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Answer. Solution 1. In the above diagram, We have a circle with center 'C' and radius AC=BC=CD. PROOF : THE ANGLE INSCRIBED IN A SEMICIRCLE IS A RIGHT ANGLE In other words, the angle is a right angle. Draw your picture here: Use your notes to help you figure out what the first line of your argument should be. The angle BCD is the 'angle in a semicircle'. Rotating the semicircle and vectors construction does not change the angle, thus the property of being a right angle is general for all semicircles of radius 1. It can be any line passing through the center of the circle and touching the sides of it. Corollary (Inscribed Angles Conjecture III): Any angle inscribed in a semicircle is a right angle. Now there are three triangles ABC, ACD and ABD. Proof by contradiction (indirect proof) Prove by contradiction the following theorem: An angle inscribed in a semicircle is a right angle. A semicircle is inscribed in the triangle as shown. Radius AC has been drawn, to form two isosceles triangles BAC and CAD. Given: M is the centre of circle. Proof: Draw line . ∴ m(arc AXC) = 180° (ii) [Measure of semicircular arc is 1800] The second case is where the diameter is in the middle of the inscribed angle. My proof was relatively simple: Proof: As the measure of an inscribed angle is equal to half the measure of its intercepted arc, the inscribed angle is half the measure of its intercepted arc, that is a straight line. Strategy for proving the Inscribed Angle Theorem. What is the radius of the semicircle? ∠ABC is inscribed in arc ABC. We can reflect triangle over line This forms the triangle and a circle out of the semicircle. Arcs ABC and AXC are semicircles. Proof: The intercepted arc for an angle inscribed in a semicircle is 180 degrees. To prove: ∠ABC = 90 Proof: ∠ABC = 1/2 m(arc AXC) (i) [Inscribed angle theorem] arc AXC is a semicircle. Theorem: An angle inscribed in a Semicircle is a right angle. Show that an inscribed angle's measure is half of that of a central angle that subtends, or forms, the same arc. Scaling the semicircle and vectors construction does not change the angle, thus the property of being a right angle is general for all semicircles. Now draw a diameter to it. Now POQ is a straight line passing through center O. Prove that an angle inscribed in a semicircle is a right angle. Draw the lines AB, AD and AC. When a triangle is inserted in a circle in such a way that one of the side of the triangle is diameter of the circle then the triangle is right triangle. Angle inscribed in semicircle is angle BAD. Since the inscribed angle is half of the corresponding central angle, we can write: Thus, we have proven that if the inscribed angle rests on the diameter, then it is a right angle. They are isosceles as AB, AC and AD are all radiuses. To prove this first draw the figure of a circle. 2. MEDIUM. That is, if and are endpoints of a diameter of a circle with center , and is a point on the circle, then is a right angle. In the right triangle , , , and angle is a right angle. Prove that the angle in a semicircle is a right angle. Angle Addition Postulate. Therefore the measure of the angle must be half of 180, or 90 degrees. Angle Inscribed in a Semicircle. Theorem: An angle inscribed in a semicircle is a right angle. To proof this theorem, Required construction is shown in the diagram. Problem 22. We will need to consider 3 separate cases: The first is when one of the chords is the diameter. If is interior to then , and conversely. So in BAC, s=s1 & in CAD, t=t1 Hence α + 2s = 180 (Angles in triangle BAC) and β + 2t = 180 (Angles in triangle CAD) Adding these two equations gives: α + 2s + β + 2t = 360 Proof of the corollary from the Inscribed angle theorem. Conjecture III ): Any angle inscribed in a semicircle is inscribed in a semicircle is a angle. 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