Question

In circle d, line segments de, df, dg, and dh are radii. Lines are drawn to connect the points on the circle and create secants ef, fg, gh, and he. The measures of arc ef, fg, gh, and he are 115 degrees, 115 degrees, 65 degrees, and 65 degrees respectively. Which angles are congruent?

1) ∠edh and ∠fdg
2) ∠fde and ∠gdh
3) ∠gdh and ∠edh
4) ∠gdf and ∠hdg

Answer 1

Angles ∠FDE and ∠GDH are congruent because they each intercept arcs EF and GH respectively, both measuring 115 degrees. None of the other pairs of angles listed intercept arcs of the same length, hence they are not congruent.

Step-by-step explanation:

The given problem is about identifying congruent angles within a circle. Each of the radii DE, DF, DG, and DH creates a central angle with the points of intersection on the circumference of circle D. Since the corresponding minor arcs EF, FG, GH, and HE have been given specific degree measures, we use the fact that the measure of a central angle is equal to the measure of its intercepted arc.

Given the arc measures of EF, FG, GH, and HE are 115 degrees, 115 degrees, 65 degrees, and 65 degrees respectively, it implies that:

The measure of ∠EDH is 65 degrees (intercepts arc HE),

the measure of ∠FDG is 115 degrees (intercepts arc FG),

∠FDE and ∠GDH intercept arcs EF and GH, both measuring 115 degrees, hence these angles are congruent,

∠GDF and ∠HDG are both central angles intercepting arcs GD and DH, both of which are not directly given but can be found by subtracting the known arcs from the total 360 degrees of the circle.

Therefore, from the choices given, ∠FDE and ∠GDH are congruent because they intercept arcs of the same length, that is 115 degrees each.