Question

problem 24 given: ∠f≅∠j, (overline{fd}perpoverline{dh}), (overline{jh}perpoverline{dh}), (overline{fg}congoverline{jg}) prove: g is the mid - point of (overline{dh}). 1. (overline{fd}perpoverline{dh}), (overline{jh}perpoverline{dh}) 1. given 2. perpendicular lines intersect to form right angles. 3. ∠fdg≅∠jhg 3. all right angles are congruent. 4. given

Explanation:

Step1: Identify given information

$\angle F\cong\angle J$, $\overline{FD}\perp\overline{DH}$, $\overline{JH}\perp\overline{DH}$, $\overline{FG}\cong\overline{JG}$

Step2: Determine right - angle measures

Since $\overline{FD}\perp\overline{DH}$ and $\overline{JH}\perp\overline{DH}$, $\angle FDG = 90^{\circ}$ and $\angle JHG = 90^{\circ}$

Step3: Use right - angle congruence

All right angles are congruent, so $\angle FDG\cong\angle JHG$

Step4: Prove triangle congruence

With $\angle F\cong\angle J$, $\angle FDG\cong\angle JHG$, and $\overline{FG}\cong\overline{JG}$, by AAS, $\triangle FDG\cong\triangle JHG$

Step5: Apply CPCTC

Since the triangles are congruent, corresponding parts $\overline{DG}$ and $\overline{HG}$ are congruent

Step6: Define mid - point

As $\overline{DG}\cong\overline{HG}$, $G$ is the mid - point of $\overline{DH}$

Answer:

1. Statement: $\angle F\cong\angle J$, $\overline{FD}\perp\overline{DH}$, $\overline{JH}\perp\overline{DH}$, $\overline{FG}\cong\overline{JG}$

• Reason: Given

1. Statement: $\angle FDG = 90^{\circ}$, $\angle JHG=90^{\circ}$

• Reason: Perpendicular lines intersect to form right - angles

1. Statement: $\angle FDG\cong\angle JHG$

• Reason: All right angles are congruent

1. Statement: $\triangle FDG\cong\triangle JHG$

• Reason: Angle - Angle - Side (AAS) congruence criterion ($\angle F\cong\angle J$, $\angle FDG\cong\angle JHG$, $\overline{FG}\cong\overline{JG}$)

1. Statement: $\overline{DG}\cong\overline{HG}$

• Reason: Corresponding parts of congruent triangles are congruent (CPCTC)

1. Statement: $G$ is the mid - point of $\overline{DH}$

• Reason: A point that divides a line segment into two congruent line segments is the mid - point of the line segment