Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Give three people the same three side lengths and ask them to build a triangle with straws, and they will all produce the same shape. Give them three angle measures instead, and they may produce triangles of very different sizes. Seventh graders explore exactly this: which sets of conditions pin down one unique triangle, which allow many, and which make a triangle impossible.
Students draw shapes freehand, with a ruler and protractor, and with geometry software. Through these constructions they discover the triangle inequality, that any two sides must add up to more than the third (sides of 3, 4 and 8 cannot meet), and that the three angles of a triangle must total 180 degrees, so 90°, 60° and 40° fail. Three given sides, or two sides with the angle between them, determine a unique triangle. Three angles alone fix the shape but not the size, giving infinitely many similar triangles. The focus is on noticing and explaining these patterns rather than on formal proof, which comes in high school.
Students often assume a triangle always exists. Building 2, 3 and 6 cm with strips of paper, and watching the short sides fail to meet, makes the triangle inequality concrete.
Lengths 1, 4 and 5 cm only reach as a flat line, since 1 + 4 = 5. The two shorter sides must be strictly longer than the third.
Matching angles produce the same shape, but students can draw a tiny and a huge triangle with the same angles. Overlaying the two drawings shows they are similar, not identical.
Can a triangle be built with sides of 5, 6 and 9 cm? What about 3, 4 and 8 cm? If a triangle exists, is it unique?
Answer: 5, 6 and 9 cm make exactly one triangle; 3, 4 and 8 cm make no triangle.
Let students discover the rules before naming them. Give pairs a bag of straws or paper strips in several lengths and a table to record which combinations close up into triangles. Dynamic geometry software is ideal for the 'how many triangles' question because students can drag vertices and watch whether the conditions still hold.
Assessment items tend to ask whether a set of measures forms a triangle, whether the triangle is unique, or what range a third side could take. Requiring a reason with each answer, such as 'because 3 + 4 is less than 8', keeps the focus on understanding.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) 4 cm, 5 cm, 7 cm
4 + 5 = 9 is more than 7, so the first set works. In the others the two shorter sides add to 5, 5 and 6, which do not exceed the longest side.
Answer: A) Infinitely many, all the same shape but different sizes
The angles add to 180°, so the triangle exists, but angles fix only the shape. You can draw it at any size, so there are infinitely many.
Answer: 55 (also accepted: 55°, 55 degrees)
The angles of a triangle add to 180°: 180 - 48 - 77 = 55°.
Answer: C) No, the angles add to more than 180°
90 + 60 + 40 = 190°, which is more than 180°, so no triangle has these angles.
Answer: D) Exactly one
Two sides and the included angle fix the third side, so every triangle drawn with these conditions is identical.
Answer: 5 (also accepted: 5 cm)
The third side plus 5 must be more than 9, so it must be more than 4 cm. The smallest whole number that works is 5 cm.
A full lesson with slides, activities and an exit ticket on drawing triangles from given conditions, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.G.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn drawing triangles from given conditions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It says the sum of any two sides of a triangle is greater than the third side. If two sides add up to the third side or less, no triangle can be formed.
Not for this standard. It asks for drawing freehand, with ruler and protractor, and with technology. Formal compass and straightedge constructions come in high school geometry.