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an equilateral triangle BCD attached along side BD.](https://assets.mathsgenie.co.uk/question-figures/v1/6ecc1c132247fab67111809a2d308ecb.svg)","difficulty":1,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"dbd27de4-05c9-41f3-a841-75a390fe6c27","label":"a","content":"Write down the size of angle $ABD$","markscheme":"- $90^\\circ$ B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Identifies the angle in a square as 90 degrees.","code":"B"}],"finalAnswers":["90"],"generatedAt":"2026-05-05T10:05:14.256Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"square_angles","label":"Use the square angle","steps":[{"type":"text","order":0,"title":"Identify the angle","content":"In $\\angle ABD$, the middle letter $B$ is the vertex. 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":"34cef9e9-c447-417b-8753-ba74243e6299","revisionId":"ded62354-da6f-4e6b-b89a-6646b278bfce","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","tier":null,"totalMarks":1,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"e24e8ddb-3bbb-41f1-8550-d3e01bf5e09b","label":"0","content":"In the space below draw an angle of $250^\\circ$ and label the angle $B$.","markscheme":"Correct reflex angle drawn within tolerance ($248^\\circ - 252^\\circ$) and labeled B. B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct reflex angle drawn within tolerance ($248^\\circ - 252^\\circ$) and labelled B.","code":"B","acceptable":["A reflex angle between $248^\\circ$ and $252^\\circ$, labelled B"]}],"finalAnswers":[],"generatedAt":"2026-09-22T14:13:02.692Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"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","tier":null,"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 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Angles

Question 9

Angles Questions

Angles

Question 9

Angles Questions