Question

What is true about the statement “If two angles are right angles, the angles have equal measure” and its converse “If two angles have equal measure then the two angles are right angles”?
(A) The statement is true but its converse is false
(B) The statement is false but its converse is true
(C) Both the statement and its converse are false
(D) Both the statement and its converse are true

Answer 1

Hint: An angle is the figure formed by two rays, called sides of an angle, sharing a common endpoint, called the vertex of the angle. A right angle is an angle of exactly ${90^ \circ }$(degrees), corresponding to a quarter turn. We can see right angles in the corners of our rooms, book, cube, windows and several other places. And two angles are called the same measures if their radiant values are exactly the same i.e. two angles $\angle A$ and $\angle B$ are equal if $\angle A = \angle B$ .

Complete step-by-step solution:
We already know that a right angle is the angle which measures exactly ${90^ \circ }$. Now let us take two right angles, $\angle P$ and $\angle Q$, say. Since both of these two are right angles then $\angle P = \angle Q = {90^ \circ }$.
This clearly says that the two right angles $\angle P$ and $\angle Q$are equal.
Since $\angle P$ and $\angle Q$ are taken arbitrarily, therefore we must have any two right angles equal i.e. they are of the same measure.
Hence the statement “If two angles are right angles the angles have equal measure” appears to be true.
Now we will examine whether the converse part of this statement is true or not. Let us take two angles $\angle R$ and $\angle S$such that $\angle R = \angle S = {60^ \circ }$.
Evidently the two angles $\angle R$ and $\angle S$ are of equal measure as $\angle R = \angle S$. But none of these two are right angles as neither the angle $\angle R$ nor the angle $\angle S$ equal to ${90^ \circ }$.
Therefore, this example disproves the converse part of the statement and hence we see that,
The converse statement “If two angles have equal measure then the two angles are right angles” is not true.
Hence the option A is correct answer.

Note: Two angles having the same measure are called Congruent Angles. Angles measuring less than ${90^ \circ }$ are called Acute Angles and Angles which measure greater than ${90^ \circ }$ are called Obtuse Angles.
When we prove a statement to be true, we take some arbitrary elements, then prove it true for all. But in case of false statements, we prove it false by taking a simple example and show the statement to be invalid.