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Q29

Expert-verifiedFound in: Page 35

Book edition
Student Edition

Author(s)
Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages
227 pages

ISBN
9780395977279

Write a definition of congruent angles as a biconditional.

The definition of congruent angles as a biconditional is: “$\angle A\cong \angle B$ **if and only if** $m\angle A=m\angle B$”.

Congruent angles are angles that have equal measures.

Let $\angle A$ and $\angle B$ both have equal measure then:

$\angle A\cong \angle B$ or $m\angle A=m\angle B$.

If a conditional and its converse are both true then they can be combined into a single statement by using connector “if and only if”. Such a statement containing the words “if and only if” is defined as a biconditional. For example, “*p* if and only if *q*”.

The definition of congruent angles tells us that the two statements $\angle A\cong \angle B$ or $m\angle A=m\angle B$ are equivalent.

Therefore, the bi-conditional statement is:

$\angle A\cong \angle B$ if and only if $m\angle A=m\angle B$.

Prove the following statement by filling in the blanks.

If *A* and *B* have coordinated *a* and *b*, with $b>a$, and the midpoint *M* of $\overline{AB}$ has coordinate *x*, then prove $x=\frac{a+b}{2}$.

Proof:

Statement | Reasons |

1. | 1. ? |

2. $AM=x-a\text{};\text{}MB=b-x$ | 2. ? |

3. | 3. ? |

4. $\overline{AM}\cong \overline{MB}$, or $AM=MB$ | 4. ? |

5. $x-a=b-x$ | 5. ? |

6. $2x=$? | 6. ? |

7. $x=\frac{a+b}{2}$ | 7. ? |

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