circle d is shown with the measures of the minor arcs.
which angles are congruent?
∠gdf and ∠hdg
∠edh and ∠fdg
∠gdh and ∠edh
∠fde and ∠gdh

Question

Accepted Answer

# Explanation:
## Step1: Recall Central Angle Theorem
The measure of a central angle is equal to the measure of its intercepted arc. So, we need to find the measures of each central angle by their intercepted arcs.
## Step2: Analyze Each Option
- For $\angle GDF$ and $\angle HDG$: Arc $GF = 115^\circ$, arc $HG=65^\circ$. So $\angle GDF = 115^\circ$, $\angle HDG = 65^\circ$. Not congruent.
- For $\angle EDH$ and $\angle FDG$: Arc $EH = 65^\circ$, so $\angle EDH = 65^\circ$. Arc $FG = 115^\circ$? Wait, no, arc $HG = 65^\circ$, arc $GF = 115^\circ$, arc $EH=65^\circ$, arc $EF = 115^\circ$. Wait, $\angle FDG$: intercepted arc $FG$? No, $\angle FDG$ intercepts arc $FG$? Wait, no, let's re - check. $\angle EDH$ intercepts arc $EH$ (measure $65^\circ$). $\angle FDG$: Let's see, arc $FG$ is $115^\circ$? No, wait the arcs: $EH = 65^\circ$, $HG = 65^\circ$, $GF = 115^\circ$, $FE=115^\circ$. Wait, $\angle FDG$: The central angle $\angle FDG$ intercepts arc $FG$ which is $115^\circ$? No, that's not right. Wait, maybe I made a mistake. Wait, $\angle EDH$: intercepted arc $EH = 65^\circ$, so $\angle EDH=65^\circ$. $\angle FDG$: Wait, what about $\angle FDE$? No, let's check the third option: $\angle GDH$ and $\angle EDH$. $\angle GDH$ intercepts arc $HG = 65^\circ$, so $\angle GDH = 65^\circ$. $\angle EDH$ intercepts arc $EH = 65^\circ$, so $\angle EDH=65^\circ$. Wait, no, wait the fourth option: $\angle FDE$ and $\angle GDH$. $\angle FDE$ intercepts arc $FE = 115^\circ$, $\angle GDH$ intercepts arc $HG = 65^\circ$. No. Wait, wait the correct one: $\angle EDH$ (intercepts $EH = 65^\circ$) and $\angle GDH$ (intercepts $HG = 65^\circ$)? Wait, no, the third option is $\angle GDH$ and $\angle EDH$. Wait, $\angle GDH$ is central angle for arc $HG$ ( $65^\circ$), $\angle EDH$ is central angle for arc $EH$ ( $65^\circ$). So they are both $65^\circ$. Wait, but also, let's check the fourth option: $\angle FDE$ and $\angle GDH$. $\angle FDE$ intercepts arc $FE = 115^\circ$, $\angle GDH$ intercepts arc $HG = 65^\circ$. No. Wait, the second option: $\angle EDH$ and $\angle FDG$: Wait, maybe I messed up the arcs. Wait, arc $EH = 65^\circ$, so $\angle EDH = 65^\circ$. Arc $FG$: Wait, no, arc $HG = 65^\circ$, arc $GF = 115^\circ$, arc $EF = 115^\circ$, arc $EH = 65^\circ$. So $\angle FDG$: Let's see, the central angle $\angle FDG$ intercepts arc $FG$ which is $115^\circ$? No, that can't be. Wait, maybe the correct pair is $\angle GDH$ and $\angle EDH$? Wait, no, let's re - examine the diagram. The arcs: $EH = 65^\circ$, $HG = 65^\circ$, $GF = 115^\circ$, $FE = 115^\circ$. So central angles: $\angle EDH$ (arc $EH$) $= 65^\circ$, $\angle GDH$ (arc $HG$) $= 65^\circ$, $\angle FDG$ (arc $FG$) $= 115^\circ$, $\angle FDE$ (arc $FE$) $= 115^\circ$, $\angle GDF$ (arc $GF$) $= 115^\circ$, $\angle EDG$? Wait, the fourth option: $\angle FDE$ and $\angle GDH$. $\angle FDE$ is $115^\circ$ (arc $FE$), $\angle GDH$ is $65^\circ$ (arc $HG$). Not congruent. The third option: $\angle GDH$ ($65^\circ$) and $\angle EDH$ ($65^\circ$): Wait, but let's check the second option again. Wait, maybe I made a mistake. Wait, $\angle EDH$ (arc $EH = 65^\circ$) and $\angle FDG$: Wait, no, $\angle FDG$ intercepts arc $FG$ which is $115^\circ$? No, that's not. Wait, the correct answer is $\angle GDH$ and $\angle EDH$? Wait, no, the third option is $\angle GDH$ and $\angle EDH$, and the fourth option: Wait, $\angle FDE$ (arc $FE = 115^\circ$) and $\angle GDH$ (arc $HG = 65^\circ$) no. Wait, the correct pair is $\angle GDH$ and $\angle EDH$? Wait, no, let's check the angles again. $\angle GDH$ is formed by radii $DH$ and $DG$, intercepting arc $HG$ ( $65^\circ$), so $\angle GDH = 65^\circ$. $\angle EDH$ is formed by radii $DH$ and $DE$, intercepting arc $EH$ ( $65^\circ$), so $\angle EDH = 65^\circ$. So they are congruent. Wait, but also, $\angle FDE$ (intercepting arc $FE = 115^\circ$) and $\angle GDF$ (intercepting arc $GF = 115^\circ$) would be congruent, but that's not an option. Wait, the options: the third option is $\angle GDH$ and $\angle EDH$, both $65^\circ$, so they are congruent. Wait, but let's check the fourth option: $\angle FDE$ and $\angle GDH$. $\angle FDE = 115^\circ$, $\angle GDH = 65^\circ$. Not congruent. The second option: $\angle EDH = 65^\circ$, $\angle FDG$: if $\angle FDG$ intercepts arc $FG = 115^\circ$, then it's $115^\circ$, not congruent. The first option: $\angle GDF = 115^\circ$, $\angle HDG = 65^\circ$, not congruent. So the correct option is $\angle GDH$ and $\angle EDH$? Wait, no, the third option is $\angle GDH$ and $\angle EDH$, which are both $65^\circ$, so they are congruent. Wait, but let's re - check the problem. Wait, the fourth option is $\angle FDE$ and $\angle GDH$. $\angle FDE$ intercepts arc $FE = 115^\circ$, $\angle GDH$ intercepts arc $HG = 65^\circ$. No. The third option: $\angle GDH$ (arc $HG = 65^\circ$) and $\angle EDH$ (arc $EH = 65^\circ$): both $65^\circ$, so congruent. Wait, but maybe I made a mistake in the arc for $\angle FDG$. Wait, no, let's go back. The correct answer is the third option: $\angle GDH$ and $\angle EDH$? Wait, no, the third option is $\angle GDH$ and $\angle EDH$, which are congruent because their intercepted arcs are both $65^\circ$.
# Answer: $\angle GDH$ and $\angle EDH$ (i.e., the option " $\angle GDH$ and $\angle EDH$")

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Central Angle Theorem

Step2: Analyze Each Option

Answer: