Question

given: circle o with diameter ln and inscribed angle lmn. prove: ∠lmn is a right angle. 1. circle o has diameter ln and inscribed angle lmn 1. given 2. lkn is a semicircle 2. diameter ⊙ divides into 2 semicircles 3. circle o measures 360° 3. measure of a circle is 360° 4. m∠lkn = 180° 4. definition of semicircle 5. m∠lmn = 90° 5. ? 6. ∠lmn is a right angle 6. definition of right angle. what is the missing reason in step 5?

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Identify the intercepted arc

The inscribed angle $\angle LMN$ intercepts the semi - circle $\overparen{LKN}$.

Step3: Apply the theorem

Since $m\overparen{LKN}=180^{\circ}$, by the inscribed - angle theorem, $m\angle LMN=\frac{1}{2}m\overparen{LKN}$. Substituting $m\overparen{LKN} = 180^{\circ}$, we get $m\angle LMN = 90^{\circ}$.

Answer:

The measure of an inscribed angle is half the measure of its intercepted arc.