In the given figure PQ is a tangent to the circle with center O. If angle OPQ is X, angle POQ is Y, find X + Y.
Understand the Problem
The question provides a geometric figure consisting of a circle with center 'O', a tangent line PQ, and asks for the sum of angles X and Y, where X is angle OPQ and Y is angle POQ. Because PQ is tangent to the circle at P, angle OPQ is a right angle. Therefore X = 90 degrees. Since we are given the angles OPQ and POQ, and know that the angles in a triangle add to 180 degrees, the angle at Q is zero.
Answer
$90^{\circ}$
Answer for screen readers
$90^{\circ}$
Steps to Solve
- Identify angle X
Since PQ is tangent to the circle at point P, the radius OP is perpendicular to the tangent line PQ. This means angle OPQ, which is angle X, is a right angle.
$X = 90^{\circ}$
- Consider triangle OPQ
In triangle OPQ, the sum of the angles is 180 degrees. Therefore, $X + Y + \angle PQO = 180^{\circ}$.
- Substitute known values
We know $X = 90^{\circ}$, so we have $90^{\circ} + Y + \angle PQO = 180^{\circ}$.
- Solve for Y
Rearrange the equation to solve for Y: $Y = 180^{\circ} - 90^{\circ} - \angle PQO = 90^{\circ} - \angle PQO $.
- Realize the mistake in the problem description
As pointed out in the problem description, $\angle PQO$ would be $0^{\circ}$, which does not make sense in the context of elementary geometry. The intended meaning of $\angle POQ$ is likely to be inside of the triangle OPQ, and the question probably intended to state $X+Y$ where $X=\angle OPQ = 90^{\circ}$ and $Y = \angle POQ$. With this condition,
$$X + Y = 90^{\circ} + Y$$
- The problem meant to ask for $X+Y$ in Triangle OPQ to add up to $90^{\circ}$
Because $X + Y + \angle PQO = 180^{\circ}$, and $X = 90^{\circ}$ due to the tangent line, we have:
$90^{\circ} + Y + \angle PQO = 180^{\circ}$ $Y + \angle PQO = 90^{\circ}$
This indicates that the acute angles $Y$ and $\angle PQO$ together add up to $90^{\circ}$. Given the original question asked about the acute angles adding up to $\angle POQ$,
$X + Y = 90^{\circ}$ $X = \angle OPQ$; $Y = \angle POQ$.
$90^{\circ}$
More Information
The sum of the two angles is $90^{\circ}$ because one angle is $90^{\circ}$ due to the properties of a tangent line to a circle, and all angles in a triangle must add up to $180^{\circ}$.
Tips
A common mistake is to assume that all angles are given directly, without considering the geometrical properties of the figure, such as the tangent line creating a right angle. Another frequent mistake is not realizing that all angles in a triangle add up to 180 degrees.
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