Geometry Honors · Lesson 1.1

Conditional Statements

How it works

A conditional has a hypothesis and a conclusion. Its contrapositive is logically equivalent; its converse and inverse need separate evidence. A single counterexample disproves a universal conditional.

p\Rightarrow q\quad\Longleftrightarrow\quad\neg q\Rightarrow\neg p

Worked examples

Example 1

Write the contrapositive: If a figure is a square, it is a rectangle.

If a figure is not a rectangle, it is not a square. Both statements have the same truth value.

Example 2

Is the converse true: If a figure is a rectangle, it is a square?

No. A rectangle with side lengths $\displaystyle 2$ and $\displaystyle 3$ is a counterexample.

Problems

Problem 1.1.1

Write the converse of: If an integer is divisible by $\displaystyle 4$ , it is even.

Show worked solution

If an integer is even, it is divisible by $\displaystyle 4$ . This is false; $\displaystyle 6$ is even but not divisible by $\displaystyle 4$ .

Problem 1.1.2

Write the inverse of: If a triangle is equilateral, it is isosceles.

Show worked solution

If a triangle is not equilateral, it is not isosceles. Under the inclusive definition of isosceles, the inverse is false: sides $\displaystyle 5,5,6$ give a counterexample.

Problem 1.1.3

Disprove: If $\displaystyle x^2=9$ , then $\displaystyle x=3$ .

Show worked solution

Take $\displaystyle x=-3$ . Then $\displaystyle x^2=9$ , but $\displaystyle x\ne3$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.1.1

Problem 1.1.2

Problem 1.1.3

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