1. In figure 3, if m∠BDE = 38° and m∠DEK = 78°, then m∠L = ? 2. In figure 4, if m∠JKL = 29° and m∠MN = 26°, then m∠M2 = ?

Steps to Solve

1. Understanding Angles in a Circle

In a circle, angles can be determined using relationships derived from the positions of the points involved. The angles around a point or along a line can provide necessary information.

2. Finding m∠L from Figure 3

Given:

• $m∠BDE = 38°$
• $m∠DEK = 78°$

To find $m∠L$ which can be calculated by using the exterior angle theorem which states that the measure of an exterior angle (here angle $DEK$) is equal to the sum of the measures of the two opposite interior angles.

Thus:

$$ m∠L = m∠BDE + m∠DEK $$

Substituting the values:

$$ m∠L = 38° + 78° = 116° $$

3. Finding m∠M2 from Figure 4

Given:

• $m∠JKL = 29°$
• $m∠MN = 26°$

To find $m∠M2$ which can be determined based on the same principle of exterior angle theorem. In this case, $m∠M2$ is obtained by adding $m∠JKL$ and $m∠MN$.

Thus:

$$ m∠M2 = m∠JKL + m∠MN $$

Substituting the values:

$$ m∠M2 = 29° + 26° = 55° $$