Find the measure of ∠ AOB using the fact that two angles in a rightisosceles triangle measure 45^(∘).
mAB=90^(∘)
Practice makes perfect
We are given the following diagram.
We know that chords OM and MB are congruent, which makes the right triangle △ OMB isosceles. Thus, angles ∠ MOB and ∠ MBO are congruent and their measures can be represented by x. Since the sum of angles in any triangle is 180^(∘), we can write the following equation.
90^(∘)+x+x=180^(∘)Let's solve it and find the value of x.
In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.
In our case diameter CD is perpendicular to chord AB, so by the theorem M is its midpoint. Hence, segments AM and MB are congruent.
Similarly to △ OMB, triangle △ OMA is also an isosceles right triangle and its angles ∠ AOM and ∠ OAM are congruent and measure 45^(∘).
Adding the measures of ∠ AOM and ∠ BOM, we can find the measure of ∠ AOB.
m∠ AOB=45^(∘)+45^(∘)=90^(∘)
The measure of an arc is the same as the measure of its central angle. Angle ∠ AOB is the central angle of the arc AB, which allows us to conclude that mAB=90^(∘).
