Let's study each statement to see which one is true.

A right angle has a complement

An angle $A$ has a complement if there is an angle $B$ such that

$m ∠ B > 0,$ and
$m ∠ A + m ∠ B = 90 .$

A right angle is an angle whose measure is $90 .$ There is no angle $B,$ with $m ∠ B > 0,$ so that $90 + m ∠ B = 90 .$ Therefore, a right angle does not have a complement and thus statement A is false.

An obtuse angle has a complement

An angle $A$ has a complement if there is an angle $B$ such that

$m ∠ B > 0,$ and
$m ∠ A + m ∠ B = 90 .$

An obtuse angle is an angle whose measure is greater than

90 .

Let

A

be an obtuse angle and

B

it's complement. Therefore, we have

m ∠ A + m ∠ B = 90 \Leftrightarrow m ∠ B = 90 - m ∠ A

Since

A

is an obtuse angle, it is

m ∠ A > 90

and thus

90 - m ∠ A < 0 .

Hence

m ∠ B < 0,

which cannot happen. Therefore, an obtuse angle does not have a complement and thus statement B is false.

The supplement of a right angle is a right angle

The supplement of an angle $A$ is another angle $B$ such that

$m ∠ B > 0,$ and
$m ∠ A + m ∠ B = 180 .$

Hence, the supplement of a right angle is an angle

B,

with

m ∠ B > 0

such that the following is true.

90 + m ∠ B = 180 \Leftrightarrow m ∠ B = 90

Therefore,

∠ B

is a right angle and thus statement C is true.

Every angle has a supplement

The supplement of an angle $A$ is another angle $B$ such that

$m ∠ B > 0,$ and
$m ∠ A + m ∠ B = 180 .$

Let $A$ be a straight angle. Thus $m ∠ A = 180 .$ There is no angle $B,$ with $m ∠ B > 0,$ so that $180 + m ∠ B = 180 .$ Therefore, a straight angle does not have a supplement and thus it is not true that every angle has a supplement. Hence statement D is false.

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