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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":"$$\\color{@green}{x=180^\\circ-118^\\circ=62^\\circ}$"}]},"order":1,"title":"Calculate the angle","description":"The direct calculation 180 − 118 = 62° earns B1."}]}],"generatedAt":"2026-07-19T01:56:08.144Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"5f7cc59c-157a-4f35-bdbc-09b0a4029ce7","label":"b","content":"Work out the size of the angle marked $y$.","markscheme":"- $180 - (62 + 62) = 56^\\circ$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Calculates y = 56 using isosceles triangle properties.","code":"B"}],"finalAnswers":["56"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_facts","label":"Use angle facts","steps":[{"type":"text","order":0,"title":"Use equal base 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From part (a), one base angle is $62^\\circ$, so the other base angle is also $62^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\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 answer.","location":"specification"}]}]}],"generatedAt":"2026-07-19T02:07:03.722Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"c369008e-1db0-43bd-8d6d-5a9c7f34e2f6","revisionId":"40069e43-a76b-4004-96f7-c723403b1c1b","questionNumber":"14","questionType":"LA","specification":"$$ABC$ is a straight line.\n\n![Triangle ABD with exterior angle 124 at B and angle 68 at D.](https://assets.mathsgenie.co.uk/booklet/f1e5c16456acc479c9ffc99763a925a16887eb760f5f2035e0733cdb616f3bdb.jpg)","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"0c23d77c-1de7-4897-8623-0f9dddf125a4","label":"0","content":"Show that $ABD$ is an isosceles triangle","markscheme":"- $\\angle ABD = 180 - 124 = 56^\\circ$ M1\n- Reason: Angles on a straight line add to $180^\\circ$ C1\n- $\\angle DAB = 180 - (56 + 68) = 56^\\circ$ M1\n- Conclusion: Two angles are equal ($56^\\circ$), therefore it is an isosceles triangle. C1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculates angle ABD = 56.","code":"M"},{"id":"p2","marks":1,"description":"States reason for angle ABD (straight line).","code":"C"},{"id":"p3","marks":1,"description":"Calculates angle DAB = 56.","code":"M"},{"id":"p4","marks":1,"description":"Concludes triangle is isosceles because two angles are equal.","code":"C"}],"finalAnswers":[],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_reasoning","label":"Use angle facts","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=abd_sum}{\\color{@sky}{\\angle ABD+124^\\circ=180^\\circ}}$"},{"latex":"$$\\htmlData{anchor=abd_subtract}{\\color{@orange}{\\angle ABD=180^\\circ-124^\\circ}}$"},{"latex":"$$\\htmlData{anchor=abd_result}{\\color{@green}{\\angle ABD=56^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"abd_sum"},{"color":"orange","style":"text-color","anchor":"abd_subtract"},{"color":"green","style":"text-color","anchor":"abd_result"}],"connections":[{"to":"abd_subtract","from":"abd_sum","color":"sky","style":"solid"}]},"order":1,"title":"Calculate angle ABD","description":"Angles on a straight line add to a straight angle. Rearranging to find the interior angle earns M1; stating this angle fact earns C1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=dab_sum}{\\color{@sky}{\\angle DAB+56^\\circ+68^\\circ=180^\\circ}}$"},{"latex":"$$\\htmlData{anchor=dab_subtract}{\\color{@orange}{\\angle DAB=180^\\circ-(56^\\circ+68^\\circ)}}$"},{"latex":"$$\\htmlData{anchor=dab_result}{\\color{@green}{\\angle DAB=56^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"dab_sum"},{"color":"orange","style":"text-color","anchor":"dab_subtract"},{"color":"green","style":"text-color","anchor":"dab_result"}],"connections":[{"to":"dab_subtract","from":"dab_sum","color":"sky","style":"solid"}]},"order":2,"title":"Calculate angle DAB","description":"The interior angles of a triangle add to a straight angle. Substitute the two known angles, then find the remaining angle. This earns M1."},{"type":"text","order":3,"title":"Conclude it is isosceles","content":"We have shown that\n\n$$\\angle ABD=\\angle DAB=56^\\circ.$$\n\nA triangle with two equal angles has two equal opposite sides. Here, $AD=BD$, so $ABD$ is an isosceles triangle. Identifying the two equal angles and giving this conclusion earns C1.","highlights":[{"text":"Show that $ABD$ is an isosceles triangle","location":"specification"}]}]}],"generatedAt":"2026-07-20T10:40:32.852Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0c23d77c-1de7-4897-8623-0f9dddf125a4:angles-straight-line-and-triangle","label":"Angles on a Straight Line and Triangle Sum","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"To show that triangle $ABD$ is an isosceles triangle, we need to find its interior angles and demonstrate that two of them are equal."},{"type":"text","order":1,"title":"Find angle ABD","content":"Since $ABC$ is a straight line, the angles on it add up to $180^\\circ$.\n\nWe can calculate the interior angle $\\angle ABD$ by subtracting the given exterior angle $124^\\circ$ from $180^\\circ$:\n\n$$\\angle ABD = 180^\\circ - 124^\\circ = 56^\\circ$$\n\n**Reason:** Angles on a straight line add up to $180^\\circ$. *(This step and reason earn the first 2 marks)*","highlights":[{"text":"ABC is a straight line.","location":"specification"}]},{"type":"text","order":2,"title":"Find angle DAB","content":"Now we look at the triangle $ABD$. The sum of angles in any triangle is $180^\\circ$. \n\nWe are given that $\\angle ADB = 68^\\circ$, and we just found that $\\angle ABD = 56^\\circ$. Let's calculate the final angle, $\\angle DAB$:\n\n$$\\angle DAB = 180^\\circ - (56^\\circ + 68^\\circ)$$\n$$\\angle DAB = 180^\\circ - 124^\\circ = 56^\\circ$$\n\n**Reason:** Angles in a triangle add up to $180^\\circ$. *(This calculation earns the 3rd mark)*"},{"type":"text","order":3,"title":"Conclude the proof","content":"We have found the interior angles of triangle $ABD$:\n- $\\angle ABD = 56^\\circ$\n- $\\angle DAB = 56^\\circ$\n- $\\angle ADB = 68^\\circ$\n\nSince two angles are equal ($\\angle ABD = \\angle DAB = 56^\\circ$), triangle $ABD$ must be an **isosceles** triangle. *(This conclusion earns the final mark)*"}]},{"id":"0c23d77c-1de7-4897-8623-0f9dddf125a4:exterior-angle-theorem","label":"Using the Exterior Angle Theorem","steps":[{"type":"text","order":0,"title":"Identify the exterior angle","content":"An **exterior angle** of a triangle is formed by extending one of its sides. In this diagram, the line $ABC$ is a straight line, which makes the $124^\\circ$ angle at $B$ (specifically $\\angle DBC$) the exterior angle to the triangle $ABD$ at vertex $B$."},{"type":"text","order":1,"title":"Apply Exterior Angle Theorem","content":"The **Exterior Angle Theorem** states that the exterior angle of a triangle is equal to the sum of its two opposite interior angles.\n\nFor $\\triangle ABD$, this means:\n$$\\text{Exterior Angle } (\\angle DBC) = \\angle DAB + \\angle ADB$$\n\nSubstituting our known values:\n$$124^\\circ = \\angle DAB + 68^\\circ$$"},{"type":"text","order":2,"title":"Calculate angle DAB","content":"We rearrange the equation to find the interior angle $\\angle DAB$:\n\n$$\\angle DAB = 124^\\circ - 68^\\circ$$\n$$\\angle DAB = 56^\\circ$$"},{"type":"text","order":3,"title":"Calculate angle ABD","content":"Next, we find the interior angle $\\angle ABD$. Since angles on a straight line add up to $180^\\circ$:\n\n$$\\angle ABD = 180^\\circ - 124^\\circ$$\n$$\\angle ABD = 56^\\circ$$\n\n*(This earns you the first method mark **M1** and communication mark **C1** for stating the straight-line rule)*."},{"type":"text","order":4,"title":"Draw the conclusion","content":"We have found two of the interior angles in $\\triangle ABD$:\n- $\\angle DAB = 56^\\circ$ (from **M1**)\n- $\\angle ABD = 56^\\circ$\n\nSince two angles in the triangle are equal ($56^\\circ$), the triangle has two equal sides opposite those angles. \n\nTherefore, $\\triangle ABD$ is an **isosceles triangle** (earning the final **C1** mark)."}]}],"generatedAt":"2026-06-13T04:07:29.576Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":32},"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":"easy","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Angles

Question 13

Angles Questions

Angles

Question 13

Angles Questions