In the given figure, 𝑋𝑌 and 𝑋
prime 𝑌 prime are two parallel tangents to circle 𝑂, and another tangent 𝐴𝐵 with
point of contact 𝐶 is intersecting 𝑋𝑌 at 𝐴 and 𝑋 prime 𝑌 prime at 𝐵. Prove that angle 𝐴𝑂𝐵 equals 90
First, let’s recognize which angle
angle 𝐴𝑂𝐵 is — that’s here. And we want to prove that this
angle measures 90 degrees. To prove this, we’ll have to start
with what we know. We know that 𝑋 and 𝑌 are tangents
to circle 𝑂. And we also know that 𝐴𝐵 is a
tangent to circle 𝑂.
Given that these are tangents, we
can say that line segment 𝑂𝐶 is perpendicular to line segment 𝐴𝐵 because tangent
lines are always perpendicular to a radius. We can also say by the same
reasoning that 𝑂𝑃 is perpendicular to 𝐴𝑃, 𝑂𝑃 being a radius and 𝐴𝑃 being
We can also say from the given
information that 𝑂𝐶 is equal to 𝑂𝑃 as they both are a radius length. We can say that 𝑂𝐴 is equal to
𝐴𝑂. And both of these lengths represent
a hypotenuse of a right triangle, which means triangle 𝐴𝑂𝐶 is congruent to
Because these triangles both have a
right angle, the same hypotenuse, and an equal side, these two triangles are
congruent by right hypotenuse side. Because these two triangles are
congruent, angle 𝐴𝑂𝑃 is equal in measure to angle 𝐴𝑂𝐶. We know this is true because
corresponding parts of congruent triangles are equal.
Similarly, we can say that 𝑂𝐶 is
perpendicular to 𝐶𝐵 because one is a radius and one is a tangent line. 𝑂𝑄 is perpendicular to 𝑄𝐵 and
these both create right angles. 𝑂𝐶 is equal to 𝑂𝑄: they’re both
a radius. And 𝑂𝐵 equals 𝐵𝑂: they’re the
same line and they represent the hypotenuse of two right triangles. Triangle 𝐵𝑂𝐶 is congruent to
triangle 𝐵𝑂𝑄 by right hypotenuse side. Because of that, we know that angle
𝐵𝑂𝐶 is equal to angle 𝐵𝑂𝑄 by corresponding parts of congruent triangles.
Remember that we’re trying to prove
angle 𝐴𝑂𝐵 measures 90 degrees. Since 𝑃𝑄 is a straight line, all
four of these angles must add up to 180 degrees. We know that angle 𝐴𝑂𝑃 is equal
to angle 𝐴𝑂𝐶. So we’re going to substitute angle
𝐴𝑂𝐶 in place of angle 𝐴𝑂𝑃. Similarly, in place of angle
𝐵𝑂𝑄, we’ll substitute angle 𝐵𝑂𝐶 because they are equal.
Looks like we need a little bit
more room. So I’m gonna move some things
around. Back to where we were. We now have two 𝐴𝑂𝐶 angles and
two 𝐵𝑂𝐶 angles. We could write this as two times
angle 𝐴𝑂𝐶 plus angle 𝐵𝑂𝐶. We can divide this equation by two
on both sides. When we simplify that, we get angle
𝐴𝑂𝐶 plus angle 𝐵𝑂𝐶 equals 90 degrees.
Let’s look at our image. We’ve just said that angle 𝐴𝑂𝐶
plus angle 𝐵𝑂𝐶 equals 90 degrees. Angle 𝐴𝑂𝐶 plus angle 𝐵𝑂𝐶 is
the same thing as saying angle 𝐴𝑂𝐵. We have thus proven that angle
𝐴𝑂𝐵 equals 90 degrees. And here is our full proof.