7.G.A.2 7th Grade Geometry

Triangle Inequality

Draw geometric shapes with given conditions, focusing on constructing triangles from three measures of angles or sides.

How to explain it

The anchor students hold onto: Check ALL THREE pairs: a+b>c, a+c>b, and b+c>a. ALL must be strictly greater. Equality means a degenerate (flat) shape — not a triangle.

The triangle inequality is a prerequisite for the Pythagorean theorem in 8.G.B.7 — students must confirm that given side lengths can form a valid triangle before applying the theorem.

Worked examples

Example 1 Can They Form a Triangle?

Check: sides 4, 7, 5 — triangle?

Step 14 + 7 = 11 > 5. Pass.

Step 24 + 5 = 9 > 7. Pass.

Step 37 + 5 = 12 > 4. Pass.

Step 4All three pass — valid triangle.

AnswerYes, these sides form a triangle.

Example 2 Find the Range of the Third Side

Given 3 and 8 — range of c?

Step 1Lower bound: |8 - 3| = 5.

Step 2Upper bound: 3 + 8 = 11.

Step 3Range: 5 < c < 11.

Step 4c must be strictly between 5 and 11.

Answer5 < c < 11.

Common mistakes

What students write Student tests only one pair and declares the triangle valid.

The fix All three pairs must pass — test a+b>c, a+c>b, AND b+c>a.

What students write Student treats a+b = c as a valid triangle.

The fix Equality means a flat, degenerate shape — the inequality must be STRICT.

Teacher tip

Head off the two predictable errors before they happen. First: All three pairs must pass — test a+b>c, a+c>b, AND b+c>a. Second: Equality means a flat, degenerate shape — the inequality must be STRICT.