2
given: $overline{hv} \perp \overline{gt}$, $overline{gh} \cong \overline{tv}$,
$i$ is the midpoint of $overline{hv}$
prove: $\triangle igh \cong \triangle itv$
(there is a diagram on the right side of the proof statement, showing triangles with right angles and congruent marks)
(then there is a two - column proof table with columns statements and reasons, and rows numbered 1 to 5 for filling in statements and corresponding reasons)

Question

2
given: $overline{hv} \perp \overline{gt}$, $overline{gh} \cong \overline{tv}$,
       $i$ is the midpoint of $overline{hv}$
prove: $\triangle igh \cong \triangle itv$
(there is a diagram on the right side of the proof statement, showing triangles with right angles and congruent marks)
(then there is a two - column proof table with columns statements and reasons, and rows numbered 1 to 5 for filling in statements and corresponding reasons)

Accepted Answer

# Answer:
To prove $\triangle IGH \cong \triangle ITV$, we follow these steps:

## Step 1: Identify Right Angles
**Statement**: $\angle IGH$ and $\angle ITV$ are right angles.
**Reason**: Given $ \overline{HV} \perp \overline{GT} $, so by definition of perpendicular lines, the angles formed are $ 90^\circ $ (right angles).

## Step 2: Midpoint Implies Equal Segments
**Statement**: $ \overline{IH} \cong \overline{IV} $
**Reason**: $ I $ is the midpoint of $ \overline{HV} $, and by the definition of a midpoint, a midpoint divides a segment into two congruent segments.

## Step 3: Given Congruent Hypotenuses
**Statement**: $ \overline{GH} \cong \overline{TV} $
**Reason**: Given in the problem.

## Step 4: Apply Hypotenuse-Leg (HL) Theorem
**Statement**: $ \triangle IGH \cong \triangle ITV $
**Reason**: The Hypotenuse-Leg (HL) Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Here, $ \overline{GH} \cong \overline{TV} $ (hypotenuses) and $ \overline{IH} \cong \overline{IV} $ (legs), and both are right triangles (from Step 1), so by HL, the triangles are congruent.

Filling in the table:

| Statements | Reasons |
| --- | --- |
| 1) $ \overline{HV} \perp \overline{GT} $, $ \overline{GH} \cong \overline{TV} $, $ I $ is the midpoint of $ \overline{HV} $ | 1) Given |
| 2) $ \angle IGH = 90^\circ $, $ \angle ITV = 90^\circ $ (or $ \angle IGH \cong \angle ITV $) | 2) Definition of perpendicular lines (forms right angles) |
| 3) $ \overline{IH} \cong \overline{IV} $ | 3) Definition of midpoint |
| 4) $ \triangle IGH $ and $ \triangle ITV $ are right triangles | 4) A triangle with a right angle is a right triangle (from Step 2) |
| 5) $ \triangle IGH \cong \triangle ITV $ | 5) Hypotenuse-Leg (HL) Congruence Theorem |

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QUESTION IMAGE

Question

Explanation:

Answer:

Step 1: Identify Right Angles

Step 2: Midpoint Implies Equal Segments

Step 3: Given Congruent Hypotenuses

Step 4: Apply Hypotenuse-Leg (HL) Theorem

Statements	Reasons
2) \( \angle IGH = 90^\circ \), \( \angle ITV = 90^\circ \) (or \( \angle IGH \cong \angle ITV \))	2) Definition of perpendicular lines (forms right angles)
3) \( \overline{IH} \cong \overline{IV} \)	3) Definition of midpoint
4) \( \triangle IGH \) and \( \triangle ITV \) are right triangles	4) A triangle with a right angle is a right triangle (from Step 2)
5) \( \triangle IGH \cong \triangle ITV \)	5) Hypotenuse-Leg (HL) Congruence Theorem