How to find the value of x in a triangle with an exterior angle

Find the value of x if interior angles are x°, 3y° and 5y° and exterior angle of a triangle is 7y°

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$\Delta ABC$ is triangle and its interior angles are $x^\circ$, $3y^\circ$ and $5y^\circ$. The side $\overrightarrow{AB}$ of the triangle is extended through a point $D$. Thus, the triangle $ABC$ has an exterior angle $CBD$ and it is $7y^\circ$.

In this geometry problem, the literals $x$ and $y$ are used to represent angles but they are symbolically written as $x^\circ$ and $y^\circ$. The meaning of representing $x$ and $y$ as $x^\circ$ and $y^\circ$ is the values of $x$ and $y$ are in degrees, which means the literal numbers $x$ and $y$ represent angles in degree measuring system.

Now, let us find the value of $x$ by using the geometrical relation between $y$ and $x$.

Forget about the triangle $ABC$ but only consider the intersecting lines $\overrightarrow{AD}$ and $\overline{BC}$. Geometrically, the line segment $\overline{BC}$ divides the angle $ABD$ into two parts and the angles are $5y$ and $7y$ in degrees.

Therefore $\angle CBA = 5y$ and $\angle CBD = 7y$ in degrees.

The line $\overrightarrow{AD}$ is a straight ray. So, the angle of the line $\overrightarrow{AD}$ is $\angle ABD$ and it is a straight angle. Therefore, the angle $ABD$ is $180^\circ$ geometrically.

Geometrically, the sum of the angles $\angle CBA$ and $\angle CBD$ is equal to $\angle ABD$.

$\angle CBD + \angle CBA = \angle ABD$

$\implies 7y + 5y = 180^\circ$

$\implies 12y = 180^\circ$

$\implies y = \dfrac{180^\circ}{12}$

$\implies \require{cancel} y = \dfrac{\cancel{180^\circ}}{\cancel{12}}$

$\therefore \,\,\,\,\,\, y = 15^\circ$

Therefore, the value of $y$ is $15^\circ$.

The value of literal $y$ in degrees is calculated in previous step and it is $15^\circ$. Now, consider only $\Delta ABC$. Geometrically, it is proved that the sum of three angles of a triangle is $180^\circ$.

It can be expressed in mathematical form as follows.

$\angle CBA + \angle BAC + \angle ACB = 180^\circ$

Substitute the values of all three angles of triangle $ABC$ in algebraic form.

$\implies 5y + x + 3y = 180^\circ$

$\implies x + 5y + 3y = 180^\circ$

$\implies x + 8y = 180^\circ$

$\implies x = 180^\circ -8y$

$\implies x = 180^\circ -8 (15^\circ)$

$\implies x = 180^\circ -120^\circ$

$\therefore \,\,\,\,\,\, x = 60^\circ$

Therefore, the value of $x$ is $60^\circ$ and it is the required solution for this geometric problem.

Solution 1:

(i) \angle A\ +\ \angle B+\ \angle C\ =\ 180\degree { angle sum property of triangle }

70\degree+\ 50\degree+\ \angle B\ =\ 180\degree

\angle B\ =\ 180\degree-\ 120\degree

\angle B\ =\ 60\degree

(ii) \angle BAC\ +\angle ACB\ =\ x\ {an exterior angle of a triangle is equal to the sum of the opposite interior angle }

65\degree+45\degree=x

110\degree=x

(iii) \angle CAB+\angle ABC\ =\ x {an exterior angle of a triangle is equal to the sum of the opposite interior angle }

30\degree+40\degree\ =\ x\

70\degree=x

(iv) \angle BCA+\angle CBA\ =\ x {an exterior angle of a triangle is equal to the sum of the opposite interior angle }

60\degree+60\degree\ =\ x

120\degree=x

(v) \angle A+\angle B=\ x {an exterior angle of a triangle is equal to the sum of the opposite interior angle }

50\degree+50\degree=\ x

100\degree=\ x\

(vi) \angle A+\angle B\ =\ x\ {an exterior angle of a triangle is equal to the sum of the opposite interior angle }

30\degree+60\degree\ =\ x\

90\degree=\ x

Properties of Triangles

Question 1: Find the value of unknown exterior angle in following diagrams.

Answer:

(i) `x = 50° + 70° = 120°`

(ii) `x = 65° + 45° = 110°`

(iii) `x = 30° + 40° = 70°`

(iv) `x = 60° + 60° = 120°`

(v) `x = 50° + 50° = 100°`

(vi) `x = 60° + 30° = 90°`

Chapter List

Question 2: Find the value of unknown interior angle in following figures:

Answer: Exterior angle of a triangle is equal to the sum of opposite interior angles.

(i) `x = 115° - 50° = 65°`

(ii) `x = 100° - 70° = 30°`

(iii) `x = 120° - 60° = 60°`

(iv) `x = 80° - 30° = 50°`

(v) `x = 75° - 35° = 40°`

Exercise 6.3

Question 1: Find the value of unknown x in following figures:

(a) Answer: `x = 180° - (50° + 60°) ``= 180° - 110° = 70°`

(b) Answer: `x = 180° - (30° + 90°)`

Since it is a right angle so the third angle is a right angle.

Or, `x = 180° - 120° = 60°`

(c) Answer: `x = 180° - (30° + 110°) ``= 180° - 140° = 40°`

(d) Answer: Here; `50° + 2x = 180°`

Or, `2x = 180° - 50° = 130°`

Or, `x = 130° ÷ 2 = 65°`

(e) Answer: This is an equilateral triangle

Hence, `3x = 180°`

Or, `x = 180° ÷ 3 = 60°`

(f) Answer: This is a right angled triangle.

Hence, `2x + x + 90°= 180°`

Or, `3x = 180° - 90°`

Or, `x = 90° ÷ 3 = 30°`

Question 2: Find the values of the unknowns x and y in the following diagrams.

(i) Answer: Since an external angle is equal to the sum of opposite exterior angles.

Hence, `120° = 50° + x`

Or, `x = 120° - 50° = 70°`

Now, `120° + y = 180°`

Because they make linear pair of angles and angles of a linear pair are always supplementary.

Or, `y = 180° - 120° = 60°`

(ii) Answer: In this case, y = 80°

Because, vertically opposite angles are always equal.

Now, `50° + y + x = 180°` (Angle sum property of triangle)

Or, `50° + 80° + x = 180°`

Or, `x + 130° = 180°`

Or, `x = 180° - 130°= 50°`

(iii) Answer: Here, x = 50° + 60° = 110°

Because , exterior angle in a triangle is equal to sum of opposite internal angles.

Now, `x + y = 180°` (Linear pair of angles are supplementary)

Or, `110° + y = 180°`

Or, `y = 180° - 110° = 70°`

(iv) Answer: Here, `x = 60°` (Vertically opposite angles are equal)

Now, `30° + x + y = 180°` (Angle sum of triangle)

Or, `30° + 60° + y = 180°`

Or, `y + 90° = 180°`

Or, `y = 180° - 90°`

Or, `x = 60°` and `y = 90°`

(v) Answer: Here, x = 90° (Vertically opposite angles are equal)

Now, `x + x + y = 180°`

Or, `2x + 90° = 180°`

Or, `2x = 180° - 90° = 90°`

Or, `x = 90° ÷ 2 = 45°`

(vi) Answer: Here, x = y (Vertically opposite angles are equal.

Thus, all angles of the given triangle are equal. It means that the given triangle is equilateral triangle and each angle has same measure.

Hence, `x = y = 60°`

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