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Its arms are $BA$ and $BD$, so this is the corner of the square at $B$.","highlights":[{"text":"angle $ABD$","location":"specification"}]},{"type":"text","order":1,"title":"Use the square fact","content":"Every interior angle of a square is a right angle, so it measures $90^\\circ$. Therefore,\n\n$$\\angle ABD=90^\\circ$$\n\nThis correct value earns **B1**. The equilateral triangle is not needed for this part.","highlights":[{"text":"square $ABDE$","location":"specification"}]}]}],"generatedAt":"2026-07-19T18:40:10.797Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"b1351396-e600-4469-b486-69dad4cf272c","label":"b","content":"Write down the size of angle $BCD$","markscheme":"- $60^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Identifies the angle in an equilateral triangle as 60 degrees.","code":"B"}],"finalAnswers":["60"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"equilateral_angle","label":"Use the triangle property","steps":[{"type":"text","order":1,"title":"Use the equilateral triangle","content":"Triangle $BCD$ is equilateral, so each interior angle is $60^\\circ$. Therefore\n\n$$\\boxed{\\angle BCD=60^\\circ}$$\n\nThis correct angle earns the **B1** mark; because the question says “write down”, no further working is required.","highlights":[{"text":"equilateral triangle $BCD$","location":"specification"},{"text":"Write down the size of angle $BCD$","location":"specification"}]}]}],"generatedAt":"2026-07-19T18:42:29.175Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"517591bc-dcf0-463d-b14a-93ae1d3e008f","label":"c","content":"Find the size of angle $ADC$","markscheme":"- $45 + 60$ M1\n- $105^\\circ$ A1","marks":2,"order":2,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Adds 45 and 60 (assuming BD is a diagonal).","code":"M"},{"id":"p2","marks":1,"description":"Final answer 105.","code":"A"}],"finalAnswers":["105"],"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":"Split the angle","content":"The line $DB$ splits angle $ADC$ into two adjacent angles:\n\n$$\\angle ADC=\\angle ADB+\\angle BDC$$\n\nSo find the square’s part and the triangle’s part, then add them.","highlights":[{"text":"square $ABDE$","location":"specification"},{"text":"equilateral triangle $BCD$","location":"specification"}]},{"type":"text","order":1,"title":"Use the square","content":"A corner of a square is $90^\\circ$. The diagonal $AD$ bisects the angle at $D$; **bisects** means divides into two equal angles. Therefore,\n\n$$\\angle ADB=45^\\circ$$"},{"type":"text","order":2,"title":"Use the triangle","content":"Every angle in an equilateral triangle is $60^\\circ$, so\n\n$$\\angle BDC=60^\\circ$$","highlights":[{"text":"equilateral triangle $BCD$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\angle ADC=\\color{@sky}{\\htmlData{anchor=sum}{45^\\circ+60^\\circ}}$"},{"latex":"$$\\angle ADC=\\color{@green}{\\htmlData{anchor=result}{105^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum"},{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"sum","color":"green","label":"add"}]},"order":3,"title":"Add the angles","description":"Writing 45 + 60 earns M1. Evaluating this correctly as 105° earns A1."}]}],"generatedAt":"2026-07-19T18:46:21.982Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"dbd27de4-05c9-41f3-a841-75a390fe6c27:square_property","label":"(a) Interior Angle of a Square","steps":[{"type":"text","order":0,"title":"Identify the shape","content":"The question tells us that $ABDE$ is a square. Let's look at where angle $ABD$ is located within that shape.","highlights":[{"text":"square ABDE","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Shape: } \\htmlData{anchor=shape}{\\text{Square } ABDE}$"},{"latex":"$$\\text{Interior Angle } \\angle ABD = \\htmlData{anchor=angle}{90^\\circ}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"shape"},{"color":"sky","style":"box","anchor":"angle"}],"connections":[{"to":"angle","from":"shape","color":"sky","label":"property","style":"solid"}]},"order":1,"title":"Recall square properties","description":"In geometry, we know that every interior angle of a square is a right angle."},{"type":"text","order":2,"title":"State the final answer","content":"Since angle $ABD$ is one of the four interior angles of the square $ABDE$, its size must be exactly $90^\\circ$."}]},{"id":"b1351396-e600-4469-b486-69dad4cf272c:direct_recall","label":"(b) Property of Equilateral Triangles","steps":[{"type":"text","order":0,"title":"Identify the shape","content":"The question asks for the size of angle $BCD$. In the diagram, this is an interior angle of the triangle $BCD$.","highlights":[{"text":"equilateral triangle $BCD$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\text{Property: An } \\htmlData{anchor=prop}{\\color{@sky}{\\text{equilateral triangle}}}"},{"latex":"\\text{has interior angles of } \\htmlData{anchor=val}{\\color{@sky}{60^\\circ}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"prop"},{"color":"sky","style":"text-color","anchor":"val"}],"connections":[{"to":"val","from":"prop","color":"sky","label":"property","style":"solid"}]},"order":1,"title":"Recall triangle properties","description":"For an equilateral triangle, we use a fundamental geometric property to find the size of its interior angles."},{"type":"text","order":2,"title":"State the angle size","content":"Since triangle $BCD$ is equilateral, every interior angle, including angle $BCD$, must be $60^\\circ$.","highlights":[{"text":"60^∘","location":"specification"}]}]},{"id":"b1351396-e600-4469-b486-69dad4cf272c:angle_sum_calculation","label":"(b) Angle Sum Calculation","steps":[{"type":"text","order":0,"title":"Identify triangle properties","content":"The question asks for the size of angle $BCD$. We are told that $BCD$ is an **equilateral triangle**. A key property of equilateral triangles is that all three interior angles are equal in size.","highlights":[{"text":"equilateral triangle BCD","location":"specification"},{"text":"angle BCD","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=angle-sum}{x + x + x} = 180^\\circ$"},{"latex":"$$\\htmlData{anchor=equation}{3x} = 180^\\circ$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle-sum"},{"color":"sky","style":"text-color","anchor":"equation"}],"connections":[{"to":"equation","from":"angle-sum","color":"sky","label":"combine like terms","style":"solid"}]},"order":1,"title":"Formulate the equation","description":"Since all three angles are equal, let's call each angle $x$. We know that the sum of angles in any triangle is always $180^\\circ$."},{"type":"text","order":2,"title":"Solve for the angle","content":"To find the value of $x$, we divide the total sum of the angles by 3:\n\n$$\\begin{aligned} 3x &= 180^\\circ \\\\ x &= \\frac{180^\\circ}{3} \\\\ x &= 60^\\circ \\end{aligned}$$\n\nSo, angle $BCD = 60^\\circ$.","calculator":[{"gdc":{"expression":"180 / 3","instructions":"Enter the division calculation on the main screen."},"ti84":{"expression":"180 / 3","instructions":"Type the division and press ENTER."},"desmos":{"expression":"180 / 3","instructions":"Type the calculation in the expression bar."},"description":"You can use a calculator to quickly perform the division."}]}]},{"id":"517591bc-dcf0-463d-b14a-93ae1d3e008f:angle_addition","label":"(c) Angle Addition","steps":[{"type":"text","order":0,"title":"Identify the angle parts","content":"The question asks for the size of angle $ADC$. Looking at the diagram, we can see that angle $ADC$ is formed by two smaller angles joined together:\n1. Angle $ADB$ (part of the square $ABDE$)\n2. Angle $BDC$ (part of the equilateral triangle $BCD$)\n\nTherefore, we can find the total by adding them: $$\\angle ADC = \\angle ADB + \\angle BDC$$","highlights":[{"text":"Find the size of angle $ADC$","location":"specification"}]},{"type":"text","order":1,"title":"Find angle ADB","content":"In the square $ABDE$, the line $AD$ is a diagonal. We know that the interior angles of a square are all $90^\\circ$. A diagonal of a square bisects (cuts in half) these $90^\\circ$ angles.\n\n$$\\angle ADB = \\frac{90^\\circ}{2} = 45^\\circ$$","highlights":[{"text":"square $ABDE$","location":"specification"}]},{"type":"text","order":2,"title":"Find angle BDC","content":"Next, we look at triangle $BCD$. The question states this is an equilateral triangle. In an equilateral triangle, all three interior angles are equal to $60^\\circ$.\n\n$$\\angle BDC = 60^\\circ$$","highlights":[{"text":"equilateral triangle $BCD$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\angle ADC = \\htmlData{anchor=adb}{\\color{@sky}{45^\\circ}} + \\htmlData{anchor=bdc}{\\color{@purple}{60^\\circ}}$"},{"latex":"$$\\angle ADC = 105^\\circ$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"adb"},{"color":"purple","style":"text-color","anchor":"bdc"}],"connections":[{"to":"bdc","from":"adb","color":"sky","label":null,"style":"solid"}]},"order":3,"title":"Calculate the final sum","description":"Now, we add the two parts together to find the total size of angle $ADC$."}]}],"generatedAt":"2026-06-08T04:29:39.783Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":39},"walkthroughVideo":null},{"id":"07a712ec-993a-4df1-af99-2c843930e58a","revisionId":"c3b82fd2-8556-48e1-ac3e-ca55db282a44","questionNumber":"10","questionType":"LA","specification":"![Diagram showing angles around a point: 90, 99, 122, and x.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v2/07a712ec-993a-4df1-af99-2c843930e58a/image-1-71f90d8f7986aac22caf0c9f.svg)","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"d4f24042-b08e-4098-a14f-d4e951a37c96","label":"a","content":"Work out the size of the angle marked $x$.","markscheme":"- $99 + 122 + 90 = 311$ M1\n- $360 - 311 = 49^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correctly calculates $x = 49$.","code":"B"}],"finalAnswers":["49"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angles_around_a_point","label":"Angles around a point","steps":[{"type":"text","order":0,"title":"Recall the angle fact","content":"Angles around a point add up to $360^\\circ$. 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","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}],"subjectCode":"gcse_maths","summaries":[{"id":"4fe0ce56-8606-4ab8-b705-f9e5419d1d0b","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"552b2cc4-39be-4258-a6ea-f6235bc1c95d","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"c519db2a-9077-4b51-b952-b3c63596bfae","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"516d1fe3-8cfb-4bd6-ae90-1ec6fe1a2ec6","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"39717d19-7e89-4142-add3-7063f2bfb9d2","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"4172501d-9c7c-4383-bd8e-4f011fe1005d","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"4f229e13-6f55-4ac4-b678-7e3031076da0","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"925da1bf-89c0-47ff-8733-87da6fce81f6","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b9c7a426-3cc7-4bd9-b9aa-bb2c12da4b3c","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c451d101-18f1-4a40-b48a-24d52d2ca5de","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"f9d420bc-2783-4d13-80fb-396a27541ab9","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"ab9a072b-3e00-4684-9566-bf7cf80f06f5","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"1702b49f-f6bd-4382-937f-8f7fcfce6196","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"5ff005bf-704c-4ed0-bab4-e41425d8f525","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b136dad8-4851-4898-9ede-c216c6c95391","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b7e5e68e-6097-4589-a75a-424308d9d22d","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"f9d8947f-7877-4d32-8759-9de6fe925bc0","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"0787502d-dc64-4e4a-a515-1ac132a03b83","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"f5c70828-a463-4b3b-a40f-980bd3465dd8","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"5a3c696b-17f2-4437-8036-20b4dbd8e633","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"fa295aae-8afc-4e8b-b521-9e0648c3d29a","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"0b525798-3060-4b48-8540-a5464043e7cd","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"07a712ec-993a-4df1-af99-2c843930e58a","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a6454e73-0734-486c-95d4-df33cf26ddda","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1076a56b-692c-4491-9c91-72780605c9e0","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"fdf48a6c-0bf2-4a89-a88d-50818f6174ee","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c369008e-1db0-43bd-8d6d-5a9c7f34e2f6","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Angles","topicId":"6362325b-dd27-49b1-a525-93b90580475e","topicSidebar":"$L3b","topicSlug":"grade-2-2-10-angles","topicTitle":"Angles"}]}],"$L3c","$L3d"]}]

Angles

Question 10

Angles Questions

Angles

Question 10

Angles Questions