d) If the angles of a triangle are in the ratio $$ 2: 3: 5 $$, find them. If two acute angles of a right angled triangle are in the ratio $$ 1: 2 $$, find them. If $$ x^{\circ} $$ is the exterior angle and $$ 70^{\circ} $$ and $$ 50^{\circ} $$ are its opposite interior angle of

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To find the angles of a triangle when they are in a given ratio, we can use the fact that the sum of the angles in a triangle is always $$ 180^{\circ} $$.

1. For the first part, where the angles of a triangle are in the ratio $$ 2: 3: 5 $$:
Let the angles be $$ 2x, 3x, $$ and $$ 5x $$ respectively.
The sum of the angles in a triangle is $$ 180^{\circ} $$, so we have:
$$ 2x + 3x + 5x = 180 $$
$$ 10x = 180 $$
$$ x = 18 $$
Therefore, the angles of the triangle are $$ 2x = 36^{\circ}, 3x = 54^{\circ}, $$ and $$ 5x = 90^{\circ} $$.

2. For the second part, where two acute angles of a right-angled triangle are in the ratio $$ 1: 2 $$:
Let the angles be $$ x $$ and $$ 2x $$ respectively.
Since it is a right-angled triangle, the sum of the two acute angles is $$ 90^{\circ} $$, so we have:
$$ x + 2x = 90 $$
$$ 3x = 90 $$
$$ x = 30 $$
Therefore, the angles of the right-angled triangle are $$ 30^{\circ} $$ and $$ 60^{\circ} $$.

3. For the third part, where $$ x^{\circ} $$ is the exterior angle and $$ 70^{\circ} $$ and $$ 50^{\circ} $$ are its opposite interior angles:
The sum of the interior and exterior angles of a triangle is always $$ 180^{\circ} $$.
So, we have:
$$ x + 70 + 50 = 180 $$
$$ x + 120 = 180 $$
$$ x = 60 $$
Therefore, the exterior angle is $$ 60^{\circ} $$ and the opposite interior angles are $$ 70^{\circ} $$ and $$ 50^{\circ} $$.
1. Triangle angles: $$ 36^{\circ}, 54^{\circ}, 90^{\circ} $$. 2. Right triangle angles: $$ 30^{\circ}, 60^{\circ} $$. 3. Exterior angle: $$ 60^{\circ} $$, opposite interior angles: $$ 70^{\circ}, 50^{\circ} $$.

d) If the angles of a triangle are in the ratio \( 2: 3: 5 \), find them. If two acute angles of a right angled triangle are in the ratio \( 1: 2 \), find them. If \( x^{\circ} \) is the exterior angle and \( 70^{\circ} \) and \( 50^{\circ} \) are its opposite interior angle of

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