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S.](https://assets.mathsgenie.co.uk/questions/7bda5f63-b6b6-43de-a856-e75645f1165f.png)","difficulty":2,"difficultyBand":"medium","totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"3c0f97a9-57c9-4942-aaa8-9af858a110f0","label":"a","content":"Show that $PQS$ is an isosceles triangle.","markscheme":"- $\\angle PQS = 180 - 115 = 65^\\circ$ M1\n- Reason: Angles on a straight line add to $180^\\circ$ C1\n- $\\angle QPS = 180 - (65 + 50) = 65^\\circ$ M1\n- Conclusion: Two angles are equal ($65^\\circ$), therefore it is an isosceles triangle. C1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculates interior angle $\\angle PQS = 65^\\circ$.","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 $\\angle QPS = 65^\\circ$.","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 (65°), therefore it is an isosceles triangle."],"generatedAt":"2026-05-06T13:58:29.034Z","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 given $115^\\circ$ angle and $\\angle PQS$ meet at $Q$ on the straight line $PQR$. Angles on a straight line add to $180^\\circ$, so subtract the exterior angle from $180^\\circ$. Stating this angle fact earns the **C1** mark.","highlights":[{"text":"$$PQR$ is a straight line.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\angle PQS+\\color{@orange}{\\htmlData{anchor=exterior_given}{115^\\circ}}=180^\\circ$"},{"latex":"$$\\angle PQS=180^\\circ-\\color{@orange}{\\htmlData{anchor=exterior_subtracted}{115^\\circ}}$"},{"latex":"$$\\angle PQS=\\color{@sky}{\\htmlData{anchor=q_result}{65^\\circ}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"exterior_given"},{"color":"orange","style":"text-color","anchor":"exterior_subtracted"},{"color":"sky","style":"text-color","anchor":"q_result"}]},"order":1,"title":"Find the angle at Q","description":"Subtracting the exterior angle from the straight-line total earns the first M1 mark."},{"type":"text","order":2,"title":"Use the triangle sum","content":"The interior angles of a triangle add to $180^\\circ$. Triangle $PQS$ now has angles $65^\\circ$ at $Q$ and $50^\\circ$ at $S$, so subtract both from $180^\\circ$ to find the angle at $P$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\angle QPS+\\color{@orange}{\\htmlData{anchor=p_known_sum}{65^\\circ+50^\\circ}}=180^\\circ$"},{"latex":"$$\\angle QPS=180^\\circ-\\color{@orange}{\\htmlData{anchor=p_subtract_sum}{(65^\\circ+50^\\circ)}}$"},{"latex":"$$\\angle QPS=\\color{@sky}{\\htmlData{anchor=p_result}{65^\\circ}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"p_known_sum"},{"color":"orange","style":"text-color","anchor":"p_subtract_sum"},{"color":"sky","style":"text-color","anchor":"p_result"}]},"order":3,"title":"Find the angle at P","description":"Correctly using the triangle angle sum earns the second M1 mark."},{"type":"text","order":4,"title":"State the conclusion","content":"We have shown that $\\angle PQS=\\angle QPS=65^\\circ$. Therefore, triangle $PQS$ has two equal angles, so it is **isosceles**. This stated conclusion earns the final **C1** mark. The equal angles are opposite the equal sides, so $PS=QS$.","highlights":[{"text":"Show that $PQS$ is an isosceles triangle.","location":"specification"}]}]}],"generatedAt":"2026-07-19T07:55:27.209Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"3c0f97a9-57c9-4942-aaa8-9af858a110f0:angle-sum-triangle","label":"Using Triangle Angle Sum","steps":[{"type":"text","order":0,"title":"Analyze the diagram","content":"To show that $PQS$ is an **isosceles triangle**, we need to find its interior angles. An isosceles triangle has at least two equal angles.\n\nFrom the given diagram, we are given:\n- The line $PQR$ is a straight line.\n- The exterior angle $\\angle SQR = 115^\\circ$.\n- The interior angle $\\angle QSP = 50^\\circ$."},{"type":"text","order":1,"title":"Find angle PQS","content":"Since $PQR$ is a straight line, the angles on a straight line add up to $180^\\circ$.\n\nWe can calculate the interior angle $\\angle PQS$:\n$$\\angle PQS = 180^\\circ - 115^\\circ = 65^\\circ$$\n\n*Note: In an exam, you must write down the reason: **\"Angles on a straight line add to $180^\\circ$\"** to earn a communication mark ($C1$).*"},{"type":"text","order":2,"title":"Calculate angle QPS","content":"Now we look at the triangle $PQS$. The interior angles of any triangle always add up to $180^\\circ$.\n\nWe know two of the angles:\n- $\\angle PQS = 65^\\circ$\n- $\\angle QSP = 50^\\circ$\n\nWe find the third angle, $\\angle QPS$:\n$$\\angle QPS = 180^\\circ - (65^\\circ + 50^\\circ)$$\n$$\\angle QPS = 180^\\circ - 115^\\circ = 65^\\circ$$"},{"type":"text","order":3,"title":"Draw conclusion","content":"We have found the interior angles of triangle $PQS$:\n- $\\angle PQS = 65^\\circ$\n- $\\angle QPS = 65^\\circ$\n- $\\angle QSP = 50^\\circ$\n\nSince two angles are equal ($\\angle PQS = \\angle QPS = 65^\\circ$), triangle $PQS$ is indeed an **isosceles triangle**."}]},{"id":"3c0f97a9-57c9-4942-aaa8-9af858a110f0:exterior-angle-theorem","label":"Using Exterior Angle Theorem","steps":[{"type":"text","order":0,"title":"Understand the Goal","content":"To show that $\\triangle PQS$ is an **isosceles triangle**, we need to find its interior angles and demonstrate that two of them are equal.","highlights":[{"text":"Show that $PQS$ is an isosceles triangle.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"The exterior angle of a triangle is equal to the sum of the two opposite interior angles."},{"latex":"$$$\\htmlData{anchor=ext}{\\color{@sky}{\\angle SQR}} = \\htmlData{anchor=opp1}{\\color{@purple}{\\angle QPS}} + \\htmlData{anchor=opp2}{\\color{@amber}{\\angle PSQ}}$$"},{"latex":"$$$\\color{@sky}{115^\\circ} = \\color{@purple}{\\angle QPS} + \\color{@amber}{50^\\circ}$$"},{"latex":"$$$\\color{@purple}{\\angle QPS} = 115^\\circ - 50^\\circ = \\color{@purple}{65^\\circ}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"ext"},{"color":"purple","style":"text-color","anchor":"opp1"},{"color":"amber","style":"text-color","anchor":"opp2"}],"connections":[{"to":"opp1","from":"ext","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Use Exterior Angle Theorem","description":"Applying the Exterior Angle Theorem gives us the interior angle at vertex $P$ directly."},{"type":"text","order":2,"title":"Find Angle PQS","content":"Since $PQR$ is a straight line, the angles on a straight line add up to $180^\\circ$:\n\n$$\\begin{aligned} \\angle PQS &= 180^\\circ - 115^\\circ \\\\ &= 65^\\circ \\end{aligned}$$\n\n*Note: Writing down that angles on a straight line add to $180^\\circ$ earns you reasoning marks ($C1$ and $M1$).*"},{"type":"text","order":3,"title":"State the Conclusion","content":"We have found the following interior angles for $\\triangle PQS$:\n- $\\angle QPS = 65^\\circ$\n- $\\angle PQS = 65^\\circ$\n\nSince two of the interior angles are equal ($\\angle QPS = \\angle PQS = 65^\\circ$), we have proven that $\\triangle PQS$ is an **isosceles triangle**."}]}],"generatedAt":"2026-06-13T05:06:11.889Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":24},"walkthroughVideo":null},{"id":"925da1bf-89c0-47ff-8733-87da6fce81f6","revisionId":"f185d1d6-9132-4d8d-a108-81a7e357f310","questionNumber":null,"questionType":"LA","specification":"In triangle $XYZ$, angle $YXZ = 112^\\circ$ and angle $XZY = 34^\\circ$.","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"70fc31d2-7a47-456b-879f-8bf75e48908f","label":"a","content":"Work out the size of the angle $XYZ$.","markscheme":"- $112 + 34 = 146$ M1\n- $180 - 146 = 34^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct calculation and result of $34^\\circ$","code":"A"}],"finalAnswers":["34","34°"],"generatedAt":"2026-05-06T13:59:09.023Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"triangle_angle_sum","label":"Use the angle sum","steps":[{"type":"text","order":0,"title":"Recall the angle sum","content":"The interior angles of any triangle add to $180^\\circ$. The two known angles are $112^\\circ$ and $34^\\circ$, so first add these together.","highlights":[{"text":"angle $YXZ = 112^\\circ$ and angle $XZY = 34^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\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}],"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 1

Angles Questions

Angles

Question 1

Angles Questions