The problem involves a circle with diameters $AC$ and $BD$ that intersect at point $E$, and these diameters are perpendicular. This setup forms right angles at $E$ and divides the circle into four right-angled triangles.

Let's go through each question:

31. $m\angle AED$

Since $AC$ and $BD$ are diameters and are perpendicular, they form a right angle at $E$. Thus: $m\angle AED = 90^\circ$

32. $m\overset{\frown}{AB}$

The arc $AB$ is intercepted by the central angle $\angle AEB$, which measures $90^\circ$ (since $\angle AEB$ is part of the perpendicular diameters). Therefore: $m\overset{\frown}{AB} = 90^\circ$

33. $m\angle ADC$

Since $A$, $D$, and $C$ are points on the circle and $AC$ is a diameter, $\angle ADC$ is an inscribed angle that intercepts a semicircle (180° arc). By the inscribed angle theorem, an angle that intercepts a semicircle is a right angle: $m\angle ADC = 90^\circ$

34. $m\overset{\frown}{BCD}$

The arc $BCD$ is the major arc that covers three of the four 90° quadrants in the circle. Thus: $m\overset{\frown}{BCD} = 270^\circ$

35. $m\angle BDC$

This is an inscribed angle that intercepts the arc $BC$, which is 90°. By the inscribed angle theorem, $m\angle BDC$ is half of the intercepted arc: $m\angle BDC = \frac{90^\circ}{2} = 45^\circ$

Summary of Answers

• 31. $m\angle AED = 90^\circ$
• 32. $m\overset{\frown}{AB} = 90^\circ$
• 33. $m\angle ADC = 90^\circ$
• 34. $m\overset{\frown}{BCD} = 270^\circ$
• 35. $m\angle BDC = 45^\circ$

Would you like any additional details or explanations?

Related Questions:

1. Why is an inscribed angle half of the intercepted arc's angle?
2. What are the properties of diameters that intersect at right angles in a circle?
3. How does the central angle relate to its intercepted arc in a circle?
4. Can you apply the same concepts if the diameters were not perpendicular?
5. What happens to inscribed angles in a circle when they intercept the same arc?

Tip: In a circle, an inscribed angle is always half the measure of the arc it intercepts. This is helpful for quickly finding angle measures based on arc lengths.