Incenter Angle Equality 750472

1. **Problem Statement:** We have $\triangle HJK$ with points $D$, $E$, $F$ on sides $HJ$, $HK$, and $JK$ respectively. Point $G$ is the incenter of the triangle, and segments $DG$, $GF$, and $GE$ are drawn perpendicular to the sides at $D$, $F$, and $E$. We are asked which angle equalities among the given options are true. 2. **Key Concept:** The incenter $G$ is the point where the angle bisectors of $\triangle HJK$ intersect. It is equidistant from all sides, and the segments $GD$, $GE$, and $GF$ are perpendicular to the sides, representing the inradius segments. 3. **Important Properties:** - Since $G$ is the incenter, $GD$, $GE$, and $GF$ are angle bisectors meeting the sides at right angles. - Angles formed by the incenter and points on the sides relate to the bisected angles of the triangle. 4. **Analyze each statement:** - $m\angle DHG = m\angle EHG$: These angles are adjacent to vertex $H$ and involve points $D$ and $E$ on sides $HJ$ and $HK$. Since $G$ lies on the angle bisector at $H$, it divides the angle at $H$ into two equal parts. Therefore, $m\angle DHG = m\angle EHG$ is true. - $m\angle GEK = m\angle GKF$: These angles are at points $E$ and $F$ on sides $HK$ and $JK$. Since $GE$ and $GF$ are perpendicular to sides and $G$ is the incenter, these angles correspond to angles formed by the incenter and points on the sides but are not necessarily equal. This equality is generally not true. - $m\angle DHG = m\angle EGH$: $m\angle DHG$ is at $H$ involving $D$ and $G$, while $m\angle EGH$ is at $G$ involving $E$ and $H$. These angles are not equal by the properties of the incenter and angle bisectors. 5. **Conclusion:** Only the first statement is true. Therefore, the answer is that exactly one of these is true, not two. **Final answer:** $$m\angle DHG = m\angle EHG$$