Suppose two adjacent angles are supplementary. Show that if one of them is an obtuse angle, then the other angle must be acute.

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Solution

Let $x$ and $y$ be two adjacent supplementary angles
$\Rightarrow x + y = 180^{\circ}$
$\Rightarrow x = 180 - y$ .......(i)
$∠ x$ is obtuse.
So, $90^{\circ} < x < 180^{\circ}$
Substitute equation(i) in above inequality, we get
$\Rightarrow 90^{\circ} < 180^{\circ} - y < 180^{\circ}$
Subtracting $180^{\circ}$ , we get
$\Rightarrow 90^{\circ} - 180^{\circ} < 180^{\circ} - 180^{\circ} - y < 180^{\circ} - 180^{\circ}$
$\Rightarrow - 90^{\circ} < - y < 0^{\circ}$
$\Rightarrow 0^{\circ} < y < 90^{\circ}$ [ multiplying $- 1$ ]
Hence, the other angle $y$ is an acute angle.

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Answer the following questions and justify.
(i) Can two acute angles he complement to each other?
(ii) Can two obtuse angles be complement to each other?
(iii) Can two right angles he complement to each other?
(iv) Can two acute angles be supplementary?
(v) Can two obtuse angles be supplementary?
(vi) Can two right angles be supplementary?
(vii) Can two adjacent angles be supplementary?
(viii) Can two adjacent angles be complementary?
(ix) Can two obtuse angles be adjacent angles?
(x) Can an acute angle be adjacent to an obtuse angle?
(xi) Can two acute angles form a linear pair?
(xii) can two obtuse angles form a linear pair?
(xiii) Can two right angles form a linear pair?

Can two angles be supplementary if both of them are:

(i) Acute? (ii) Obtuse? (iii) Right?

Q. The supplementary angle of an acute angle will always be an _______ angle, whereas the supplementary angle of an obtuse angle will always be an _______ angle.

If true then enter

1

and if false then enter

0

Can an acute angle be adjacent to an obtuse angle?

Q. Consider the following statements:

Statement I: One of the supplementary angles is always an obtuse angle.

Statement II: One of the complementary angles is always an acute angle.

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