Consider the given diagram. On the figure, the red marker indicates that the two lines are perpendicular. This means that ∠ 4, ∠ 7, and ∠ 9 are all right angles.
m∠ 4=m∠ 7=m∠ 9=90
Let's shade two of the right triangles on the figure.
Now, to find which angle has the greatest measure, we will first estimate the measures of ∠ 8 and ∠ 3.
Estimating m∠ 8
According to the Exterior Angle Theorem, the measure of an exterior angle is the sum of the measures of the two remote interior angles. Let's apply this to the second right triangle.
Notice that ∠ 8 is an exterior angle and ∠ 5 and ∠ 7 are the corresponding remote interior angles.
m∠ 8=m∠ 5+ m∠ 7
Because ∠ 7 is a right angle, ∠ 8 is an obtuse angle.
Estimating m∠ 3
In the first right triangle, ∠ 4 is a right angle.
Since ∠ 9 is a right angle, ∠ 8 is an obtuse angle, and ∠ 3 is an acute angle, out of these three angles ∠ 8 has the greatest measure.
m∠ 8 > m∠ 9 > m∠ 3
