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The four angles shown are therefore parts of one complete turn.","highlights":[{"text":"angle marked $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=known_angles}{\\color{@sky}{99^\\circ+122^\\circ+90^\\circ}}$"},{"latex":"$$\\htmlData{anchor=known_total}{\\color{@sky}{311^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"known_angles"}],"connections":[{"to":"known_total","from":"known_angles","color":"sky","label":"add"}]},"order":1,"title":"Add known angles","description":"First add the three given angles. This correct calculation earns the method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\htmlData{anchor=subtraction}{\\color{@orange}{360^\\circ-311^\\circ}}$"},{"latex":"$$x=\\htmlData{anchor=answer}{\\color{@green}{49^\\circ}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"subtraction"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"subtraction","color":"orange","label":"calculate"}]},"order":2,"title":"Subtract from full turn","description":"Subtract the known total from the complete turn. Giving the correct angle with its degree sign earns A1."}]}],"generatedAt":"2026-07-19T08:29:01.657Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"14aeb394-516b-43ea-9443-bd6470c87f66","label":"b","content":"Give a reason for your answer.","markscheme":"- Angles around a point add to $360^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States that angles around a point sum to 360 degrees.","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_fact","label":"Use the angle fact","steps":[{"type":"text","order":0,"title":"State the angle fact","content":"The labelled angles make one complete turn around the same point, so write: **Angles around a point add to $360^\\circ$.** This earns the **B1 mark**.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]}]}],"generatedAt":"2026-07-19T08:35:47.585Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"sum_subtract","label":"Sum and Subtract","steps":[{"type":"text","order":0,"title":"Identify angle properties","content":"The angles around a point always add up to $360^\\circ$. Looking at the diagram, we have three known angles and one unknown angle, $x$. Note that the square symbol represents a right angle, which is $90^\\circ$.","highlights":[{"text":"90","location":"specification"},{"text":"99","location":"specification"},{"text":"122","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=a1}{\\color{@sky}{90^\\circ}}, \\htmlData{anchor=a2}{\\color{@amber}{99^\\circ}}, \\htmlData{anchor=a3}{\\color{@purple}{122^\\circ}}$"},{"latex":"$$\\text{Sum} = \\htmlData{anchor=s1}{\\color{@sky}{90}} + \\htmlData{anchor=s2}{\\color{@amber}{99}} + \\htmlData{anchor=s3}{\\color{@purple}{122}} = \\htmlData{anchor=sum_res}{311^\\circ}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a1"},{"color":"amber","style":"text-color","anchor":"a2"},{"color":"purple","style":"text-color","anchor":"a3"}],"connections":[{"to":"s1","from":"a1","color":"sky","label":null,"style":"solid"},{"to":"s2","from":"a2","color":"amber","label":null,"style":"solid"},{"to":"s3","from":"a3","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Sum the known angles","description":"First, we calculate the total of the three angles we already know. This earns the first mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Total Sum} = \\htmlData{anchor=prev_sum}{\\color{@teal}{311^\\circ}}$"},{"latex":"$$x = 360^\\circ - \\htmlData{anchor=subtract_sum}{\\color{@teal}{311^\\circ}} = \\htmlData{anchor=final_x}{49^\\circ}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"prev_sum"},{"color":"green","style":"box","anchor":"final_x"}],"connections":[{"to":"subtract_sum","from":"prev_sum","color":"teal","label":null,"style":"solid"}]},"order":2,"title":"Subtract from 360 degrees","description":"To find $x$, we subtract our sum from the total degrees in a circle ($360^\\circ$). This earns the final mark (A1)."},{"type":"text","order":3,"title":"Verify using a calculator","content":"You can quickly verify your answer by adding all four angles together to ensure they equal $360^\\circ$.","calculator":[{"gdc":{"expression":"360 - (90 + 99 + 122)","instructions":"Use the standard calculation mode to subtract the sum of the angles from 360."},"ti84":{"expression":"360 - (90 + 99 + 122)","instructions":"Type the subtraction directly into the main screen."},"desmos":{"expression":"360 - 311","instructions":"Input the expression to see the result immediately."},"description":"Calculate the final angle size."}]}]},{"id":"algebraic","label":"Algebraic Equation","steps":[{"type":"text","order":0,"title":"Recall the geometric rule","content":"First, we need to remember that the angles around a single point always sum to $360^\\circ$. Looking at the diagram, we have four angles: $99^\\circ$, $122^\\circ$, a right angle (indicated by the square symbol), and the unknown angle $x$. Since a right angle is exactly $90^\\circ$, we have all the components needed to form an equation.","highlights":[{"text":"90","location":"specification"},{"text":"99","location":"specification"},{"text":"122","location":"specification"},{"text":"x","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=var}{x} + \\htmlData{anchor=const1}{90} + \\htmlData{anchor=const2}{99} + \\htmlData{anchor=const3}{122} = 360$"},{"latex":"$$x + \\htmlData{anchor=sum}{311} = 360$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"const1"},{"color":"sky","style":"text-color","anchor":"const2"},{"color":"sky","style":"text-color","anchor":"const3"},{"color":"sky","style":"text-color","anchor":"sum"}],"connections":[{"to":"sum","from":"const1","color":"sky","label":null,"style":"solid"},{"to":"sum","from":"const2","color":"sky","label":null,"style":"solid"},{"to":"sum","from":"const3","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Form the equation","description":"We set the sum of all four angles equal to $360^\\circ$."},{"type":"text","order":2,"title":"Solve for x","content":"To find the value of $x$, we subtract the sum of the known angles ($311^\\circ$) from the total ($360^\\circ$):\n\n$$\\begin{aligned} x &= 360 - 311 \\\\ x &= 49 \\end{aligned}$$\n\nAccording to the markscheme, finding the sum $311$ earns you the first mark (M1), and the final result of $49^\\circ$ earns the second mark (A1).","calculator":[{"gdc":{"expression":"360 - (90 + 99 + 122)","instructions":"On the main calculation screen, sum the three known values and subtract the result from 360."},"ti84":{"expression":"360 - (90 + 99 + 122)","instructions":"Enter 90 + 99 + 122 then press ENTER to find the sum. Then, type 360 - ANS (or 360 - 311) and press ENTER."},"desmos":{"expression":"360 - (90 + 99 + 122)","instructions":"Type the full expression into a single bar to get the result instantly."},"description":"Use your calculator to quickly sum the known angles and subtract from 360 to avoid arithmetic errors."}]},{"type":"text","order":3,"title":"State the final answer","content":"The size of the angle marked $x$ is $49^\\circ$."}]},{"id":"geometric_property","label":"State Geometric Property","steps":[{"type":"text","order":0,"title":"Identify the angle configuration","content":"In the provided diagram, the angles $90^\\circ$, $99^\\circ$, $122^\\circ$, and $x$ all meet at a single central point to complete a full turn."},{"type":"text","order":1,"title":"State the geometric property","content":"The standard geometric property required here is that **angles around a point add to $360^\\circ$**. \n\nThis theorem allows us to form an equation to solve for $x$:\n\n$$90^\\circ + 99^\\circ + 122^\\circ + x = 360^\\circ$$","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]}]}],"generatedAt":"2026-06-08T02:39:39.929Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"a6454e73-0734-486c-95d4-df33cf26ddda","revisionId":"e51c8d12-11ef-4158-b569-246a1bc51636","questionNumber":"11","questionType":"LA","specification":"![Triangle ABC with angles 64 and 32 degrees.](https://assets.mathsgenie.co.uk/booklet/de15fc8c0288f7b3983c1ac8c9b77f109848cc5e0c179501c8b288f3689bf56e.jpg)","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"ff3436cd-0d5c-474a-a2ab-ddd779b04092","label":"a","content":"Work out the size of the angle $ABC$.","markscheme":"- $64 + 32 = 96$ M1\n- $180 - 96 = 84^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correctly calculates angle ABC = 84.","code":"B"}],"finalAnswers":["84"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triangle_angles","label":"Use angles in a triangle","steps":[{"type":"text","order":0,"title":"Use the triangle rule","content":"The three interior angles in any triangle add to $180^\\circ$. The known angles are at $A$ and $C$, so add them first to find the total known angle.","highlights":[{"text":"Work out the size of the angle $ABC$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=given_angles}{\\color{@sky}{64^\\circ+32^\\circ}}$"},{"latex":"$$\\htmlData{anchor=known_total}{\\color{@green}{96^\\circ}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"known_total"}],"connections":[{"to":"known_total","from":"given_angles","color":"green","label":"add"}]},"order":1,"title":"Add known angles","description":"Add the two given angles. This correct calculation earns the method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=full_subtraction}{\\color{@green}{180^\\circ-96^\\circ}}$"},{"latex":"$$\\htmlData{anchor=angle_answer}{\\color{@red}{\\angle ABC=84^\\circ}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"full_subtraction"},{"color":"red","style":"text-color-with-background","anchor":"angle_answer"}],"connections":[{"to":"angle_answer","from":"full_subtraction","color":"green","label":"subtract"}]},"order":2,"title":"Subtract from the total","description":"Subtract the known total from the angle sum of a triangle. This gives the accuracy mark."}]}],"generatedAt":"2026-07-20T10:49:56.101Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"e98ad270-6bff-4e6a-a942-4f20434d8259","label":"b","content":"Give a reason for your answer.","markscheme":"- Angles in a triangle add to $180^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States that angles in a triangle sum to 180 degrees.","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"state_angle_fact","label":"State the angle fact","steps":[{"type":"text","order":1,"title":"State the reason","content":"Angles in a triangle add to $180^\\circ$. This precise reason earns the **B1** mark.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]}]}],"generatedAt":"2026-07-19T18:08:19.223Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"ff3436cd-0d5c-474a-a2ab-ddd779b04092:triangle_angle_sum","label":"(a) Triangle Angle Sum","steps":[{"type":"text","order":0,"title":"Identify the given angles","content":"From the given triangle, we are given two interior angles:\n- Angle $CAB = 64^\\circ$\n- Angle $BCA = 32^\\circ$\n\nWe need to find the size of the third angle, $ABC$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=angle1}{\\color{@sky}{64^\\circ}} + \\htmlData{anchor=angle2}{\\color{@amber}{32^\\circ}} = \\htmlData{anchor=sum-val}{\\color{@purple}{96^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle1"},{"color":"amber","style":"text-color","anchor":"angle2"},{"color":"purple","style":"box","anchor":"sum-val"}],"connections":[{"to":"sum-val","from":"angle1","color":"sky","label":null,"style":"solid"},{"to":"sum-val","from":"angle2","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Sum the given angles","description":"First, we add the two known interior angles together. This step is worth the first method mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Angle } ABC = 180^\\circ - \\htmlData{anchor=sum-sub}{\\color{@purple}{96^\\circ}}$"},{"latex":"$$\\phantom{\\text{Angle } ABC} = \\htmlData{anchor=ans-val}{\\color{@green}{84^\\circ}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"sum-sub"},{"color":"green","style":"box","anchor":"ans-val"}],"connections":[{"to":"ans-val","from":"sum-sub","color":"purple","label":null,"style":"solid"}]},"order":2,"title":"Subtract from 180°","description":"Since the interior angles of any triangle always add up to $180^\\circ$, we subtract our sum from $180^\\circ$ to find the remaining angle. This calculation gives us the final accuracy mark (**A1**)."},{"type":"text","order":3,"title":"State final answer","content":"The size of angle $ABC$ is **$84^\\circ$**."}]},{"id":"e98ad270-6bff-4e6a-a942-4f20434d8259:triangle_angle_sum","label":"(b) Triangle Angle Sum Theorem","steps":[{"type":"text","order":0,"title":"Identify the requirement","content":"The question asks us to provide the geometric reason that justifies how we find the missing angle in triangle $ABC$. We need to state the rule clearly to earn the mark.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]},{"type":"text","order":1,"title":"State the geometric rule","content":"To find the missing angle, we use the property that the three interior angles of any triangle always add up to $180^\\circ$.\n\nAccording to the markscheme, the standard way to write this reason is:\n\n$$\\text{Angles in a triangle add to } 180^\\circ$$\n\nStating this rule clearly will earn you the $1$ mark."}]}],"generatedAt":"2026-06-13T00:43:23.358Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"fdf48a6c-0bf2-4a89-a88d-50818f6174ee","revisionId":"235581ec-4d8c-434d-a3f9-18445964f100","questionNumber":"13","questionType":"LA","specification":"![Isosceles triangle with exterior angle 118 degrees.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v2/fdf48a6c-0bf2-4a89-a88d-50818f6174ee/image-1-d0812890a204d4393f2819ef.svg)","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"86ef9fb4-11d0-469a-9f75-1a7f1061298f","label":"a","content":"Work out the size of the angle marked $x$.","markscheme":"- $180 - 118 = 62^\\circ$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Calculates x = 62 using angles on a straight line.","code":"B"}],"finalAnswers":["62"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"supplementary_angles","label":"Use supplementary angles","steps":[{"type":"text","order":0,"title":"Identify the straight line","content":"The angle marked $x$ and the $118^\\circ$ exterior angle form a straight line, so they add to $180^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"text":"Use the straight-line total of 180°."},{"latex":"$$\\color{@green}{x=180^\\circ-118^\\circ=62^\\circ}$"}]},"order":1,"title":"Calculate the angle","description":"The direct calculation 180 − 118 = 62° earns B1."}]}],"generatedAt":"2026-07-19T01:56:08.144Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"5f7cc59c-157a-4f35-bdbc-09b0a4029ce7","label":"b","content":"Work out the size of the angle marked $y$.","markscheme":"- $180 - (62 + 62) = 56^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Calculates y = 56 using isosceles triangle properties.","code":"B"}],"finalAnswers":["56"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_facts","label":"Use angle facts","steps":[{"type":"text","order":0,"title":"Use equal base angles","content":"The triangle is isosceles, so its two base angles are equal. From part (a), one base angle is $62^\\circ$, so the other base angle is also $62^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=eq1}{y+62^\\circ+62^\\circ=180^\\circ}}$"},{"latex":"$$\\color{@amber}{\\htmlData{anchor=eq2}{y=180^\\circ-(62^\\circ+62^\\circ)}}$"},{"latex":"$$\\color{@green}{y=56^\\circ}$"}],"connections":[{"to":"eq2","from":"eq1","color":"sky","style":"solid"}]},"order":1,"title":"Use the angle sum","description":"Angles inside a triangle add to 180°. Subtracting the two known angles from 180° gives the B1 mark."}]}],"generatedAt":"2026-07-19T02:05:03.340Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"34c5dddb-95ae-43e7-8d4a-c77288cdfb43","label":"c","content":"Give reasons for your answer.","markscheme":"- Angles at the base of an isosceles triangle are equal AND angles in a triangle add to $180^\\circ$ B1","marks":1,"order":2,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Provides both required geometric reasons.","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_reasons","label":"Use angle facts","steps":[{"type":"text","order":0,"title":"Identify the equal angles","content":"The two angles labelled $x$ are equal because **angles at the base of an isosceles triangle are equal**. This is the first reason the examiner requires.","highlights":[{"text":"Give reasons for your answer.","location":"specification"}]},{"type":"text","order":1,"title":"State the second reason","content":"The second reason is that the three interior angles of a triangle add to $180^\\circ$. Together with equal base angles, these are both required for the B1 mark.","highlights":[{"text":"Give reasons for your 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Angles

Question 11

Angles Questions

Angles

Question 11

Angles Questions