Diagonals quadrilateral worksheet

I'm sort of flatlining in Quant...I used to be able to get 170s (including on the real ETS exams) on several practice exams, and now am getting in the mid-160s more consistently than not (including on the retake of those exams).
How do you prevent stupid mistakes? For instance, in one question I knew that the degrees of a quadrilateral was 360, and the degrees of a triangle was 180 (it was a two triangles leading to a quadrilateral figure question). An example of this stupid mistake (I've been doing more of this in the past few months than in the first 8) is that I said a right angle + 40 degrees was 140 :\... how do I prevent this (other than practice)?!? Usually stupid mistakes account for about 1/2 of my misses.
The other half is going to be almost always geometry. Are there specific questions from resources where we can just focus on diagonals within shapes? Not too many of these come up, but usually I miss it every time it does.

The diagonals of a right trapezoid intersect at right angles. Prove that the lengths of the diagonal quadrilaterals into which the diagonals are divided by their point of intersection form a geometric progression. I don't really have any ideas, I found 2 similar triangles and tried writing out similarity coefficients.

ABCD is given a quadrilateral his diagonals divide it into four triangles The area of AOB is 2 The area of AOD is 3 THE area of BOC is 1 What is the area of COD?

I get confused in finding the order of which you prove first, We are learning the 2 Collum Proof in proving Quadrilaterals and idk which lines, angles or diagonals to give statements and reasons first. I know that you have to State the given first but idk what to do next, My teacher only accepts Statements and Reasons only in order so I fail every quiz.

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Hello,
I have provided the prompt and response of the GPT model I designed, " Lesson Planner Pro", from the Teachers Pack for those who are not able to access GPT +.
https://chat.openai.com/g/g-Av7maprPq-els-lesson-planner-pro-aus-edition
Prompt:
Design me a lesson plan for Year 7 maths on Demonstrating that the angle sum of a triangle is 180°
Lesson Planner Pro:
For a Year 7 mathematics lesson focused on demonstrating that the angle sum of a triangle is 180°, we will incorporate innovative teaching methods, adhere to the curriculum, and engage students actively. The plan will include a teaching hook, video recommendation, key vocabulary, reading materials, structured lesson outline, and integration of assessment methods and gamification options.

Lesson Plan: Understanding the Angle Sum of a Triangle

Grade: Year 7

Subject: Mathematics

Topic: Angle Sum of a Triangle

Duration: 60 minutes

Australian Curriculum Alignment: Investigate angle properties of triangles and quadrilaterals (ACMMG141)

Objectives:

• Students will understand that the sum of the angles in any triangle is 180°.
  
• Students will be able to demonstrate and justify the angle sum property of triangles through investigation and reasoning.
  
• Students will develop problem-solving skills by applying the angle sum property to find unknown angles in various triangles.
  

Teaching Hook:

Start the lesson with a hands-on activity: provide students with various cut-out triangle shapes. Ask them to tear off each corner and then try to align the corners to form a straight line. This visual and tactile experience will intrigue students and serve as a concrete representation of the angle sum property.

Key Vocabulary:

• Angle
  
• Triangle
  
• Vertex (Vertices)
  
• Sum
  
• Straight Angle (180 degrees)
  

Video Recommendation:

Use a short, engaging educational video that visually demonstrates the angle sum property in triangles. A recommendation would be a video from a reputable educational source such as Khan Academy or BBC Bitesize that explains the concept with animations and examples.

Reading Materials:

Provide a handout with diagrams of different types of triangles (acute, obtuse, and right-angled) showing how the angle sum property applies to each. Include a section with historical context, mentioning that this property was known to ancient mathematicians like Euclid.

Structured Lesson Outline:

Introduction (10 minutes)

• Begin with the teaching hook activity.
  
• Discuss the results of the activity and introduce the concept that all triangles' angles add up to 180°.
  
• Introduce key vocabulary.
  

Direct Instruction (10 minutes)

• Show the recommended video to reinforce the concept.
  
• Discuss the video and clarify any misconceptions.
  

Guided Practice (15 minutes)

• Provide worksheets with different triangles where students measure angles using protractors and calculate the missing angles using the angle sum property.
  
• Circulate the room, offering support and answering questions.
  

Independent Practice (15 minutes)

• Students attempt to solve more complex problems involving the angle sum property, including problems that involve algebraic expressions.

Closure (10 minutes)

• Review key concepts and vocabulary.
  
• Highlight the importance of the angle sum property in understanding geometry.
  

Scaffolds for Diverse Learning Needs:

• Provide modified worksheets with guiding questions for students who need extra support.
  
• Offer extension tasks that challenge students to apply the angle sum property in different contexts, such as finding the angle sums of polygons by dividing them into triangles.
  

Assessment Methods:

• Use the guided practice worksheets as a formative assessment to gauge student understanding.
  
• At the end of the lesson, conduct a quick quiz involving a few problems that require students to apply the angle sum property.
  

Gamification Options:

• Incorporate a "Triangle Challenge" where students work in teams to solve a series of angle sum puzzles. Points are awarded for correct solutions, and the team with the most points wins.
  
• Use digital tools like Kahoot for the closing quiz to make the assessment more engaging.
  

Integration of Social-Emotional Learning (SEL) in Academic Instruction:

• Encourage teamwork and collaboration during the guided practice and gamification activities.
  
• Promote a growth mindset by highlighting the importance of effort and persistence in solving geometric problems.
  

Developmentally Appropriate Activities:

• Ensure the hands-on activities are suitable for Year 7 students, providing clear instructions and support where needed.
  
• Design the independent practice problems to be challenging but achievable, catering to the developmental stage of Year 7 students.
  

This lesson plan is designed to be interactive, engaging, and aligned with the Australian Curriculum, ensuring a comprehensive understanding of the angle sum property of triangles among Year 7 students.

Before renaissance Europe is it safe to argue that it was Indian civilization that contributed the most to the field of mathematics?
Indians contributed to every field that existed at that time from Number theory, geometry, trigonometry, algebra, arithmetic, series, differential and integral calculus, combinatorics, magic squares, spherical trigonometry and many more and in non of these fields can we say that India lacked behind but this can be said for many other civilizations, some of these fields originated in India like differential calculus, combinatorics and many subfields within it
i) Babylonians did not have trigonometry, their geometry was basic and many of these field just about absent, they were however advance in arithmetic and algebra
ii) Greeks, supposedly invented trigonometry and also revolutionized geometry to an extent no other civilization could catch up but had terrible arithmetic, but had no notion of combinatorics, magic squares etc arithmetic was not good and furthermore algebra relied on ad hoc methods but diophantus is considered father of algebra inspite of that
iii) Egypt had more advanced geometry as compared to babylon but in other mathematical fields lagged behind but in arithmetic they made much better progress with fractions as compared to even greeks and chinese
iv) Chinese contributed in multiple fields like algebra, arithmetic and so on and inventing magic squares but fell significantly behind islamic world and india in the later field and trigonometric contributions were minor, they introduced tangent function but the rationale for it already existed in brahmagupta's shadow function which is ration of sine and cosine and the methods resonate aryabhata's difference method but their approach fraud and trigonometric second order schemes but this was not built upon and faded until contacts with europe and combinatorics contributions even fewer till european contacts same for spherical trigonometry
v) Islamic mathematicians well al khwarizmi is called father of algebra but undeservedly so as it was a retrogression to the work of brahmagupta and diophantus and it involves rules like reduction and restoration due to him refusing to accept negative numbers without which modern algebra is impossible and their contributions to algebra also lacks, combinatorics they are behind India in this field and also arithmetic their own originality lacks
But if we consider Indian civilization there is no lack in any field
Starting with some of the achievements is
Kuttaka- first general solution to linear determinate equations and so called chinese remainder problem which is equivalent to bachet's method, chinese general method came later
Bhavana- First algorithm for so called pell's equation also first studied in India
Chakravala- First general algorithm for pell's equation, used to solve so called Fermat's challenge 500 years before Fermat
Fermat factorization- One of biggest achievements in arithmetic, discovered by sripati, elaborated by narayana before fermat
Fibonacci sequence- Unique to Indian mathematics before europe, one of the biggest achievements in combinatorics
Pascal's triangle- Oldest evidence of this anywhere in the world is India and it was linked to combinatorics which appeared elsewhere much later
Sine, cosine, versine and inverse sine- Yes this appear in India for the first time, greeks did it in chord, Indians did it in half chord i.e. the sine and made it more effiecient
Projection of planar triangles for solution of spherical triangles- this approach is uniquely Indian and is simpler than using menelaus theorem or analemma and indians obtained general solutions using this approach another way of describing this is rule of three approach, which by itself is another indian discovery
In geometry there are a list of results I wish to list out - Brahmagupta's formula. brahmagupta's theorem, brahmagupta's trapezium, brahmagupta's quadrilateral, parameshvara's formula, third diagonal of cyclic quadrilateral(narayana's formula), nilakantha's chord formula and many more such results and this was the only work on euclidean geometry before renaissance again
Algebraic and algorithmic geometry- sulba sutras present many algorithms for construction of geometric figures due to which they discovered many new values for pi and these algorithms for construction don't occur in any other culture and another unique feature of Indian mathematics geometric representation of series which aryabhata refers to as piling and same definition used in china a 1000 years later and sripati, narayana, nilakantha all give methods to implement this, mahavira solves many problems in constructions using algebraic approach
Geometry in Ancient and Medieval India - T. A. Sarasvati Amma : Free Download, Borrow, and Streaming : Internet Archive
Combinatorics- nCr, nPr, n factorial, n raised to power r and other results such as sum of all permutations, systematic generation of permutations, generalized fibonacci sequence, combinatorial sequences generation algorithms, narayana's cow sequence all appearing for the first time in Indian mathematics and global mathematics
Algebra- Brahmagupta first to use systematic symbolism for this and solved many types of diophantine equations for the first time and gave general solutions as well most equation of multiple variables using principles radically different to diophantus and mahavira used algebra to construct different types of geometric figures, classified equation in one variable into ten types and solved them all in ingenious ways and crown jewel was bhaskara for extending all of it to new heights with new inputs from narayana later on
History Of Hindu Mathematics A Source Book, Part 2: Algebra : Singh, Avadhesh Narayan : Free Download, Borrow, and Streaming : Internet Archive
Calculus: Major field Indians excelled in, it started of with the derivation of rate of change of sine as cosine and the rate of change of cosine being negative of sine, this was proved geometrically by bhaskara and he gave many theorems related to this, one of which is the rolle's theorem, another being maxima and minima in rudimentary form, these results were used to obtain infinite series and further the derivative of inverse sine and two ratios by nilakantha and achyuta respectively and paramesvara gave the mean value theorem for this, no other country contributed in this field. Integral calculus came in the form of deriving various volumes, while different countries contributed differently to this field Bhaskara's approach was uniques, he solves volume and surface area of sphere in two different ways using trigonometry in both in a method resembling integral calculus and similar to rudimentary Reimann integration.
Trigonometry; Crown jewel of Indian mathematics, taylor series upto two and three digits, infinite series for sine and cosine along with inverse tangent, finite difference equation for sine, second order finite difference interpolation formulas for trigonometric functions where, newton stirling, newton gauss and newton gauss backward interpolation formulas were discovered upto second order and furthermore iterative interpolation methods were also discovered, new formulas for arc of a function such as bhaskara II and kamalara's approaches where bhaskara I gave the best one with pade's approximation being part of it, aryabhata's approach gave rise to two new recursive methods by nilakantha somayaji and parameshvara discovered what we now call secant method, geometric trigonometry also appears first time only in Indian texts and besides many other innovatiions
Bharatiya Trikonamiti Sastra (Indian Trigonometry) Exotic India Art
of course what every body is taught in school, modern number system and the zero and other arithmetic discoveries like vertical representation of fractions, decomposition of fractions where mahavira discovered fibonacci's greedy algorithm for fractions, and virasena pioneered logarithms to base two, three and four, and besides bhaskara first gave rule of anything by zero is infinity and also the fact that zero zero can be cancelled in limiting case though not fully correct and another big discovery is rule of three which first appears in vedanga jyotisha and it most likely spread to china from here and this rule was developed into many new ways
History of Hindu Mathematics - 1 - Bibhutibhusan Datta, Avadesh Narayan Singh : Free Download, Borrow, and Streaming : Internet Archive
Mathematics In Ancient And Medieval India Bag A. K. Chowkambha : Pramoda Chowdhury : Free Download, Borrow, and Streaming : Internet Archive
so considering how much each field grew due to Indians and how much contributions existed while so many other civilizations made such minor contributions in several other fields concentrating on only a few, isn't is safe to say that Indian civilization contribution most compared to any civilization to math compared to any ancient civilization? What are your view
Modern times also we are not far behind with Ramanujan giving 4000 theorems and being asia' greatest mathematician ever produced but it does not end with him we have
CR Rao
RC bose
SR srinivasa varadhan
Harish chandra
Ganesh prasad
Asutosh mukherjee
Veeravalli s varadharajan
PC mahalanobis
subhash khot
kannan soundararajan
madhu sudan
bhargav bhatt and many more, modern contributions are themselves huge
notable-modern-indian-mathematicians-and-statisticians-during-the-19th-and-20th-centuries-of-bengal-1st-ed-2022-981196131x-9789811961311.pdf
so it is not a thing of the past, so do you agree?

This Sunday is International Kite Day! A kite is a quadrilateral with reflection symmetry across one of its diagonals, usually convex. This is named after the wind-blown flying kite which is often traditionally shaped like this; which in turn is named after the species of bird. The international celebration is about the middle one, but really it ought to be about the birds, which are much more interesting. "Kite" is also the B-side to the Wuthering Heights single by Kate Bush, released in 1978, so watch out for that on the 70s round...
Come join us at the Canonbury Tavern for the pub quiz!
What:
Pub Quiz - Exactly as it sounds! A quizmaster asks the questions and teams write down their answers. Questions cover a range of subjects and the quiz features picture and music rounds. It is all for fun and there is no pressure to memorise the encyclopaedia before coming.
Where:
The Canonbury Tavern https://maps.app.goo.gl/xocQRtc3PDJrQF676?g\_st=ic
Our tables are usually in the dining area at the back of the pub, but you can ask staff for LSC when you arrive; There is usually someone with a laptop who has a record of bookings. We also post on the WhatsApp group where we are sitting (more on that below).
When:
Every Sunday, the pub quiz starts at 7pm, with the picture sheet being given out a little earlier. Tables are booked under "LSC" from 6:30pm. Two tables are booked but we will try to make any number work.
How much:
£2.50 per person payable to the pub quiz people (Cashless only).
Who:
All are welcome. We typically make two teams with around 10 - 15 people.
Do I need to sign up / am I too late?:
There is no need to sign up, but feel free to say you're coming! It's not a problem to turn up after it starts, but try to be there on time to avoid missing the fun!
Do I need to be a whizz-kid, intellectual genius, or quizzing-extraordinaire?:
Nope, we do the quiz for fun.
Any Questions:
Write them into a comment and we will answer them - we are kind of into that.
We also have a WhatsApp group chat. Ask around on the day to be added.

Let's consider two triangles ABC and DEF.
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All the six angles of both the triangles are known. And, BC = DE. Now, I form a quadrilateral by combining these triangles such that the side BC or DE is now one of the diagonals of the quadrilateral.
So, what's the angle of intersection of the diagonals of this quadrilateral?
I think this question must be solvable because the shape of both the triangles is fixed and thus the shape of the quadrilateral is also fixed so the angle of intersection of the triangles must also remain fixed.
If both of these triangles are right angled triangles, then the quadrilateral formed by these triangles taking their hypotenuses as diagonal is a cyclic quadrilateral and the angle of intersection can easily be calculated. This is how I got this question, what if the triangles aren't right angled then how I will calculate the angle of intersection.

I’ve been stuck on this for an hour and I’m trying to proof MOPR is a parallelogram so I can lead to “If a quadrilateral is a parallelogram whose diagonals are congruent, then it is a rectangle”. That’s where I got stuck.

I'm trying to make a program where i input coordinates for the 4 vertices of a quadrilateral and my output will be the 4 side lengths and the 2 diagonals of the quadrilateral, and I want the output to go to L3, and I input x values into L1 and y values into L2. I know how to do the distance formula programming, but how do I get the program to draw values from the list and put the output values in another list. It's ok if my inputs don't have to be in a list, but I need the outputs in a list.

I’m trying to make a program where i input coordinates for the 4 vertices of a quadrilateral and my output will be the 4 side lengths and the 2 diagonals of the quadrilateral, and I want the output to go to L3, and I input x values into L1 and y values into L2. I know how to do the distance formula programming, but how do I get the program to draw values from the list and put the output values in another list. It’s ok if my inputs don’t have to be in a list, but I need the outputs in a list.