1. Complementary Angles?

,.,good day,.,
,.somethings been buggling my mind lately,.,.i have encountered this theorem on geometry,.,it says "If two angles are complementary, then both are acute"
so i have this question "is zero degree the complement of 90 degrees???if so,.then both are not acute since acute angles are angles less than 90,.,please help me with this,.,.thnx

2. Re: Complementary Angles?

Originally Posted by aldrincabrera
theorem on geometry,.,it says "If two angles are complementary, then both are acute" so i have this question "is zero degree the complement of 90 degrees???if
In a course in axiomatic geometry, most authors use this definition:
An angle is the union of two non-collinear rays with a common endpoint.
So that rules out both a degenerate angle and a so-called ‘straight angle’.
In fact, Jay Greenberg has the following theorem in his text:
$\displaystyle \left( {m\angle A} \right)^ \circ \text{ is a real number and }0<\left( {m\angle A} \right)^ \circ<180.$

So strictly speaking we would not have an angle of measure zero.

3. Re: Complementary Angles?

,.,sir,.,i was proving the theorem and i ended up with this,.plss tell me if im doing it correctly,.
here is my proof.,
let alpha and beta be complementary angles,
then alpha + beta = 90.
we show that alpha < 90 and beta<90.
Suppose alpha >equal to 90 or beta >equal to 90,
then alpha+beta >equal to 90 in either cases.
thus alpha<90 and beta <90,.

am i wrong sir??thnx

4. Re: Complementary Angles?

Originally Posted by aldrincabrera
,.,sir,.,i was proving the theorem and i ended up with this,.plss tell me if im doing it correctly,.
here is my proof.,
let alpha and beta be complementary angles,
then alpha + beta = 90.
we show that alpha < 90 and beta<90.
Suppose alpha >equal to 90 or beta >equal to 90,
then alpha+beta >equal to 90 in either cases.
thus alpha<90 and beta <90,.
If the sum of two positive numbers is 90 then each must be less than 90.

5. Re: Complementary Angles?

,.but sir,.,am i on the right track??

6. Re: Complementary Angles?

Let $\displaystyle \alpha$ and $\displaystyle \beta$ be complementary angles so indeed:
$\displaystyle \alpha+\beta=90$
If you want to show: $\displaystyle \alpha<90$ and $\displaystyle \beta<90$.
You have to do that for positive numbers!

Because if you take $\displaystyle \alpha=-240°$ and $\displaystyle \beta=330$
Then $\displaystyle \alpha+\beta=90$ but $\displaystyle \beta$ is not $\displaystyle <90$.

So you have to assure you work with positive numbers.

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theorem 4-3 if two angles are complementary, then both are acute

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