight line is half a full turn. Turning from facing right to facing left is a rotation of 180°, so angles on one side of the line at a point share that half-turn.
What reasons should you give for angles at points and lines?
The reasons to give are the angle facts: 'angles at a point sum to 360°', 'angles on a straight line sum to 180°' and 'vertically opposite angles are equal'. Writing the fact next to each step earns method marks.