Question

Given: Circle O with diameter LN and inscribed angle LMN
Prove: Angle L M N is a right angle.

Circle O is shown. Line segment L N is a diameter. Points L, M, N, and K are on the circle. Lines connect each point.

What is the missing reason in step 5?


Statements

Reasons
1. circle O has diameter LN and inscribed angle LMN 1. given
2. Arc L K N is a semicircle 2. diameter Circle divides into 2 semicircles
3. circle O measures 360o 3.
measure of a circle is 360o

4. m Arc L K N = 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 ar

Answer 1

Reason: Inscribed angle theorem — an inscribed angle measures half its intercepted arc.

Explanation: ∠LMN intercepts arc LKN, so m∠LMN = 1/2 · m(arc LKN) = 1/2 · 180° = 90°.