T P is a line that tangent to a circle centered at O . If P Q ∥ T O and ∠ O T P = 2 5 ∘ , find the measure of ∠ P O Q in degrees.
Note: The diagram above is not drawn to scale.
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It's beautiful
The diagram above is drawn to scale
∠ O P T is obviously 9 0 ∘ because it’s tangent to a circle
∠ T O P = 1 8 0 ∘ − ∡ O T P − ∡ O P T = 1 8 0 ∘ − 2 5 ∘ − 9 0 ∘ = 6 5 ∘
Since P Q is parallel to C D , so we can conclude that ∡ T O P = ∠ O P Q 6 5 ∘ = ∡ O P Q
Since O P and O Q is the same length, we can conclude that ∡ O P Q = ∠ O Q P 6 5 ∘ = ∡ O Q P
Finally, ∠ P O Q = 1 8 0 ∘ − ∡ O P Q − ∡ O Q P = 1 8 0 ∘ − 6 5 ∘ − 6 5 ∘ = 1 8 0 ∘ − 1 3 0 ∘ = ♣ ♠ 5 0 ∘ ♣ ♠
The Spadesuit( ♠ )and the Clubsuit( ♣ ) are just for the decoration. Want more ? :)
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∠ Q R P = ∠ Q P S ⇒ ∠ P O Q = 5 0 °