A triangle is specified by side lengths 2 cm, 4 cm, and 9 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
CCSS 7.G.A.2 · unique, multiple, or no triangle
Triangle Construction: Unique, Multiple, or Impossible Worksheets
Classify whether given measurements determine one triangle, more than one triangle, or no triangle. The generator uses valid construction logic instead of decorative triangles that may contradict the measurements.
Preview · US Letter
Triangle Construction
Diagrams are part of the problem data, not decorative assets. Student pages keep solution-specific marks hidden; reveal answers or use the separate PDF key to check work.
Problem 1
Problem 2
A triangle is specified by side lengths 3 cm, 5 cm, and 11 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 3
A triangle is specified by side lengths 7 cm, 8 cm, and 7 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 4
A triangle is specified by angle measures 30°, 60°, and 90°. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 5
A triangle is specified by side lengths 3 cm, 2 cm, and 7 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 6
A triangle is specified by side lengths 6 cm, 5 cm, and 13 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 7
A triangle is specified by angle measures 45°, 70°, and 65°. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Problem 8
A triangle is specified by side lengths 5 cm, 5 cm, and 10 cm. Classify the construction as unique, more than one triangle, or impossible.
Show reasoning / computation: __________________________________________
Deciding whether a triangle can be built
Three side lengths make a triangle only if the sum of any two sides is greater than the third. The angles of a triangle always add to 180 degrees. Two sides and the angle between them give exactly one triangle, while three angles alone do not fix the size, so many triangles can have the same angles.
Worked examples
- Sides 3, 4, 5: 3 + 4 > 5, 3 + 5 > 4 and 4 + 5 > 3, so a triangle can be built.
- Sides 2, 3, 6: 2 + 3 = 5, which is less than 6, so these sides cannot form a triangle.
- Angles 50° and 60° leave 180 − 50 − 60 = 70° for the third angle.
Common slips
- Checking only one pair of sides. All three comparisons must hold.
- Using 180 for the side lengths. The 180 applies to angles, not to sides.
- Assuming three given angles fix one triangle. Different-sized triangles can have the same angles.
How to use this worksheet
Ask students to show all three side comparisons before deciding. The answer key says whether a triangle is possible, which makes it easy to find a missed comparison.
Common Core connection
- 7.G.A.2 · Draw geometric shapes from stated conditions and explain when the conditions determine a figure.
A practical match for the skill this worksheet practises, not a claim that one worksheet teaches or assesses the whole standard. Check the official math standards.
Classroom note 1
Three valid side lengths determine a unique triangle up to reflection.
Classroom note 2
Three angles can fix a triangle’s shape without fixing its size, so more than one similar triangle is possible.
Classroom note 3
If the two shorter sides do not total more than the longest side, no triangle can close. Example: sides 3, 4, 9 fail because 3 + 4 is not greater than 9.
Classroom note 4
Same three measurements given in a different arrangement can still describe the same triangle, so check before assuming a case has multiple answers.
Continue the 7.G progression
Move to another Grade 7 geometry skill
Questions people ask
Do these require a compass and protractor?
They can be used as classification practice on their own; teachers can also have students construct selected items with drawing tools.
How do the worksheets show a 'more than one triangle' case?
AAA cases use three valid angle measures: the angles fix shape but not size, so multiple similar triangles satisfy the conditions.
Which standard does this practice?
CCSS 7.G.A.2: draw geometric shapes with given conditions and decide whether the conditions determine a unique triangle, more than one, or none.