Evaluating Conditional Statements
Use examples and counterexamples to evaluate conditional statements.
Students evaluate conditional statements by identifying the hypothesis and conclusion, proving true statements with definitions or theorems, and disproving false statements with a single counterexample.
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.
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Hypothesis and conclusion
A conditional statement has the form “If p, then q.” The p part is the hypothesis, and the q part is the conclusion. The statement claims q happens every time p happens.
Disprove with a counterexample
One case where the hypothesis is true and the conclusion is false proves the statement false. Search special cases, such as equilateral triangles, right triangles, squares, or zero.
Prove with reasoning
To show a statement is always true, use definitions, postulates, and theorems rather than a few examples. For instance, the triangle angle sum limits how many obtuse or right angles a triangle can have.
Worked example
Evaluate: “If a quadrilateral has four congruent sides, then it is a square.”
- Hypothesis: the quadrilateral has four congruent sides. Conclusion: it is a square.
- Look for a quadrilateral with four congruent sides that is not a square.
- A rhombus with angles of 60°, 120°, 60°, and 120° satisfies the hypothesis but has no right angles.
False; a non-square rhombus is a counterexample.
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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 · Geometry (TX TEKS) courseIdentify the hypothesis and conclusion and decide which related statements must be true.
Watch on YouTubeGeometry: Deductive Reasoning in Quadrilaterals
UT IPSI · Kid2Kid / Texas Education AgencyUse definitions and properties to reason deductively about geometric figures.
Watch on YouTubeTry it yourself
TEKSForge practice · unscored
1. Evaluate: “If , then .”
Show answer and reasoning
False
The value satisfies the hypothesis because , but it does not satisfy the conclusion.
2. Evaluate: “If two angles are vertical angles, then they are congruent.”
Show answer and reasoning
True
The vertical angles theorem guarantees it. Each angle forms a linear pair with the same adjacent angle, so both are supplements of that angle.
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 11 · 1 point
A conditional statement is given: ‘If two interior angles of a triangle are acute, then the third interior angle must be obtuse.’ Which of the following best describes this statement?
- This statement is true because all obtuse triangles have two acute interior angles.
- This statement is false because the third interior angle must also be acute.
- This statement is true because a triangle can have at most one interior obtuse angle.
- This statement is false because the third interior angle can be acute, right, or obtuse.
Show STAAR answer and explanation
This statement is false because the third interior angle can be acute, right, or obtuse.
Two acute angles do not force the third to be obtuse. An equilateral triangle has three acute angles, and a right triangle has two acute angles plus a right angle, so the conditional is false.
This statement is true because all obtuse triangles have two acute interior angles.: The converse can be true without making the original conditional true; an acute triangle is a counterexample.
This statement is false because the third interior angle must also be acute.: The third angle need not be acute: it can also be right or obtuse.
This statement is true because a triangle can have at most one interior obtuse angle.: Having at most one obtuse angle does not require every triangle with two acute angles to have one.
Archived released STAAR Geometry EOC · 2013 · Item 18 · 1 point
A statement is given: The number of square units in the area of a square is greater than or equal to the number of units in the perimeter of the square. Which side length of a square provides a counterexample to the given statement?
- 6 units
- 4 units
- 10 units
- 2 units
Show STAAR answer and explanation
2 units
For side s, compare with . At s = 2, area is 4 and perimeter is 8, so the claimed inequality is false.
6 units: At s = 6, 36 ≥ 24, so the statement holds.
4 units: At s = 4, 16 = 16, so the statement holds.
10 units: At s = 10, 100 ≥ 40, so the statement holds.
Learning objectives
By the end of the lesson, students can:
- Identify the hypothesis and conclusion of a conditional statement.
- Write a counterexample that satisfies the hypothesis but not the conclusion.
- Explain why examples alone do not prove a statement is always true.
Instructional sequence
- 1
Engage
5 minPresent “If an animal has four legs, then it is a dog.” Ask students to disprove it in one example.
- 2
Direct instruction
10 minLabel the hypothesis and conclusion, and define a counterexample as a case where the hypothesis is true and the conclusion is false.
- 3
Guided practice
12 minEvaluate geometry statements about triangles, angles, and quadrilaterals, supplying a counterexample or a reason.
- 4
Justify true statements
12 minUse definitions and theorems, such as the triangle angle sum, to explain why true statements always hold.
- 5
Closure
6 minStudents complete the exit ticket with a drawn counterexample.
Misconceptions to address
Teacher move: Examples can support a conjecture, but a proof must explain why every case works. One counterexample is enough to prove a statement false.
Teacher move: A counterexample must satisfy the hypothesis and fail the conclusion; otherwise, it does not test the statement.