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34^\\circ$.","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"70fc31d2-7a47-456b-879f-8bf75e48908f","label":"a","content":"Work out the size of the angle $XYZ$.","markscheme":"- $112 + 34 = 146$ M1\n- $180 - 146 = 34^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct calculation and result of $34^\\circ$","code":"A"}],"finalAnswers":["34","34°"],"generatedAt":"2026-05-06T13:59:09.023Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triangle_angle_sum","label":"Use the angle sum","steps":[{"type":"text","order":0,"title":"Recall the angle sum","content":"The interior angles of any triangle add to $180^\\circ$. The two known angles are $112^\\circ$ and $34^\\circ$, so first add these together.","highlights":[{"text":"angle $YXZ = 112^\\circ$ and angle $XZY = 34^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=known_angles}{\\color{@sky}{112^\\circ+34^\\circ}}$"},{"latex":"$$\\htmlData{anchor=known_total}{\\color{@sky}{146^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"known_angles"},{"color":"sky","style":"text-color","anchor":"known_total"}],"connections":[{"to":"known_total","from":"known_angles","color":"sky","style":"solid"}]},"order":1,"title":"Add known angles","description":"This calculation earns the method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\angle XYZ=180^\\circ-\\htmlData{anchor=known_total_sub}{\\color{@orange}{146^\\circ}}$"},{"latex":"$$\\htmlData{anchor=missing_angle}{\\color{@green}{\\angle XYZ=34^\\circ}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"known_total_sub"},{"color":"green","style":"text-color","anchor":"missing_angle"}],"connections":[{"to":"missing_angle","from":"known_total_sub","color":"green","style":"solid"}]},"order":2,"title":"Subtract from total","description":"Subtract the known-angle total from the angle sum of a triangle. The correct missing angle earns A1."}]}],"generatedAt":"2026-07-19T07:02:39.242Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"4274a316-5618-4b12-ad13-a47725d2bed0","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":"State that angles in a triangle sum to 180 degrees","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-06T13:59:09.023Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_sum_reason","label":"Use the triangle angle sum","steps":[{"type":"text","order":0,"title":"State the reason","content":"**Model response:** Angles in a triangle add up to $180^\\circ$.\n\nThis correct reason earns **B1**.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]}]}],"generatedAt":"2026-07-19T07:04:12.783Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"70fc31d2-7a47-456b-879f-8bf75e48908f:sum_and_subtract","label":"(a) Sum and Subtract","steps":[{"type":"text","order":0,"title":"Identify the given angles","content":"We are given a triangle $XYZ$ with two known interior angles:\n- $\\text{Angle } YXZ = 112^\\circ$\n- $\\text{Angle } XZY = 34^\\circ$\n\nWe need to find the remaining angle, $XYZ$. Recall that the sum of all interior angles in any triangle is always $180^\\circ$. Using the **Sum and Subtract** approach, we will first add the two known angles together and then subtract their sum from $180^\\circ$.","highlights":[{"text":"angle YXZ = 112°","location":"specification"},{"text":"angle XZY = 34°","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=angle1}{\\color{@sky}{112^\\circ}} + \\htmlData{anchor=angle2}{\\color{@purple}{34^\\circ}} = \\htmlData{anchor=sum}{\\color{@teal}{146^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle1"},{"color":"purple","style":"text-color","anchor":"angle2"},{"color":"teal","style":"box","anchor":"sum"}],"connections":[]},"order":1,"title":"Sum the given angles","description":"First, we add the two given interior angles together to find their combined sum: $$112^\\circ + 34^\\circ = 146^\\circ$$ This step earns the first mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$180^\\circ - \\htmlData{anchor=sum-sub}{\\color{@teal}{146^\\circ}} = \\htmlData{anchor=ans}{\\color{@green}{34^\\circ}}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"sum-sub"},{"color":"green","style":"box","anchor":"ans"}],"connections":[{"to":"ans","from":"sum-sub","color":"green","label":null,"style":"dashed"}]},"order":2,"title":"Subtract from 180°","description":"Now, subtract this combined sum from $180^\\circ$ to find the size of the remaining angle $XYZ$: $$180^\\circ - 146^\\circ = 34^\\circ$$ This step earns the final accuracy mark (A1)."},{"type":"text","order":3,"title":"State final answer","content":"The size of angle $XYZ$ is **$34^\\circ$**."}]},{"id":"70fc31d2-7a47-456b-879f-8bf75e48908f:sequential_subtraction","label":"(a) Sequential Subtraction","steps":[{"type":"text","order":0,"title":"Recall total triangle angles","content":"The angles in any triangle always add up to $180^\\circ$.\n\nTo find the size of the missing angle $XYZ$, we can start with this total of $180^\\circ$ and subtract the known angles one by one."},{"type":"text","order":1,"title":"Subtract the first angle","content":"We begin by subtracting the first known angle, $\\text{angle } YXZ = 112^\\circ$, from our total of $180^\\circ$:\n\n$$180^\\circ - 112^\\circ = 68^\\circ$$","highlights":[{"text":"angle YXZ = 112°","location":"specification"}]},{"type":"text","order":2,"title":"Subtract the second angle","content":"Next, we take the remaining $68^\\circ$ and subtract the second known angle, $\\text{angle } XZY = 34^\\circ$:\n\n$$68^\\circ - 34^\\circ = 34^\\circ$$","highlights":[{"text":"angle XZY = 34°","location":"specification"}]},{"type":"text","order":3,"title":"State the final answer","content":"After subtracting both known angles sequentially, we are left with the size of the missing angle $XYZ$:\n\n$$\\text{Angle } XYZ = 34^\\circ$$"}]},{"id":"4274a316-5618-4b12-ad13-a47725d2bed0:angle_sum_theorem","label":"(b) Angle Sum Theorem","steps":[{"type":"text","order":0,"title":"Identify the geometric rule","content":"To justify how we calculate the missing angle in triangle $XYZ$, we need to state the fundamental geometric property of triangles, known as the **Angle Sum Theorem**."},{"type":"text","order":1,"title":"State the reason","content":"The Angle Sum Theorem states that the sum of the interior angles in any triangle is always $180^\\circ$.\n\nTo secure the mark, we state this property clearly:\n\n**\"Angles in a triangle add up to $180^\\circ$\"**"}]}],"generatedAt":"2026-06-12T23:47:40.190Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":63},"walkthroughVideo":null},{"id":"b9c7a426-3cc7-4bd9-b9aa-bb2c12da4b3c","revisionId":"bdcb6439-8a3f-4115-92f3-c00c3e3b5643","questionNumber":null,"questionType":"LA","specification":"$$EFG$ is a straight line.\n\n![Triangle EFH with exterior angle 140 at F and angle 100 at H.](https://assets.mathsgenie.co.uk/questions/4b881f70-73aa-4cd8-a931-4c709582dedc.png)","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"af0da346-a3cb-426b-84f0-b29b15e2112e","label":"a","content":"Show that $EFH$ is an isosceles triangle.","markscheme":"- $\\angle EFH = 180 - 140 = 40^\\circ$ M1\n- Reason: Angles on a straight line add to $180^\\circ$ C1\n- $\\angle HEF = 180 - (40 + 100) = 40^\\circ$ M1\n- Conclusion: Two angles are equal ($40^\\circ$), therefore it is an isosceles triangle. C1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculates interior angle $\\angle EFH = 40^\\circ$.","code":"M"},{"id":"p2","marks":1,"description":"States that angles on a straight line sum to $180^\\circ$.","code":"C"},{"id":"p3","marks":1,"description":"Calculates third interior angle $\\angle HEF = 40^\\circ$.","code":"M"},{"id":"p4","marks":1,"description":"Concludes that the triangle is isosceles because two angles are equal.","code":"C"}],"finalAnswers":["Two angles are equal (40°), therefore it is an isosceles triangle."],"generatedAt":"2026-05-06T13:58:30.750Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_facts","label":"Use angle facts","steps":[{"type":"text","order":0,"title":"Use the straight line","content":"The $140^\\circ$ angle and $\\angle EFH$ meet at $F$ on the straight line $EFG$. Angles on a straight line add to $180^\\circ$. Stating this reason earns **C1**.","highlights":[{"text":"$$EFG$ is a straight line.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$180^\\circ=\\angle EFH+\\htmlData{anchor=f_given}{\\color{@orange}{140^\\circ}}$"},{"latex":"$$\\angle EFH=180^\\circ-\\htmlData{anchor=f_subtract}{\\color{@orange}{140^\\circ}}$"},{"latex":"$$\\angle EFH=\\color{@green}{40^\\circ}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"f_given"},{"color":"orange","style":"text-color","anchor":"f_subtract"}],"connections":[{"to":"f_subtract","from":"f_given","color":"orange","label":"subtract"}]},"order":1,"title":"Find the angle at F","description":"Subtract the exterior angle from the angle on a straight line. This calculation earns M1."},{"type":"text","order":2,"title":"Use the triangle sum","content":"The interior angles of a triangle add to $180^\\circ$. Triangle $EFH$ now has angles $40^\\circ$ at $F$ and $100^\\circ$ at $H$, so subtract both to find the angle at $E$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$180^\\circ=\\angle HEF+\\htmlData{anchor=e_known}{\\color{@orange}{(40^\\circ+100^\\circ)}}$"},{"latex":"$$\\angle HEF=180^\\circ-\\htmlData{anchor=e_subtract}{\\color{@orange}{(40^\\circ+100^\\circ)}}$"},{"latex":"$$\\angle HEF=\\color{@green}{40^\\circ}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"e_known"},{"color":"orange","style":"text-color","anchor":"e_subtract"}],"connections":[{"to":"e_subtract","from":"e_known","color":"orange","label":"subtract"}]},"order":3,"title":"Find the angle at E","description":"Subtract the two known interior angles from the triangle total. This calculation earns the second M1."},{"type":"text","order":4,"title":"State the conclusion","content":"We have shown that $\\angle EFH=\\angle HEF=40^\\circ$. A triangle with two equal angles is **isosceles**, so triangle $EFH$ is isosceles. This explicit comparison and conclusion earns the final **C1**.","highlights":[{"text":"Show that $EFH$ is an isosceles triangle.","location":"specification"}]}]}],"generatedAt":"2026-07-19T16:12:01.862Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"af0da346-a3cb-426b-84f0-b29b15e2112e:interior_angles","label":"Using Interior Angles","steps":[{"type":"text","order":0,"title":"Find angle EFH","content":"We are given that $EFG$ is a straight line. \n\nAngles on a straight line always add up to $180^\\circ$. This allows us to calculate the interior angle $\\angle EFH$:\n\n$$\\angle EFH = 180^\\circ - 140^\\circ = 40^\\circ$$\n\n*(This earns the first method mark **M1** and the explanation mark **C1**)*","highlights":[{"text":"EFG is a straight line.","location":"specification"}]},{"type":"text","order":1,"title":"Find angle HEF","content":"Next, let's look at triangle $EFH$. The sum of the interior angles in any triangle is always $180^\\circ$.\n\nWe know two of the angles:\n- $\\angle EHF = 100^\\circ$\n- $\\angle EFH = 40^\\circ$\n\nNow, we can find the third angle, $\\angle HEF$:\n\n$$\\angle HEF = 180^\\circ - (100^\\circ + 40^\\circ)$$\n$$\\angle HEF = 180^\\circ - 140^\\circ = 40^\\circ$$\n\n*(This earns the second method mark **M1**)*"},{"type":"text","order":2,"title":"State the conclusion","content":"An **isosceles triangle** is defined as a triangle with two equal sides and two equal angles.\n\nSince we have shown that:\n$$\\angle EFH = 40^\\circ \\quad \\text{and} \\quad \\angle HEF = 40^\\circ$$\n\nTwo of the interior angles are equal, which proves that $EFH$ is indeed an isosceles triangle.\n\n*(This earns the final communication mark **C1**)*","highlights":[{"text":"Show that EFH is an isosceles triangle.","location":"specification"}]}]},{"id":"af0da346-a3cb-426b-84f0-b29b15e2112e:exterior_angle_theorem","label":"Using Exterior Angle Theorem","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"To show that triangle $EFH$ is an isosceles triangle, we need to show that two of its interior angles are equal. We can find these angles using the geometric properties shown in the diagram.","highlights":[{"text":"Show that $EFH$ is an isosceles triangle.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=ext-angle}{\\color{@sky}{\\angle HFG}} = \\htmlData{anchor=opp-1}{\\color{@purple}{\\angle HEF}} + \\htmlData{anchor=opp-2}{\\color{@amber}{\\angle EHF}}$$"},{"text":"Substitute the known values from the diagram:"},{"latex":"$$$\\htmlData{anchor=ext-val}{\\color{@sky}{140^\\circ}} = \\htmlData{anchor=opp-1-var}{\\color{@purple}{\\angle HEF}} + \\htmlData{anchor=opp-2-val}{\\color{@amber}{100^\\circ}}$$"},{"text":"Solve for the unknown angle:"},{"latex":"$$$\\htmlData{anchor=final-val}{\\color{@purple}{\\angle HEF}} = 140^\\circ - 100^\\circ = 40^\\circ$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"ext-angle"},{"color":"amber","style":"text-color","anchor":"opp-2"}],"connections":[{"to":"ext-val","from":"ext-angle","color":"sky","label":"Substitute $140^\\circ$","style":"solid"},{"to":"opp-2-val","from":"opp-2","color":"amber","label":"Substitute $100^\\circ$","style":"solid"}]},"order":1,"title":"Find angle HEF","description":"According to the **Exterior Angle Theorem**, the exterior angle of a triangle is equal to the sum of the two opposite interior angles. This gives us a direct way to calculate $\\angle HEF$."},{"type":"text","order":2,"title":"Find angle EFH","content":"Since $EFG$ is a straight line, the angles on a straight line add up to $180^\\circ$:\n\n$$\\angle EFH = 180^\\circ - 140^\\circ = 40^\\circ$$\n\n*Note: Calculating $40^\\circ$ earns **1 method mark (M1)**, and stating the reason \"Angles on a straight line add to $180^\\circ$\" earns **1 communication mark (C1)**.*"},{"type":"text","order":3,"title":"Conclude the proof","content":"We have found the following interior angles for triangle $EFH$:\n- $\\angle HEF = 40^\\circ$\n- $\\angle EFH = 40^\\circ$\n\nSince two angles in the triangle are equal ($\\angle HEF = \\angle EFH = 40^\\circ$), triangle $EFH$ must be an **isosceles** triangle. \n\n*Note: Explaining that the two equal angles make it an isosceles triangle earns the final **1 communication mark (C1)**.*"}]}],"generatedAt":"2026-06-13T02:55:55.598Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":44},"walkthroughVideo":null},{"id":"c451d101-18f1-4a40-b48a-24d52d2ca5de","revisionId":"711f995f-1914-4539-a601-af06a33da2a7","questionNumber":null,"questionType":"LA","specification":"Four angles are formed around a central point. The sizes of three of these angles are $147^\\circ$, $91^\\circ$, and $52^\\circ$. The size of the remaining angle is $y$.","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"72df2ac0-4c7f-4bcd-a50e-fc61c282d980","label":"a","content":"Work out the value of $y$.","markscheme":"- $147 + 91 + 52 = 290$\n- $360 - 290 = 70^\\circ$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct calculation leading to 70.","code":"B"}],"finalAnswers":["70"],"generatedAt":"2026-05-06T13:58:51.969Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angles_around_point","label":"Angles around a point","steps":[{"type":"text","order":0,"title":"Recall the angle fact","content":"Angles around a point add up to $360^\\circ$. Therefore, the three given angles and $y$ must have a total of $360^\\circ$.","highlights":[{"text":"Four angles are formed around a central point.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=known_angles}{\\color{@sky}{147+91+52}}$"},{"latex":"$$\\htmlData{anchor=known_total}{\\color{@sky}{290}}$"}],"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 angles whose sizes are given."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=\\htmlData{anchor=subtraction}{\\color{@orange}{360^\\circ-290^\\circ}}$"},{"latex":"$$\\htmlData{anchor=result}{\\color{@green}{y=70^\\circ}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"subtraction"},{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"subtraction","color":"orange","label":"subtract"}]},"order":2,"title":"Subtract from full turn","description":"Angles around a point total a full turn. Subtract the known total to find the remaining angle; the correct final value earns the B1 mark."}]}],"generatedAt":"2026-07-19T16:44:15.614Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"8e81982a-c585-4da9-b6f6-b2f34013c8db","label":"b","content":"Give a reason for your answer.","markscheme":"- Angles at a point sum to $360^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States that the sum of angles at a point is 360 degrees.","code":"B"}],"finalAnswers":["Angles at a point sum to 360 degrees"],"generatedAt":"2026-05-06T13:58:51.969Z","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":"Angles at a point sum to $360^\\circ$. This exact reason earns the **B1** mark.","highlights":[{"text":"around a central point","location":"specification"}]}]}],"generatedAt":"2026-07-19T16:45:17.708Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"72df2ac0-4c7f-4bcd-a50e-fc61c282d980:sum_and_subtract","label":"(a) Sum and Subtract","steps":[{"type":"text","order":0,"title":"Understand the angle rule","content":"Angles around a central point always add up to a full turn of $360^\\circ$. We are given three angles: $147^\\circ$, $91^\\circ$, and $52^\\circ$. We can find the missing angle $y$ using the **Sum and Subtract** method.","highlights":[{"text":"Four angles are formed around a central point.","location":"specification"},{"text":"147^\\circ","location":"specification"},{"text":"91^\\circ","location":"specification"},{"text":"52^\\circ","location":"specification"}]},{"type":"text","order":1,"title":"Sum the known angles","content":"First, calculate the sum of the three known angles:\n\n$$147^\\circ + 91^\\circ + 52^\\circ = 290^\\circ$$","calculator":[{"gdc":{"expression":"147 + 91 + 52","instructions":"Type `147 + 91 + 52` and press [EXE] or [ENTER]."},"ti84":{"expression":"147 + 91 + 52","instructions":"Type `147 + 91 + 52` on the home screen and press [ENTER]."},"desmos":{"expression":"147 + 91 + 52","instructions":"Type `147 + 91 + 52` in the first expression line."},"description":"Calculate the sum of the three given angles."}]},{"type":"text","order":2,"title":"Subtract from 360","content":"Next, subtract this total from $360^\\circ$ to find the size of the remaining angle $y$:\n\n$$y = 360^\\circ - 290^\\circ = 70^\\circ$$","calculator":[{"gdc":{"expression":"360 - 290","instructions":"Type `360 - 290` and press [EXE] or [ENTER]."},"ti84":{"expression":"360 - 290","instructions":"Type `360 - 290` on the home screen and press [ENTER]."},"desmos":{"expression":"360 - 290","instructions":"Type `360 - 290` in the second expression line."},"description":"Subtract the sum from 360."}]}]},{"id":"72df2ac0-4c7f-4bcd-a50e-fc61c282d980:sequential_subtraction","label":"(a) Sequential Subtraction","steps":[{"type":"text","order":0,"title":"Identify the total angle","content":"Angles around a central point always add up to $360^\\circ$. \n\nTo find the remaining angle $y$, we will start with $360^\\circ$ and subtract each of the given angles ($147^\\circ$, $91^\\circ$, and $52^\\circ$) one by one."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$360^\\circ - \\htmlData{anchor=first-angle}{147^\\circ} = \\htmlData{anchor=first-diff}{213^\\circ}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"first-angle"},{"color":"pink","style":"box","anchor":"first-diff"}],"connections":[]},"order":1,"title":"Subtract the first angle","description":"Subtract the first angle of $147^\\circ$ from the starting total of $360^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=prev-diff-1}{213^\\circ} - \\htmlData{anchor=second-angle}{91^\\circ} = \\htmlData{anchor=sec-diff}{122^\\circ}$$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"prev-diff-1"},{"color":"amber","style":"text-color","anchor":"second-angle"},{"color":"purple","style":"box","anchor":"sec-diff"}],"connections":[]},"order":2,"title":"Subtract the second angle","description":"Now, take the remaining $213^\\circ$ and subtract the second angle of $91^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=prev-diff-2}{122^\\circ} - \\htmlData{anchor=third-angle}{52^\\circ} = \\htmlData{anchor=final-ans}{70^\\circ}$$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"prev-diff-2"},{"color":"teal","style":"text-color","anchor":"third-angle"},{"color":"green","style":"box","anchor":"final-ans"}],"connections":[]},"order":3,"title":"Subtract the third angle","description":"Finally, subtract the third angle of $52^\\circ$ from the remaining $122^\\circ$ to find the value of $y$."},{"type":"text","order":4,"title":"State the final answer","content":"After sequentially subtracting each of the given angles, we find the remaining value:\n\n$$y = 70^\\circ$$"}]},{"id":"72df2ac0-4c7f-4bcd-a50e-fc61c282d980:algebraic_equation","label":"(a) Algebraic Equation","steps":[{"type":"text","order":0,"title":"Recall the geometric fact","content":"Recall that angles around a central point always sum to $360^\\circ$:\n\n$$\\text{Sum of angles} = 360^\\circ$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=angle1}{\\color{@sky}{147^\\circ}} + \\htmlData{anchor=angle2}{\\color{@amber}{91^\\circ}} + \\htmlData{anchor=angle3}{\\color{@purple}{52^\\circ}} + \\htmlData{anchor=angle4}{\\color{@green}{y}} = 360^\\circ$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle1"},{"color":"amber","style":"text-color","anchor":"angle2"},{"color":"purple","style":"text-color","anchor":"angle3"},{"color":"green","style":"text-color","anchor":"angle4"}],"connections":[]},"order":1,"title":"Set up the equation","description":"We can set up an algebraic equation by adding all four angles together and setting the sum equal to $360^\\circ$."},{"type":"text","order":2,"title":"Simplify the equation","content":"Combine the known constant values on the left-hand side:\n\n$$147 + 91 + 52 = 290$$\n\nSubstituting this back into our equation gives:\n\n$$290 + y = 360$$"},{"type":"text","order":3,"title":"Solve for y","content":"Subtract $290$ from both sides of the equation to isolate $y$:\n\n$$\\begin{aligned} y &= 360 - 290 \\\\ y &= 70^\\circ \\end{aligned}$$\n\nThis matches the markscheme requirement and awards the full mark."}]},{"id":"8e81982a-c585-4da9-b6f6-b2f34013c8db:geometric_reasoning","label":"(b) Geometric Angle Fact","steps":[{"type":"text","order":0,"title":"Identify the requirement","content":"In Part 1, you calculated the value of the missing angle $y$. Now, this part of the question asks you to write down the geometric rule or reason that justifies your calculation.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]},{"type":"text","order":1,"title":"State the geometric fact","content":"The fundamental geometric rule we used is that a complete turn around a single point forms a full circle, which measures $360^\\circ$.\n\nTo secure the mark, state this key angle fact clearly:\n\n**Angles at a point sum to $360^\\circ$** (or **angles around a point add up to $360^\\circ$**)."}]}],"generatedAt":"2026-06-13T04:18:08.213Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":96},"walkthroughVideo":null}],"subjectCode":"gcse_maths__ocr","summaries":[{"id":"13cdd186-bf30-4266-afbb-122fa98d9787","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"2f00fad1-d3d2-44e4-aa62-cbdd23004c30","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"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":"0cc17d63-46da-46bc-a901-e04d234a1944","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"39717d19-7e89-4142-add3-7063f2bfb9d2","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"3f097d3b-09c5-475a-b3a0-8d94507691f0","difficultyBand":"easy","calculatorUse":null,"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":"ab9a072b-3e00-4684-9566-bf7cf80f06f5","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"16776ceb-ea68-4180-ada2-b1dc52f9cef1","difficultyBand":"easy","calculatorUse":null,"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":"ab10334c-e6c5-4a91-8c7f-7a2c6cfd827a","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"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":"f118d9a2-a0ee-4008-8760-51385dcee0c2","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"0787502d-dc64-4e4a-a515-1ac132a03b83","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"3c03a014-1bea-47ed-9967-4fcea060b5e8","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"d2336959-71bc-488e-8550-5ebe93356c6d","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"f5c70828-a463-4b3b-a40f-980bd3465dd8","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"0d3f8dfb-be98-434d-aeeb-e54bdd9b7c44","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"f8cbf2cb-c177-45bf-b07a-074b443fa4da","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"5a3c696b-17f2-4437-8036-20b4dbd8e633","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e8983317-2270-4e43-bd95-fbaa3c66d815","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"fb8f2835-873c-427c-a927-f14c138b777e","difficultyBand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Angles

Question 3

Angles Questions

Angles

Question 3

Angles Questions