Question

inscribed angles
given: circle o with diameter ln and inscribed angle lmn
prove: ∠lmn is a right angle

what is the missing reason in step 5?

statements | reasons
--- | ---

1. circle o has diameter \\(\overline{ln}\\) and inscribed angle lmn | 1. given
2. \\(\overarc{lkn}\\) is a semicircle | 2. diameter ⊙ divides into 2 semicircles
3. circle o measures \\(360^\circ\\) | 3. measure of a circle is \\(360^\circ\\)
4. \\(m\overarc{lkn} = 180^\circ\\) | 4. definition of semicircle
5. \\(m\angle lmn = 90^\circ\\) | 5.?
6. \\(\angle lmn\\) is a right angle | 6. definition of right angle

options (partially visible):

• diagonals of a rhombus are perpendicular
• ∠ formed by a tangent and a chord is half the measure of the intercepted arc.

Explanation:

Step 5 concludes the inscribed angle ∠LMN is 90° because it intercepts a semicircle (arc LKN = 180°). The key geometric principle here is that an inscribed angle is half the measure of its intercepted arc. Since the intercepted arc is 180°, the angle is half that, or 90°.

Answer:

An inscribed angle is half the measure of its intercepted arc.