lesson
Enough information can fix a triangle
1 · Learn
Congruent triangles have the same shape and size, though their orientation can differ. Three corresponding sides determine a triangle up to reflection; two sides and their included angle also provide a standard congruence condition.
Accurate ruler-and-compass construction uses intersections of arcs rather than guessing a vertex's position. Not every collection of three positive lengths forms a triangle: the two shorter lengths must sum to more than the longest.
2 · Worked example
Lengths 3, 4 and 5 can form a triangle. Lengths 2, 3 and 6 cannot because 2 + 3 is less than 6.
3 · Your turn
Can lengths 4 cm, 5 cm and 10 cm form a triangle? Explain.
Check your answer
No. 4 + 5 = 9 is less than 10, so the shorter sides cannot meet to form the triangle.
4 · Apply your learning
With safe ruler-and-compass instruments, draw AB=6 cm. Draw an arc of radius 5 cm from A and one of radius 7 cm from B. Choose an intersection C and join AC and BC. Compare with the other intersection on the opposite side of AB: explain why reflection gives a congruent triangle rather than a new set of side lengths.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
Open your writing and drawing pad
Write, draw or show your working. Use a mouse, pen or finger. This pad is not a submitted answer.
Not saved to your account. Download your drawing as PNG and your typed notes as TXT before leaving or refreshing. They are separate files. Do not enter private information.
Open a picture saved on this device to draw on it again. Opening replaces the drawing, but Undo can restore it. PNG only, up to 4 MB and 4096 × 4096 pixels. The picture stays in this browser; it is not uploaded. Paper lines in a saved picture become part of that picture.
Clear the drawing? Your typed notes will stay. You can undo this action.
Your pad is empty.
Open a UTF-8 TXT file from this device, up to 40 KB and 10,000 characters. It stays in this browser. Opening can be undone until you type again. Downloads keep a new copy; they do not change the file you opened.
Replace the notes on this page with the opened file? Download the current notes first if you want to keep them.
Typed notes are not submitted answers.
Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
Practice this lesson
Lesson resources
Browse the available practice activities and worksheets for this lesson.
Your progress