1. **Problem Statement:** Prove that in a circle with center O and diameter AB, the angle at the circumference subtended by AB is a right angle, i.e., $\angle ARB = 90^\circ$. 2. **Formula and Theorem:** The angle subtended by a diameter at the circumference of a circle is a right angle. This is a direct consequence of the theorem that the measure of an angle at the center of a circle is twice the measure of the angle at the circumference subtended by the same arc. 3. **Step-by-step Proof:** - Let $\angle AOE$ be the angle at the center subtended by arc AE. - By the theorem, $\angle AOE = 2 \times \angle ABP$ where $\angle ABP$ is the angle at the circumference subtended by the same arc. - Since AB is a diameter, $\angle AOE = 180^\circ$ (a straight angle). - Therefore, $180^\circ = 2 \times \angle ABP$. - Solving for $\angle ABP$, we get $\angle ABP = \frac{180^\circ}{2} = 90^\circ$. - Hence, $\angle ARB = 90^\circ$. 4. **Explanation:** This means any triangle inscribed in a circle where one side is the diameter will always be a right triangle with the right angle opposite the diameter. 5. **Summary:** We used the central angle theorem and the property of a diameter to prove that $\angle ARB = 90^\circ$. --- **Multiple Choice Questions:** 1. What is the measure of the angle subtended by a diameter at the circumference? A) 45° B) 90° C) 180° D) 60° 2. If $\angle AOE = 180^\circ$, what is $\angle ABP$? A) 60° B) 90° C) 120° D) 45° 3. The central angle theorem states that the angle at the center is: A) Equal to the angle at the circumference B) Twice the angle at the circumference C) Half the angle at the circumference D) Four times the angle at the circumference 4. In a circle, the angle subtended by an arc at the center is 100°. What is the angle subtended by the same arc at the circumference? A) 50° B) 100° C) 200° D) 25° 5. Which of the following is true for a triangle inscribed in a circle with one side as the diameter? A) It is an equilateral triangle B) It is a right triangle C) It is an isosceles triangle D) It is an obtuse triangle 6. If $\angle ABP = 90^\circ$, what type of triangle is $\triangle ARB$? A) Acute B) Right C) Obtuse D) Equilateral 7. The angle at the center of a circle is 2 times the angle at the circumference subtended by the same arc. This is an example of: A) Pythagoras theorem B) Central angle theorem C) Alternate segment theorem D) Angle sum property 8. If $\angle AOE = 180^\circ$, what kind of angle is it? A) Acute B) Right C) Straight D) Reflex 9. The angle subtended by a chord at the center is 80°. What is the angle subtended by the same chord at the circumference? A) 40° B) 80° C) 160° D) 20° 10. In the proof, which property of the diameter is used? A) Diameter subtends a right angle at the circumference B) Diameter is the longest chord C) Diameter divides the circle into two equal parts D) Diameter is perpendicular to the radius