EDGE Math 8

Example

State the (1) converse, (2) inverse, and (3) contrapositive of the statement "If $∠ A$ and $∠ B$ are right angles, then $∠ A ≅ ∠ B . "$ Determine the truth value of each statement.

Solution

The implication is "If $∠ A$ and $∠ B$ are right angles, then $∠ A ≅ ∠ B . "$ This statement is true since $m ∠ A = m ∠ B = 90.$

1.	Converse: If $∠ A ≅ ∠ B,$ then $∠ A$ and $∠ B$ are right angles.
	The converse statement is false since congruence of angles does not guarantee that they are right angles. A counterexample for this statement is the case when $m ∠ A = m ∠ B = 30.$ Angles A and B are congruent but they are not right angles.

2.	Inverse: If $∠ A$ and $∠ B$ are not right angles, then $∠ A ≅ ∠ B .$
	(The symbol $≅$ is read as "not congruent.") The inverse statement is false because the fact that $∠ A$ and $∠ B$ are not right angles does not guarantee that the two angles are not congruent. The same counterexample may be used when $m ∠ A = m ∠ B = 30.$ The two angles are acute (not right angles) but they are congruent.

3.	Contrapositive: If $∠ A ≅ ∠ B,$ then $∠ A$ and $∠ B$ are not right angles.
	The contrapositive statement is true since you know that all right angles have the same measure. Since $∠ A$ and $∠ B$ are not congruent, it follows that the two angles do not have the same measure and both are not right angles.

Types of Statements