Question

explain why an obtuse angle does not have a complement.
an obtuse angle does not have a complement because an angle cannot have a measure of 0°.
an obtuse angle does not have a complement because the measures of two obtuse angles cannot be added to get 180°.
the measure of an obtuse angle is greater than 90°, and cannot be added to the measure of another angle to get 90°.
the measure of an obtuse angle is greater than 90°, and you cannot add it to the measure of another obtuse angle and get 180°.

Explanation:

To determine why an obtuse angle has no complement, we first recall the definition of complementary angles: two angles are complementary if their measures add up to \(90^\circ\). An obtuse angle is defined as an angle with a measure greater than \(90^\circ\) and less than \(180^\circ\) (i.e., \(90^\circ < \text{obtuse angle} < 180^\circ\)).

Now, suppose an obtuse angle \(\theta\) (where \(90^\circ < \theta < 180^\circ\)) had a complement, say \(\alpha\). By the definition of complementary angles, we would have \(\theta+\alpha = 90^\circ\). Rearranging this formula to solve for \(\alpha\), we get \(\alpha=90^\circ - \theta\). But since \(\theta > 90^\circ\), substituting \(\theta\) into the equation for \(\alpha\) gives \(\alpha=90^\circ - \theta < 90^\circ - 90^\circ = 0^\circ\). However, the measure of an angle cannot be negative (angles have non - negative measures, typically between \(0^\circ\) and \(180^\circ\) for the types we are considering here). So, there is no non - negative angle \(\alpha\) that can satisfy the complementary angle relationship with an obtuse angle \(\theta\).

Looking at the options:

• Option 1: Talks about an angle not having a measure of \(0^\circ\), which is not relevant to the definition of complementary angles.
• Option 2: Mentions the sum of two obtuse angles not being \(180^\circ\), but complementary angles sum to \(90^\circ\), not \(180^\circ\) (supplementary angles sum to \(180^\circ\)), so this is incorrect.
• Option 3: Says an obtuse angle (greater than \(90^\circ\)) cannot be added to another angle to get \(90^\circ\). Since the obtuse angle is already more than \(90^\circ\), adding any non - negative angle to it will result in a sum greater than \(90^\circ\), so it can't have a complement. This is correct.
• Option 4: Talks about the sum of two obtuse angles not being \(180^\circ\), which is about supplementary angles, not complementary angles, so it is incorrect.

Answer:

The measure of an obtuse angle is greater than \(90^\circ\), and cannot be added to the measure of another angle to get \(90^\circ\) (the third option).