Sunday, August 14, 2022

# What Does Converse Mean In Geometry

## What Is A Conditional Statement

What is the Converse of a Conditional Statement?

A conditional statement takes the form If p, then q where p is the hypothesis while q is the conclusion. A conditional statement is also known as an implication.

Sometimes you may encounter the words antecedent for the hypothesis and consequent for the conclusion. Dont worry, they mean the same thing.

In addition, the statement If p, then q is commonly written as the statement p implies q which is expressed symbolically as p} \to q}.

Given a conditional statement, we can create related sentences namely: converse, inverse, and contrapositive. They are related sentences because they are all based on the original conditional statement.

Lets go over each one of them!

## What Are The Converse Contrapositive And Inverse

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Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesnt take long to run into something of the form If P then Q. Conditional statements are indeed important. What is also important are statements that are related to the original conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse.

## Geometry And Conditional Statements

Many times in geometry we see postulates and theorems that seem like they could become conditional statements and converse conditional statements:

• Parallel lines never meet. Postulate
• If two lines are parallel, then they are lines that never meet. Conditional Statement
• If two lines never meet, then they are parallel. Converse

Example #2

• Adjacent angles share a common side. Postulate
• If angles share a common side, then they are adjacent. Conditional Statement
• If angles are adjacent, then they share a common side. Converse

Some postulates are even written as conditional statements:

• If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
• If two points lie in a plane, then the line joining them lies in that plane.

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## How To Use ‘if And Only If’ In Mathematics

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When reading about statistics and mathematics, one phrase that regularly shows up is if and only if. This phrase particularly appears within statements of mathematical theorems or proofs. But what, precisely, does this statement mean?

## Converse Inverse And Contrapositive Of A Conditional Statement What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive.

But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.

But we saw earlier that there are infinitely many answers, and the dotted line on the graph shows this.

So yes there are infinitely many answers …

… but imagine you type 0.5 into your calculator, press cos-1 and it gives you a never ending list of possible answers …

So we have this rule that a function can only give one answer.

So, by chopping it off like that we get just one answer, but we should remember that there could be other answers.

## What Does If And Only If Mean In Mathematics

To understand if and only if, we must first know what is meant by a conditional statement. A conditional statement is one that is formed from two other statements, which we will denote by P and Q. To form a conditional statement, we could say if P then Q.

The following are examples of this kind of statement:

• If it is raining outside, then I take my umbrella with me on my walk.
• If you study hard, then you will earn an A.
• If n is divisible by 4, then n is divisible by 2.

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## How Do You Write Converse In Math

If the converse is true, then the inverse is also logically true. If two angles are congruent, then they have the same measure. If two angles have the same measure, then they are congruent. Converse, Inverse, Contrapositive.

Statement
If not p, then not q.
Contrapositive

## Necessary And Sufficient Conditions

8.1 Geometry – Pythagorean Theorem and its Converse

Biconditional statements are related to conditions that are both necessary and sufficient. Consider the statement if today is Easter, then tomorrow is Monday. Today being Easter is sufficient for tomorrow to be Monday, however, it is not necessary. Today could be any Sunday other than Easter, and tomorrow would still be Monday.

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## Does Converse Mean Opposite

contrary, converse, opposite, reverse Contrary describes something that contradicts a proposition, converse is used when the elements of a proposition are reversed, opposite pertains to that which is diametrically opposed to a proposition, and reverse can mean each of those. See also related terms for reverse.

## Truth Tables Of A Conditional Statement And Its Converse Inverse And Contrapositive

Now that we know how to symbolically write the converse, inverse, and contrapositive of a given conditional statement, it is time to state some interesting facts about these logical statements.

To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table.

Here are some of the important findings regarding the table above:

• The conditional statement is NOT logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. Thus, p} \to q} \equiv ~\colorq\to ~\colorp.
• The converse is logically equivalent to the inverse of the original conditional statement. Therefore, q} \to p} \equiv ~\colorp\to ~\colorq.

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## What Is An Inverse Statement In Math

The inverse of a conditional statement is when both the hypothesis and conclusion are negated the If part or p is negated and the then part or q is negated. In Geometry the conditional statement is referred to as p q. The Inverse is referred to as ~p ~q where ~ stands for NOT or negating the statement.

## Converse Of A Theorem In mathematics, the converse of a theorem of the form PQ will be QP. The converse may or may not be true, and even if true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse was proved only in 1997.

In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of “Given P, if Q then R will be “Given P, if R then Q. For example, the Pythagorean theorem can be stated as:

Given a triangle with sides of length a

 Some S is not P not valid

The validity of simple conversion only for E and I propositions can be expressed by the restriction that “No term must be distributed in the converse which is not distributed in the convertend.” For E propositions, both subject and predicate are distributed, while for I propositions, neither is.

In first-order predicate calculus, All S are P can be represented as

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## Converse Of A Conditional Statement

The converse of a true conditional statement does not automatically produce another true statement. It might create a true statement, or it could create nonsense:

• If a polygon is a square, then it is also a quadrilateral.

That statement is true. But the converse of that is nonsense:

• If a polygon is a quadrilateral, then it is also a square.

We know it is untrue because plenty of quadrilaterals exist that are not squares.

## Converse Contrapositive And Inverse

Now we can define the converse, the contrapositive and the inverse of a conditional statement. We start with the conditional statement If P then Q.

• The converse of the conditional statement is If Q then P.
• The contrapositive of the conditional statement is If not Q then not P.
• The inverse of the conditional statement is If not P then not Q.

We will see how these statements work with an example. Suppose we start with the conditional statement If it rained last night, then the sidewalk is wet.

• The converse of the conditional statement is If the sidewalk is wet, then it rained last night.
• The contrapositive of the conditional statement is If the sidewalk is not wet, then it did not rain last night.
• The inverse of the conditional statement is If it did not rain last night, then the sidewalk is not wet.

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## Exchanging Parts Of Conditional Statements

You can switch the hypothesis and conclusion of a conditional statement. You take the conclusion and make it the beginning, and take the hypothesis and make it the end:

• If my dog observes something that excites him, then he barks.
• If triangles have equal corresponding sides, then they are congruent.

## The Contrapositive Of A Conditional Statement

Converse, Inverse, & Contrapositive – Conditional & Biconditional Statements, Logic, Geometry

Suppose you have the conditional statement p} \to q}, we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.

In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap the roles of the hypothesis and conclusion. Therefore, the contrapositive of the conditional statement p} \to q} is the implication ~\colorq\to ~\colorp.

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## What Is The Difference Between Inverse And Converse

As nouns the difference between converse and inverse is that converse is familiar discourse free interchange of thoughts or views conversation chat or converse can be the opposite or reverse while inverse is the opposite of a given, due to contrary nature or effect.

## Contrapositive Of Conditional Statement

To get the contrapositive of a conditional statement, we negate the hypothesis and conclusion and exchange their position.

 if \ then, \

For example,

Emily’s dad watches a movie if he has time. – Conditional statement

If Emily’s dad does not have time, then he does not watch a movie. – Contrapositive of a conditional statement

• In a conditional statement “if p then q,” ‘p’ is called the hypothesis and ‘q’ is called the conclusion.
• There can be three related logical statements for a conditional statement.
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## The Inverse Of A Conditional Statement

When youre given a conditional statement p} \to q}, the inverse statement is created by negating both the hypothesis and conclusion of the original conditional statement. Thus, the inverse is the implication ~\colorp\to ~\colorq.

The symbol ~\colorp is read as not p while ~\colorq is read as not q .

## Conditional & Converse Statements Geometry is a wonderful part of mathematics for people who don’t like a lot of numbers. It has shapes and angles, and it also has logic. Logic is formal, correct thinking, reasoning, and inference. Logic is not something humans are born with we have to learn it, and geometry is a great way to learn to be logical.

• Practice Conditional Statements
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## Conditional Statements Lesson & Examples

1 hr 5 min

• What are conditional statements, converses, and biconditional statements?
• 00:05:21 Understanding venn diagrams
• 00:11:07
• Write the statement and converse then determine if they are reversible
• 00:29:17 Understanding the inverse, contrapositive, and symbol notation
• 00:35:33 Write the statement, converse, inverse, contrapositive, and biconditional statements for each question
• 00:45:40 Using geometry postulates to verify statements
• 00:53:23 What are perpendicular lines, perpendicular planes and the perpendicular bisector?
• 00:56:26 Using the figure, determine if the statement is true or false
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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## Does The Converse Of Angle Bisector Theorem True

My question is clearly stated in the title.

The proof is not that difficult but I find it was hardly mentioned. Even using converse of angle bisector theorem as search keyword in WIKI, the reply is The page . does not exist.

• 1$\begingroup$What do you mean by “divides the opposite side” in your question? As in an equilateral triangle, angle bisectors will bisect the side opposite the angle, but there are clear counterexamples if this is not what you mean .$\endgroup$ ClaytonJul 15 ’14 at 18:55
• $\begingroup$I think he’s talking about triangles. But as you’ve said, i don’t understand the question myself.$\endgroup$Jul 15 ’14 at 18:58
• 2$\begingroup$The theorem in question is probably this: In $\triangle ABC$ with $D$ on $BC$ such that $AD$ bisects $\angle A$, $$\frac = \frac$$ The easiest proof involves a simple application of the Law of Sines that can be reversed , so the converse holds.$\endgroup$Jul 15 ’14 at 19:06

You can find the proof in Classical Geometry: Euclidean, Transformational, Inversive, and Projective, page 144. It uses corollary 5.2.4:

If $B$, $C$, and $D$ are collinear points and if $A$ is a point not collinear with them, then $$\frac=\frac\ .$$

as follows:

So to be explicit about the question you ask, here is this converse theorem as I understand it:

Given a triangle $\triangle ABC$ and a point $D$ on $BC$ which satisfies $$\frac=\frac\ .$$ Then the line $AD$ will be a bisector of the angle $\angle CAB$.

$$\frac=a$$

translates into

$$\frac=a$$

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## What Is The Converse Of A Statement In Geometry

4.2/5hypothesis

Similarly, it is asked, what is the converse of a statement?

To form the converse of the conditional statement, interchange the hypothesis and the conclusion. The converse of “If it rains, then they cancel school” is “If they cancel school, then it rains.” To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

Subsequently, question is, what is a inverse statement in geometry? Inverse of a Conditional. Negating both the hypothesis and conclusion of a conditional statement. For example, the inverse of “If it is raining then the grass is wet” is “If it is not raining then the grass is not wet”. Note: As in the example, a proposition may be true but its inverse may be false.

One may also ask, what is the meaning of Converse in geometry?

Answered May 27, 2017 · Author has 3.8k answers and 3.3m answer views. A converse in geometry is when you take an conditional statement and reverse the premise if p and the conclusion then q. Given a polygon, if it is a square then it has 4 sides. This statement is true.

What is a Biconditional statement in geometry?

A biconditional statement is a combination of a conditional statement and its converse written in the if and only if form. Two line segments are congruent if and only if they are of equal length. A biconditional is true if and only if both the conditionals are true.

## How Do You Prove Contrapositive

Geometry 3.3a, Converse of the Corresponding angles postulate

In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. In other words, the conclusion if A, then B is inferred by constructing a proof of the claim if not B, then not A instead.

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## The Converse Of A Conditional Statement

For a given the conditional statement p} \to q}, we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. Therefore, the converse is the implication q} \to p}.

Notice, the hypothesis \largep} of the conditional statement becomes the conclusion of the converse. On the other hand, the conclusion of the conditional statement \largep} becomes the hypothesis of the converse.

## What Is A Inverse Statement In Geometry

4.7/5Inversestatementinverseinverse

Then, how do you find the inverse of a statement?

To form the converse of the conditional statement, interchange the hypothesis and the conclusion. The converse of “If it rains, then they cancel school” is “If they cancel school, then it rains.” To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

Similarly, what is a Biconditional statement in geometry? A biconditional statement is a combination of a conditional statement and its converse written in the if and only if form. Two line segments are congruent if and only if they are of equal length. A biconditional is true if and only if both the conditionals are true.

Also to know is, what is a converse statement in geometry?

Converse. Switching the hypothesis and conclusion of a conditional statement. For example, the converse of “If it is raining then the grass is wet” is “If the grass is wet then it is raining.” Note: As in the example, a proposition may be true but have a false converse. See also.

Which is the inverse of P Q?

The converse of p q is q p. The inverse of p q is p q. A conditional statement and its converse are NOT logically equivalent.

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## What Is A Converse In Geometry

The converse in geometry applies to a conditional statement. In a conditional statement, the words “if” and “then” are used to show assumptions and conclusions that are to be arrived at using logical reasoning. This is often used in theorems and problems involving proofs in geometry.

If two parallel lines are intersected by a third line in two points, then the pairs of alternate interior angles are congruent. This is a conditional statement and uses the word “if” followed by the word “then” in the same sentence. When this relationship is reversed, the result is a converse statement. If a pair of alternate interior angles is congruent, then the two lines are parallel. The assumptions and the required conclusions are interchanged.