Question

Given: Circle O with diameter LN and inscribed angle LMN Prove: is a right angle. What is the missing reason in step 5? Statements Reasons 1. circle O has diameter LN and inscribed angle LMN 1. given 2. is a semicircle 2. diameter divides into 2 semicircles 3. circle O measures 360o 3. measure of a circle is 360o 4. m = 180o 4. definition of semicircle 5. m∠LMN = 90o 5. ? 6. ∠LMN is a right angle 6. definition of right angle HL theorem inscribed angle theorem diagonals of a rhombus are perpendicular. formed by a tangent and a chord is half the measure of the intercepted arc.

Answer 1

To complete steps 5 - 6 of the proof, refer to the diagram shown below.

Because OL = OM = ON = the radius, therefore each of ΔOLM and ΔONM is an isosceles triangle.
ΔOLM has two equal angles denoted by a, and ΔONM has two equal angles denoted by b.

The central angles x and y add up to 180° on a straight line, so
x + y = 180           (1)

Because angles in a triangle sum to 180°, therefore
x + 2a = 180        (2)
y + 2b = 180        (3)

Add (2) and (3) to obtain
x + y + 2(a + b) = 360

From (1), obtain
180 + 2(a + b) = 360
2(a + b) = 180
a + b = 90

Answer:
Because (a + b) = ∠LMN,  it proves that ∠LMN = 90°

Answer 2

The answer is B) Inscribed Angle Theorem