/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Rewrite the following sentence i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Rewrite the following sentence in conditional form and find its converse, inverse, and contrapositive: "A square is a quadrilateral with four congruent sides."

Short Answer

Expert verified
The conditional form of the sentence is 'If a figure is a square, then it is a quadrilateral with four congruent sides.' The converse is 'If a figure is a quadrilateral with four congruent sides, then it is a square.' The inverse is 'If a figure is not a square, then it is not a quadrilateral with four congruent sides.' The contrapositive is 'If a figure is not a quadrilateral with four congruent sides, then it is not a square.'

Step by step solution

01

Put Into Conditional Form

The given sentence can be put into conditional form by identifying the hypothesis and conclusion. In this case, the hypothesis is 'A figure is a square' and the conclusion is 'It is a quadrilateral with four congruent sides.' Thus, the conditional sentence is: 'If a figure is a square, then it is a quadrilateral with four congruent sides.'
02

Find the Converses

We can find the converse of the statement by swapping the hypothesis and conclusion. So the converse is: 'If a figure is a quadrilateral with four congruent sides, then it is a square.'
03

Find the Inverse

The inverse of this statement can be found by negating both the hypothesis and the conclusion. So, the inverse is: 'If a figure is not a square, then it is not a quadrilateral with four congruent sides.'
04

Find the Contrapositive

To find the contrapositive, we need to swap and negate both the hypothesis and the conclusion. So, the contrapositive is: 'If a figure is not a quadrilateral with four congruent sides, then it is not a square.'

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Converse
When we talk about the concept of the "converse" in mathematics, especially regarding conditional statements, it involves swapping the hypothesis and conclusion.
The converse of a statement plays a critical role in logical reasoning.
For any given conditional statement, such as 鈥淚f a figure is a square, then it is a quadrilateral with four congruent sides,鈥� its converse will be achieved by simply flipping the hypothesis and conclusion.
Thus, the converse would be: 鈥淚f a figure is a quadrilateral with four congruent sides, then it is a square.鈥�
  • The original statement may be true while the converse is not.
  • Considering both the original and its converse helps in understanding the relationship from both directions.
  • The concept of the converse is fundamental in geometry and logic.
Inverse
An inverse related to a given conditional statement is formulated by negating both the hypothesis and conclusion of the original statement.
To construct an inverse, simply place the word "not" in front of both parts of the conditional statement.
Let鈥檚 consider the statement, 鈥淚f a figure is a square, then it is a quadrilateral with four congruent sides.鈥� The inverse would be, 鈥淚f a figure is not a square, then it is not a quadrilateral with four congruent sides.鈥�
  • The inverse does not always follow the truth value of the original statement.
  • Using inverses helps in deducing logical relationships by understanding what is genuinely confirmed or denied.
  • Knowing how to find an inverse is valuable when analyzing conditional statements.
Contrapositive
The contrapositive of a conditional statement is essentially a mix of finding the converse and the inverse.
This means we switch the hypothesis and conclusion, just like in a converse, while also negating both, as in an inverse.
Take the conditional statement: 鈥淚f a figure is a square, then it is a quadrilateral with four congruent sides.鈥� Its contrapositive would be, 鈥淚f a figure is not a quadrilateral with four congruent sides, then it is not a square.鈥�
  • Contrapositives hold a unique property鈥攖hey always share the truth value with the original statement.
  • If the contrapositive of a statement is true, then the original statement must also be true, and vice versa.
  • This makes contrapositive a key technique in proofs, particularly in mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(\mathrm{AB}=16, \mathrm{BC}=8,\) and \(\mathrm{AC}=24,\) which point is between the other two?

Study the following five statements. 1 Spoof is the set of all purrs. 2 Spoof contains at least two distinct purrs. 3 Every lilt is a set of purrs and contains at least two distinct purrs. 4 If \(A\) and \(B\) are any two distinct purrs, there is one and only one lilt that contains them. 5 No lilt contains all the purrs. a. Show that each of the following statements is true. i There is at least one lilt. ii There are at least three purrs. iii There are at least three lilts. b. If the lilt "girt" contains the purr "pil" and the purr "til" and if the lilt "mirt" contains the purr "pil" and the purr "til" then the lilt "girt" is the same as the lilt "mirt" except in one case. What is this case?

Study each of the arguments and state whether or not the conclusion is deducible. If it is not, comment on the error in the reasoning. If two angles of a triangle are congruent, then the sides opposite them are congruent. In \(\triangle \mathrm{ABC}, \angle \mathrm{A} \cong \angle \mathrm{B} .\) Therefore, in \(\triangle \mathrm{ABC}\), \(\overline{\mathrm{BC}} \cong \overline{\mathrm{AC}}\).

Write the converse, the inverse, and the contrapositive of each statement. a If a ray bisects an angle, it divides the angle into two congruent angles. b If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Write the converse, the inverse, and the contrapositive of each statement. Determine the truth of each of the new statements. a If each side of a triangle has a length of \(10,\) then the triangle's perimeter is 30 b-If an angle is acute, then it has a measure greater than 0 and less than 90
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.