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58^\\circ$.","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"6c4e292f-9af7-49e9-9f06-3fcba8d2b371","label":"a","content":"Work out the size of the angle $DEF$.","markscheme":"- $64 + 58 = 122$ M1\n- $180 - 122 = 58^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct calculation and result of $58^\\circ$","code":"A"}],"finalAnswers":["58","58°"],"generatedAt":"2026-05-06T13:59:11.120Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_sum","label":"Use the triangle angle sum","steps":[{"type":"text","order":0,"title":"Use the angle sum","content":"The three interior angles in a triangle add to $180^\\circ$. The angle to find is $\\angle DEF$.","highlights":[{"text":"Triangle $DEF$ has angles $\\angle EDF = 64^\\circ$ and $\\angle DFE = 58^\\circ$.","location":"specification"},{"text":"Work out the size of the angle $DEF$.","location":"specification"}]},{"type":"text","order":1,"title":"Add known angles","content":"Add the two known angles. This earns M1 for finding their total.\n\n$$64^\\circ+58^\\circ=122^\\circ$$","highlights":[{"text":"$$\\angle EDF = 64^\\circ$ and $\\angle DFE = 58^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$180^\\circ-122^\\circ=58^\\circ$"},{"text":"Therefore, ∠DEF = 58°."}]},"order":2,"title":"Find the missing angle","description":"Subtract the known total from 180°. This earns A1 for obtaining the missing angle."}]}],"generatedAt":"2026-07-19T18:35:01.553Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"47352429-f454-4795-af64-fc624bf47bea","label":"b","content":"Give a reason for your answer.","markscheme":"- Angles in a triangle add up 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":"2026-05-06T13:59:11.120Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"angle_sum_reason","label":"Use the triangle angle sum","steps":[{"type":"text","order":0,"title":"Give the reason","content":"**Angles in a triangle add up to $180^\\circ$.** This earns the **B1 mark**.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]}]}],"generatedAt":"2026-07-19T18:35:47.717Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"6c4e292f-9af7-49e9-9f06-3fcba8d2b371:angle_sum","label":"(a) Angle Sum of a Triangle","steps":[{"type":"text","order":0,"title":"Recall angle sum rule","content":"The angles inside any triangle always add up to $180^\\circ$.\n\nTo find the missing angle, we can first find the sum of the two angles we already know: $\\angle EDF = 64^\\circ$ and $\\angle DFE = 58^\\circ$.","highlights":[{"text":"\\angle EDF = 64^\\circ","location":"specification"},{"text":"\\angle DFE = 58^\\circ","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=angle1}{\\color{@sky}{64^\\circ}} + \\htmlData{anchor=angle2}{\\color{@amber}{58^\\circ}} = \\htmlData{anchor=sum-val}{\\color{@purple}{122^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle1"},{"color":"amber","style":"text-color","anchor":"angle2"},{"color":"purple","style":"box","anchor":"sum-val"}],"connections":[]},"order":1,"title":"Add the known angles","description":"Adding these two angles together gives us the total measure of the two known corners of the triangle.\n\nThis step earns the first method mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$180^\\circ - \\htmlData{anchor=sum-sub}{\\color{@purple}{122^\\circ}} = \\htmlData{anchor=ans}{\\color{@green}{58^\\circ}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"sum-sub"},{"color":"green","style":"box","anchor":"ans"}],"connections":[{"to":"ans","from":"sum-sub","color":"green","label":"subtract","style":"solid"}]},"order":2,"title":"Subtract from 180","description":"Subtract this sum from $180^\\circ$ to find the final remaining angle, $\\angle DEF$.\n\nThis final calculation completes the problem and earns the accuracy mark (**A1**)."}]},{"id":"47352429-f454-4795-af64-fc624bf47bea:angle_sum_theorem","label":"(b) Angle Sum Theorem","steps":[{"type":"text","order":0,"title":"Recall the geometric property","content":"To give a reason for how we calculated the missing angle in triangle $DEF$, we need to state the fundamental geometric property of triangles.\n\nThis property is known as the **Angle Sum Theorem**, which states that the sum of the interior angles in any triangle is always $180^\\circ$.","highlights":[{"text":"Give a reason for your answer.","location":"specification"}]},{"type":"text","order":1,"title":"State the reason","content":"To secure the mark, we write down the property clearly and concisely as our reason:\n\n**\"Angles in a triangle add up to $180^\\circ$\"** (or \"the sum of angles in a triangle is $180^\\circ$\").","highlights":[{"text":"Angles in a triangle add up to $180^circ$","location":"specification"}]}]}],"generatedAt":"2026-06-13T02:00:35.499Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0787502d-dc64-4e4a-a515-1ac132a03b83","revisionId":"0607f954-4795-4024-82c8-571222c1ccd5","questionNumber":"5","questionType":"LA","specification":"Work out the size of the angle marked $x$.\n\n![Diagram showing angles around a point including 90 and 145 degrees.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/0787502d-dc64-4e4a-a515-1ac132a03b83/image-1-597d442630f33ff43ceb9b34.svg)","difficulty":1,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Angles","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"9bff6448-f9ff-496b-a7a4-321cadf35f9e","label":"0","content":"","markscheme":"- $90 + 145 = 235$ M1\n- $360 - 235 = 125^\\circ$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Sums known angles around the point ($90 + 145$).","code":"M"},{"id":"p2","marks":1,"description":"Subtracts sum from 360 to find 125.","code":"A"}],"finalAnswers":["125"],"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 full turn","content":"Angles around one point make a full turn, so their total is $360^\\circ$. Here, the three angles are $90^\\circ$, $145^\\circ$ and $x$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=known_angles}{\\color{@sky}{90^\\circ+145^\\circ}}$"},{"latex":"$$\\htmlData{anchor=known_total}{\\color{@sky}{235^\\circ}}$"}],"connections":[{"to":"known_total","from":"known_angles","color":"sky","label":"add"}]},"order":1,"title":"Add known angles","description":"First combine the two angles whose sizes are given. This correct calculation earns the method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\htmlData{anchor=subtraction}{\\color{@orange}{360^\\circ-235^\\circ}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{x=125^\\circ}}$"}],"connections":[{"to":"answer","from":"subtraction","color":"green","label":"subtract"}]},"order":2,"title":"Subtract from full turn","description":"Subtract the known total from the full turn. Giving the correct angle with its degree symbol earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-19T18:45:35.228Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"arithmetic","label":"Sum and Subtract Approach","steps":[{"type":"text","order":0,"title":"Identify the total angle","content":"The angles shown in the diagram are arranged around a single point. Angles around a point always sum to $360^{\\circ}$. To find the missing angle $x$, we will follow the Sum and Subtract approach."},{"type":"text","order":1,"title":"Identify the right angle","content":"One of the angles is marked with a square symbol. This symbol represents a **right angle**, which is exactly $90^{\\circ}$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=a1}{145^{\\circ}} + \\htmlData{anchor=a2}{90^{\\circ}}$"},{"latex":"$$\\htmlData{anchor=sum-res}{235^{\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a1"},{"color":"amber","style":"text-color","anchor":"a2"}],"connections":[{"to":"sum-res","from":"a1","color":"sky","label":null,"style":"solid"},{"to":"sum-res","from":"a2","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Sum the known angles","description":"First, we add the two known angles together to find their total."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=sum-prev}{235^{\\circ}}$"},{"latex":"$$x = 360^{\\circ} - \\htmlData{anchor=sum-sub}{235^{\\circ}}$"},{"latex":"$$x = 125^{\\circ}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"sum-prev"},{"color":"purple","style":"text-color","anchor":"sum-sub"}],"connections":[{"to":"sum-sub","from":"sum-prev","color":"purple","label":null,"style":"solid"}]},"order":3,"title":"Subtract from 360","description":"Finally, we subtract this sum from $360^{\\circ}$ to find the size of angle $x$."}]},{"id":"algebraic","label":"Algebraic Equation","steps":[{"type":"text","order":0,"title":"Identify the given angles","content":"From the diagram, we can see three angles meeting at a single point:\n1. An angle of $145^\\circ$.\n2. A right angle, which we know is exactly $90^\\circ$.\n3. The unknown angle $x$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x + \\htmlData{anchor=a1}{\\color{@sky}{90}} + \\htmlData{anchor=a2}{\\color{@amber}{145}} = 360$"},{"latex":"$$x + \\htmlData{anchor=sum}{\\color{@purple}{235}} = 360$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a1"},{"color":"amber","style":"text-color","anchor":"a2"},{"color":"purple","style":"box","anchor":"sum"}],"connections":[{"to":"sum","from":"a1","color":"purple","label":"add","style":"solid"},{"to":"sum","from":"a2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the equation","description":"The sum of all angles around a point is always $360^\\circ$. We can model this relationship as a linear equation."},{"type":"text","order":2,"title":"Solve for x","content":"To find $x$, subtract the sum of the known angles ($235^\\circ$) from $360^\\circ$:\n\n$$\\begin{aligned} x &= 360 - 235 \\\\ x &= 125^\\circ \\end{aligned}$$\n\nIn an exam, you would earn **1 mark** (M1) for correctly summing the angles to 235 and **1 mark** (A1) for the final answer of $125^\\circ$.","calculator":[{"gdc":{"expression":"360 - (90 + 145)","instructions":"Enter the expression into the main calculation screen."},"ti84":{"expression":"360 - 90 - 145","instructions":"Type the subtraction directly on the home screen and press ENTER."},"desmos":{"expression":"360 - 90 - 145","instructions":"Enter the equation into a new expression line."},"description":"Calculate the missing angle by subtracting the known values from 360."}]}]},{"id":"sequential_subtraction","label":"Sequential Subtraction","steps":[{"type":"text","order":0,"title":"Recall angle facts","content":"Angles around a point always add up to a total of $360^\\circ$. We will use this as our starting point."},{"type":"text","order":1,"title":"Identify known angles","content":"In the diagram, we have two known angles. One is explicitly labeled as $145^\\circ$. The other is marked with a square symbol, which represents a right angle of $90^\\circ$.","highlights":[{"text":"145","location":"specification"},{"text":"90","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$360^\\circ - \\htmlData{anchor=angle1}{\\color{@sky}{145^\\circ}} = \\htmlData{anchor=rem1}{215^\\circ}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"angle1"}],"connections":[{"to":"rem1","from":"angle1","color":"sky","label":"first subtraction","style":"solid"}]},"order":2,"title":"Subtract the first angle","description":"Starting from the total of $360^\\circ$, we subtract the first known angle, $145^\\circ$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\htmlData{anchor=prev-rem}{215^\\circ} - \\htmlData{anchor=right-angle}{\\color{@amber}{90^\\circ}} = \\htmlData{anchor=final-x}{\\color{@green}{125^\\circ}}$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"right-angle"},{"color":"green","style":"box","anchor":"final-x"}],"connections":[{"to":"final-x","from":"prev-rem","color":"green","label":"final result","style":"solid"}]},"order":3,"title":"Subtract the second angle","description":"Now, we subtract the right angle ($90^\\circ$) from our remaining total to find $x$."},{"type":"text","order":4,"title":"Verify with a calculator","content":"You can quickly perform these subtractions on your calculator to ensure accuracy.","calculator":[{"gdc":{"expression":"360 - 145 - 90","instructions":"Enter 360 minus 145 minus 90 in the main calculation screen."},"ti84":{"expression":"360 - 145 - 90","instructions":"Type 360 - 145, press ENTER, then type - 90 and press ENTER."},"desmos":{"expression":"360 - 145 - 90","instructions":"Enter the sequential subtraction into the expression bar."},"description":"Subtract the angles sequentially from 360."}]},{"type":"text","order":5,"title":"Final Answer","content":"The size of the angle marked $x$ is $125^\\circ$."}]}],"generatedAt":"2026-06-08T02:38:27.042Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0b525798-3060-4b48-8540-a5464043e7cd","revisionId":"6ca20a76-eda3-4d75-8097-b1ab9d5a389d","questionNumber":"9","questionType":"LA","specification":"The diagram shows a square $ABDE$ and an equilateral triangle $BCD$.\n\n![Square and equilateral triangle sharing a side.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/0b525798-3060-4b48-8540-a5464043e7cd/image-1-200ccda8b53efee2f0ed2dd4.svg)","difficulty":1,"difficultyBand":"medium","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}],"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 8

Angles Questions

Angles

Question 8

Angles Questions