## Key Concepts

- Write definitions as conditional statements
- Verify statements
- Write the converse, inverse and the contrapositive of the conditional statement
- Write definitions as biconditional statements

### Definition

A **conditional statement **is a logical statement that has two parts, a hypothesis and a conclusion. When a conditional statement is written in **if-then form**, the “if” part contains the **hypothesis,** and the “then” part contains the **conclusion**.

Here is an example:

### Re-write a Statement in if-then form

**Example 1**:

Re-write the conditional statement in if-then form.

- All birds have feathers.
- Two angles are supplementary if they are a linear pair.

**Solution**:

First, identify the **hypothesis** and the **conclusion**. When you rewrite the statement in if-then form, you may need to reword the hypothesis or conclusion.

1. **All birds** have **feathers**.

If **an animal is a bird,** then** it has feathers.**

2. **Two angles are supplementary** if **they are a linear pair.**

If **two angles are a linear pair,** then **they are supplementary.**

### Negation and Verifying Statements

#### Negation

The **negation** of a statement is the *opposite* of the original statement. Notice that Statement 2 is already negative, so its negation is positive.

**Statement 1: **The ball is red. **Statement 2: **The cat is not black.

**Statement 2: **The cat is not black. **Negation 2: **The cat is black.

#### Verifying Statements

Conditional statements can be true or false. To show that a conditional statement is true, you must prove that the conclusion is true every time the hypothesis is true. To show that a conditional statement is false, you need to give *only one *counterexample.

### Related Conditionals

#### Definition

To write the **converse** of a conditional statement, exchange the **hypothesis** and **conclusion**.

To write the **inverse** of a conditional statement, negate both the hypothesis and the conclusion. To write the **contrapositive**, first, write the converse and then negate both the hypothesis and the conclusion.

#### Write four related conditional statements

**Example 2: **

Write the if-then form, the converse, the inverse, and the contrapositive of the conditional statement “Guitar players are musicians.” Decide whether each statement is *true* or *false*.

**Solution:**

**If-then form:** If you are a guitar player, then you are a musician.

True, guitars players are musicians.

**Converse:** If you are a musician, then you are a guitar player.

False, not all musicians play the guitar.

**Inverse:** If you are not guitar player, then you are not a musician.

False, even if you don’t play a guitar, you can still be a musician.

**Contrapositive:** If you are not a musician, then you are not a guitar player.

True, a person who is not a musician cannot be a guitar player.

### Equivalent Statements

A conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. Pairs of statements such as these are called *equivalent statements*. In general, when two statements are both true or both false, they are called equivalent statements.

#### Definition

You can write a definition as a conditional statement in if-then form or as its converse. Both the conditional statement and its converse are true. For example, consider the definition of *perpendicular lines*.

### Perpendicular Lines

#### Definition

If two lines intersect to form a right angle, then they are **perpendicular lines**.

The definition can also be written using the converse:

If two lines are perpendicular lines, then they intersect to form a right angle.

You can write “line l is perpendicular to line m” as l ⊥ m.

#### Use Definitions

**Example 3:**

Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned.

- −𝐀𝐂 ↔ ⊥ 𝐁𝐃 ↔

- ∠𝐀𝐄𝐁 and ∠𝐂𝐄𝐁 are a linear pair.

- →−𝐄𝐀→ and →−𝐄𝐁→ are opposite rays.

**Solution:**

This statement is true. The right angle symbol in the diagram indicates that the lines intersect to form a right angle. So you can say the lines are perpendicular.

This statement is true. By definition, if the noncommon sides of adjacent angles are opposite rays, then the angles are opposite rays, then the angles are a linear pair. Because

𝐄𝐀 → and →−𝐄𝐂→ are opposite rays,

∠𝐀𝐄𝐁 and ∠𝐂𝐄𝐁 are a linear pair.

This statement is false. Point E does not lie on the same line as A and B, so the rays are not opposite rays.

### Bi-conditional Statements

#### Definition

When a conditional statement and its converse are both true, you can write them as a single *biconditional statement*. A **biconditional statement **is a statement that contains the phrase “if and only if.”

Any valid definition can be written as a biconditional statement.

#### Write a Bi-conditional

**Example 4: **

Write the definition of perpendicular lines as a biconditional.

**Solution**

**Definition: **If **two lines intersect to form a right angle,** then **they are perpendicular.**

**Converse: **If **two lines perpendicular,** then **they intersect to form a right angle.**

**Biconditional: Two lines are perpendicular,** if and only if **they intersect to form a right angle.**

### Questions

**Question 1**:

Rewrite the conditional statement in if-then form.

*Only people who are registered are allowed to vote.*

**Solution:**

If one is allowed to vote, then one is registered.

**Question 2**:

Write the if-then form, the converse, the inverse, and the contrapositive of the following statement.

*3x + 10 = 16, because x = 2.*

**Solution:**

If-then form:

If 3x + 10 = 16, then x = 2.

Converse:

If x = 2, then 3x + 10 = 16.

Inverse:

If 3x + 10 is not equal to 16, then x is not equal to 2.

Contrapositive:

If x is not equal to 2, then 3x + 10 is not equal to 16.

**Question 3**:

Decide whether the statement is true or false. If false, provide a counterexample.

*If a polygon has five sides, then it is a regular pentagon.*

**Solution:**

False statement

Counterexample: If a polygon has five sides of unequal length, then it is not a regular pentagon. The sides of a regular pentagon should be equal in length.

**Question 4**:

Rewrite the definition as a biconditional statement.

*An angle with a measure between 90 degrees and 180 degrees is called obtuse.*

**Solution:**

An angle is called obtuse if and only if it measures between 90 degrees and 180 degrees.

### Key Concepts Covered

- To write a conditional statement in
**if-then**form, find the hypothesis and then the conclusion.

**Converse:**To write the converse of a conditional statement, exchange the hypothesis and conclusion.**Inverse:**To write the inverse of a conditional statement, negate both the hypothesis and the conclusion.**Contrapositive:**To write the contrapositive, first write the converse and then negate both the hypothesis and the conclusion.- When two statements are both true or both false, they are called
**equivalent statements**. - A
**biconditional statement**is a statement that contains the phrase “if and only if.”

## Exercise

- In triangle ABC, AD is a median. If the area of ΔABD is 15 cm sq. then find the area of ΔABC.
- ABCD is a parallelogram and BPC is a triangle with P falling on AD. If the area of parallelogram ABCD= 26 cm
^{2}, find the area of triangle BPC. - PQRS is a parallelogram and PQT is a triangle with T falling on RS. If area of triangle PQT = 18 cm
^{2}, then find the area of parallelogram PQRS. - ABCD is a parallelogram where E is a point on AD. Area of ΔBCE = 21 cm
^{2}.

If CD = 6 cm, then find the length of AF. - The area of triangle ABC is 15 cm sq. If ΔABC and a parallelogram ABPD are on the same base and between the same parallel lines then what is the area of parallelogram ABPD.
- The area of parallelogram PQRS is 88 cm sq. A perpendicular from S is drawn to intersect PQ at M. If SM = 8 cm, then find the length of PQ.
- Amy needs to order a shade for a triangular-shaped window that has a base of 6 feet and a height of 4 feet. What is the area of the shade?
- Monica has a triangular piece of fabric. The height of the triangle is 15 inches and the triangle’s base is 6 inches. Monica says that the area of the fabric is 90 square inches. What error did Monica make? Explain your answer.
- The sixth-grade art students are making a mosaic using tiles in the shape of right triangle. The two sides that meet to form a right angle are 3 centimeters and 5 centimeters long. If there are 200 tiles in the mosaic, what is the area of the mosaic?
- A parallelogram with area 301 has a base of 35. What is its height?

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