# The larger of two supplementary angles exceeds the smaller by $18{}^\circ$. Find them.

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Hint: Let the two angles be x and y and let x>y. Then write the conditions according to the question. Apply the condition for supplementary. Then solve the two equations. You will get the two angles.

Complete step-by-step answer:

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces.

These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

Angles that have the same measure (i.e. the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are equal in measure).

Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.

Supplementary angles are two angles with a sum of $180{}^\circ$. A common case is when they lie on the same side of a straight line.

So let $x$ and $y$ be the two supplementary angles.

We get $x+y=180{}^\circ $ …………. (1)

It is given in question that the larger of two supplementary angles exceeds the smaller by $18{}^\circ$.

So let $x>y$.

According to question,

$x=y+18{}^\circ $………….. (2)

Now substituting (1) in (2), we get,

$y+18{}^\circ +y=180{}^\circ $

Simplifying we get,

$\begin{align}

& 2y+18{}^\circ =180{}^\circ \\

& 2y=162{}^\circ \\

\end{align}$

$y=81{}^\circ $

So substituting $y$ in (2), we get,

$x=81{}^\circ +18{}^\circ =99{}^\circ $

$x=99{}^\circ $

So the two supplementary angles are $99{}^\circ $ and $81{}^\circ $ .

Note: Read the question carefully. Also, you must know the concept behind the supplementary angles. You should also know that “Supplementary angles are two angles with a sum of $180{}^\circ $”. Do not miss any term while simplifying.

Complete step-by-step answer:

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces.

These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

Angles that have the same measure (i.e. the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are equal in measure).

Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.

Supplementary angles are two angles with a sum of $180{}^\circ$. A common case is when they lie on the same side of a straight line.

So let $x$ and $y$ be the two supplementary angles.

We get $x+y=180{}^\circ $ …………. (1)

It is given in question that the larger of two supplementary angles exceeds the smaller by $18{}^\circ$.

So let $x>y$.

According to question,

$x=y+18{}^\circ $………….. (2)

Now substituting (1) in (2), we get,

$y+18{}^\circ +y=180{}^\circ $

Simplifying we get,

$\begin{align}

& 2y+18{}^\circ =180{}^\circ \\

& 2y=162{}^\circ \\

\end{align}$

$y=81{}^\circ $

So substituting $y$ in (2), we get,

$x=81{}^\circ +18{}^\circ =99{}^\circ $

$x=99{}^\circ $

So the two supplementary angles are $99{}^\circ $ and $81{}^\circ $ .

Note: Read the question carefully. Also, you must know the concept behind the supplementary angles. You should also know that “Supplementary angles are two angles with a sum of $180{}^\circ $”. Do not miss any term while simplifying.

Last updated date: 19th Sep 2023

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