Which triangles are possible
Decide whether given side and angle conditions determine a unique triangle, more than one triangle, or no triangle, and explain why, predicting with a freehand sketch and testing by construction (ruler and protractor, compasses, or dynamic geometry software): three sides give exactly one triangle when each pair of sides adds to more than the third, and none otherwise — if the two shorter sides add to less than the longest, the construction arcs do not meet, and if they add to exactly the longest (e.g. 3, 4 and 7), the arcs touch only on the base line, so the sides lie flat in a straight line rather than forming a triangle; three angles must add to 180° and then fix only the shape, not the size, so many triangles fit (none if the sum is not 180°); two sides with the included angle, or two angles with the included side, give exactly one triangle; and two sides with an angle that is not between them can give two, one or no triangles
Lesson
Set out your lesson
“Which triangles are possible means practising the idea at the table with counters or paper models and one clear check. We will demo once, then your child tries.”
Bring to the table
- Paper shapes or a ruler and pencil
- Shape or angle name cards
Extra for the adult
Worked example. A rectangle has four right angles. Naming the property is the check.
Optional challenge. For Which triangles are possible, add one harder example after success.
What to notice. Keep one parent note: examples tried; results; support used. This session samples Which triangles are possible, not mastery.
Share and support
Family records and proof options
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Practice this concept
Start practiceWhich triangles are possible practice
Practice key ideas from Which triangles are possible.
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Learning resources
Extra practice links — they do not replace Gakuva’s teaching approach.
PhET Interactive Simulations · Practice link — not the full lesson
Area Builder
Before you open · source notes
Area Builder supports hands-on exploration for Which triangles are possible. Not every lab worksheet.
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Before you openAsk the learner what they changed in the sim or page and what they observed.
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Source, access and review notes
Public PhET Interactive Simulations page. An anonymous GET on 2026-11-12 returned HTTP 200 with no login or payment prompt for this URL. Opening the link does not save practice or assess mastery. An adult should preview and guide the learner.
PhET Interactive Simulations. “Area Builder.” External page; Gakuva does not copy, host, embed, or redistribute materials.
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Check understanding
- Show that sides of 3 cm, 4 cm and 8 cm make no triangle (the arcs do not meet) and that 3 cm, 4 cm and 7 cm make none either (the arcs touch on the base, giving a straight line), while 3 cm, 5 cm and 7 cm make exactly one
- Explain that angles of 50°, 60° and 70° fit many triangles of different sizes, while angles of 100°, 50° and 40° fit none because they add to 190°
- Sketch freehand, then construct, two different triangles with AB = 7 cm, BC = 5 cm and angle A = 40°, and explain why making the 40° angle the one between the two sides gives only one triangle
“If your child is asked to make a triangle from straws 3 cm, 4 cm and 8 cm long, could they predict that it can't be done and explain why — and say why 'angles of 50°, 60° and 70°' could describe lots of different-sized triangles?”
Curriculum record
- Type
- Conceptual
- Subject
- Mathematics
- Domain
- Geometry
- Age range
- Ages 12–13
Standards