In figure 3, if mAB = 83 and mDE = 29, then what is m∠L? In figure 4a, if mJK = 78 and mLM = 26, then what is m∠I? In figure 4b, if mBHI = 250, mQR = 69, and mR = 37, then what are... In figure 3, if mAB = 83 and mDE = 29, then what is m∠L? In figure 4a, if mJK = 78 and mLM = 26, then what is m∠I? In figure 4b, if mBHI = 250, mQR = 69, and mR = 37, then what are m∠L2 and m∠N? Read more

Steps to Solve

1. Finding m∠L in Figure 3

In circles, the angle formed between two secants is half the difference of the measures of the intercepted arcs. Here, we can use the formula:

$$ m∠L = \frac{1}{2} \left( mAB - mDE \right) $$

Substituting the values:

$$ m∠L = \frac{1}{2} \left( 83 - 29 \right) = \frac{1}{2} \times 54 = 27 $$

2. Finding m∠I in Figure 4a

Again, using the same principle for the angle between the secant and tangent:

$$ m∠I = \frac{1}{2} \left( mJK - mLM \right) $$

Substituting the values:

$$ m∠I = \frac{1}{2} \left( 78 - 26 \right) = \frac{1}{2} \times 52 = 26 $$

3. Finding m∠L2 and m∠N in Figure 4b

From Figure 4b, we need to find two angles related to the same tangent-secant relationship:

First, use the formula for m∠L2, which is:

$$ m∠L2 = \frac{1}{2} \left( mBHI - mQR \right) $$

Substituting the values:

$$ m∠L2 = \frac{1}{2} \left( 250 - 69 \right) = \frac{1}{2} \times 181 = 90.5 $$

Now for m∠N, since it is vertically opposite to m∠L2 in Figure 4b:

$$ m∠N = m∠L2 = 90.5 $$