Converse, Inverse, and Contrapositive
Determine the validity of a conditional statement and its converse, inverse, and contrapositive.
Students write the converse, inverse, and contrapositive of a conditional, identify logically equivalent pairs, and use counterexamples to test statements that do not automatically follow.
Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.
Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.
Learn the concept
Start with two complete clauses
Write a conditional as : if p, then q. The hypothesis p describes the starting condition; the conclusion q is what follows. Keep each clause intact when rewriting.
Switch and negate
The converse is . The inverse is . The contrapositive is . Negating “all three sides are congruent” means “not all three sides are congruent,” not “no sides are congruent.”
Know what truth transfers
A conditional and its contrapositive always have the same truth value. Its converse and inverse also share a truth value with each other, but need not match the original. Test unsupported forms with definitions or counterexamples.
Worked example
Given the true statement “If a triangle is equilateral, then it is isosceles,” write and evaluate its three related forms. Use the definition that an isosceles triangle has at least two congruent sides.
- Converse: If a triangle is isosceles, then it is equilateral.
- The converse is false: a triangle with side lengths 5, 5, and 6 is isosceles but not equilateral.
- Inverse: If a triangle is not equilateral, then it is not isosceles. The same 5-5-6 triangle disproves it.
- Contrapositive: If a triangle is not isosceles, then it is not equilateral. This is true because it is equivalent to the given conditional.
Converse: false; inverse: false; contrapositive: true
Watch the lesson
Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.
Conditional statements and logical equivalence | Geometry (TX TEKS) | Khan Academy
Khan Academy · Geometry (TX TEKS) courseCompare related conditionals and recognize contrapositive equivalence.
Watch on YouTubeTry it yourself
TEKSForge practice · unscored
1. Write the contrapositive of “If two angles are vertical angles, then they are congruent.”
Show answer and reasoning
If two angles are not congruent, then they are not vertical angles.
Switch the hypothesis and conclusion and negate both. This statement is equivalent to the vertical angles theorem.
2. Is the converse “If two angles are congruent, then they are vertical angles” always true?
Show answer and reasoning
No
Two separate 35° angles are congruent but need not be formed by the same intersecting lines.
Released STAAR practice
Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.
Archived released STAAR Geometry EOC · 2013 · Item 27 · 1 point
The following conditional statement is true: If a quadrilateral is a square, then it has four congruent sides. Which statement must also be true?
- If a quadrilateral has four congruent sides, then it is a square.
- If a quadrilateral does not have four congruent sides, then it is not a square.
- If a quadrilateral is not a square, then it does not have four congruent sides.
- If a quadrilateral does not have four congruent sides, then it is a square.
Show STAAR answer and explanation
If a quadrilateral does not have four congruent sides, then it is not a square.
The contrapositive of a true conditional is true: not having four congruent sides rules out being a square.
If a quadrilateral has four congruent sides, then it is a square.: This is the converse; a nonsquare rhombus has four congruent sides.
If a quadrilateral is not a square, then it does not have four congruent sides.: This is the inverse; a nonsquare rhombus is a counterexample.
If a quadrilateral does not have four congruent sides, then it is a square.: Every square has four congruent sides, so a quadrilateral without them cannot be a square.
Learning objectives
By the end of the lesson, students can:
- Identify a conditional's hypothesis and conclusion.
- Write its converse, inverse, and contrapositive accurately.
- Explain why a true conditional guarantees its contrapositive but not its converse or inverse.
Instructional sequence
- 1
Engage
5 minPresent a true statement about equilateral and isosceles triangles. Ask whether reversing the statement keeps it true.
- 2
Direct instruction
10 minBuild the four related forms with p and q. Distinguish switching the clauses from negating them.
- 3
Guided practice
12 minRewrite geometric conditionals in all four forms, carefully preserving complete clauses.
- 4
Validity check
12 minUse counterexamples to test converses and inverses, and pair each conditional with its equivalent contrapositive.
- 5
Closure
6 minComplete the exit ticket and explain why its conclusion does not require checking every triangle.
Misconceptions to address
Teacher move: Use a non-equilateral isosceles triangle: it satisfies the reversed hypothesis without satisfying the reversed conclusion.
Teacher move: Adding not without switching creates the inverse. A contrapositive both switches and negates.