Gr 9 Maths Worksheets (Ages 14-15) | Level M24: Plane Geometry & Proof

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Help your child build absolute confidence and mastery in mathematics. This comprehensive workbook from MathSmith takes a highly structured, "one-rung-at-a-time" approach to Plane Geometry and Proof, ensuring students fully understand concepts before moving forward.

Instead of overwhelming learners with random equations, this workbook uses a carefully scaffolded, predictable learning ladder. It is perfect for homework, structured independent study, or mastery practice.

📊 What Is Included?

1 Complete Level (Level M24)

25 Structured Sub-levels for gradual progression

133 Printable Worksheets

1,124 Total Problems for thorough practice

Full Answer Key online, unlocked by the order ID on your Payhip receipt, for quick, stress-free marking

🗺️ Curriculum Breakdown

This collection focuses deeply on mastering Plane Geometry and Proof through a carefully curated system:

What a proof stands on (3 Sub-levels): Three rungs before a single thing is proved, because an argument has to start from something agreed and this level is honest about what. Euclid's definitions, which cannot quite be made to work; his axioms, which are about quantity of any sort; and his postulates, which are about the plane and would be meaningless said of apples. Then the fifth one, which is longer than the other four together, does not read like anything obvious, and resisted two thousand years of attempts to prove it away - the clearest example in school mathematics of the difference between a thing that is true and a thing that is assumed. The strand ends with two words for what a diagram already shows, collinear and concurrent, and with the observation that makes the first of them worth having: any two points are in a row, so the word says nothing until there are three.

Where the proving starts (6 Sub-levels): Six rungs, and the order is the content. Angles at a point and angles on a line are assumed and said to be assumed; everything after them is proved from them and from each other. Vertically opposite angles come out of two straight-line facts and one subtraction, which is the smallest proof worth writing and is where the frame is learnt. The transversal's three relations turn out to be one relation and two consequences. Two lines parallel to a third need a transversal drawn before there is anything to argue about. And then the angle sum of a triangle, which M17 found with a protractor and which is got here from a parallel through one corner - after which the exterior angle theorem is three lines, and read backwards it is an inequality as well.

Congruence, used as a reason (3 Sub-levels): Three rungs. M17 asked which of four rules a pair of marks amounts to, which is a question that can only be put when the answer is one of the four; here the answer may be 'not enough', and that is the difference between using a rule and giving a reason. AAS arrives on this level rather than on M17 because it is ASA once the third angle has been found, and finding the third angle is the theorem one rung back. Then the isosceles triangle and its converse, which are two different proofs and not one proof turned round, and the two inequalities - the only results here about an order rather than an equality.

Quadrilaterals, which are two triangles all along (6 Sub-levels): Six rungs, every one of them a consequence of the triangle work. The angle sum is a hundred and eighty said twice, across one diagonal. The three properties of a parallelogram are one congruence used three times. The tests for a parallelogram are a claim that four particular measurements are enough, where M20 showed that five are needed in general - and the trap is the halves of them. The mid-point theorem is proved by doubling the segment, which no child invents. And the strand closes on the one sentence both area formulas come out of: what an area depends on is the base and the distance between the parallels, and nothing at all depends on the lean.

Circles, where the reason is usually a radius (5 Sub-levels): Five rungs in which the first line of nearly every proof is that two radii are equal - which is true because the compasses were not moved, and is M13's reach model coming back as a reason. The perpendicular from the centre bisects a chord, by RHS. Equal chords make equal angles at the centre, by SSS, in ONE circle and not across two. The angle at the centre is twice the angle at the rim, out of which the angle in a semicircle falls without being a second theorem. Two angles on one chord from the same side are equal because both are half the same thing. And a cyclic quadrilateral's opposite angles add to a half turn, which settles which familiar shapes can be drawn round a circle - the rectangles, and no other parallelogram.

Two constructions, and what a drill can ask of one (2 Sub-levels): A workbook cannot mark a child's drawing, so the figure shows the given and the arcs so far and stops, and the question is what comes next or what may be measured afterwards to prove it worked - which is M13's bargain kept. The first rung is the short list of angles ruler and compasses can reach: sixty comes from the radius stepping round its own circle, and everything else is that halved or the difference between two of them split. Twenty degrees is not on the list, and that is a real impossibility. The last rung is the one construction in the book whose given is not a part of the finished shape: the two unknown sides are laid out along the ray as a single length and then cut at the place that is the same distance from both ends of a segment.

🎯 Who Is This For?

Parents looking to provide structured math support at home.

Homeschoolers searching for a sequential curriculum to guarantee zero gaps in math skills.

Teachers & Tutors looking for bite-sized, scaffolded printables for classroom morning work or targeted intervention.

📖 How To Use This Workbook

Print the workbook, read each rung's method page, and let your child work inside the printed frame. Then check the answers your way: in the printable answer-key PDF included in this download, or online, one page at a time, with the order ID on your Payhip receipt (valid for six months). Please keep the answer key for the adult marking the work and do not share it with your child: the workbook only works if they have worked each answer out for themselves.

Step-by-step manual and video guide: https://ebooks.thecodecrafters.net/manual/

Check the answers: https://ebooks.thecodecrafters.net/answers

⚙️ Digital Delivery Details

Format: four instant-download PDFs (free sample, sample answers, workbook, full answer key). The answer key is also available online.

Print Ready: Easily printable on home or school printers.

Usage: Licensed for individual family or single-classroom use.