Here's the completed proof with missing statements and reasons filled in, along with a clear explanation.
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Completed Proof Table:
| Statements | Reasons |
|-------------------------------------|------------------------------------------|
| 1. $\overline{GA}$ bisects $\angle G$ | 1.
Given |
| 2. $\angle LGA \cong \angle DGA$ | 2. Definition of Angle Bisector |
| 3. $\overline{GA} \perp \overline{LD}$ | 3.
Given |
| 4. $\angle GAL$ and $\angle GAD$ are right angles | 4.
Definition of Perpendicular Lines |
| 5. $\angle GAL \cong \angle GAD$ | 5. All right angles are congruent. |
| 6. $\overline{GA} \cong \overline{GA}$ | 6.
Reflexive Property of Congruence |
| 7. $\triangle GAL \cong \triangle GAD$ | 7.
ASA (Angle-Side-Angle) Congruence |
| 8. $\overline{GL} \cong \overline{GD}$ | 8.
CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
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✔ Explanation:
We are given:
- $\overline{GA}$ bisects $\angle G$ → So it splits $\angle G$ into two equal angles: $\angle LGA \cong \angle DGA$. (Statement 2)
- $\overline{GA} \perp \overline{LD}$ → So at point A, we have two right angles: $\angle GAL$ and $\angle GAD$. (Statement 3 & 4)
We want to prove: $\overline{GL} \cong \overline{GD}$
To do this, we show that triangles $\triangle GAL$ and $\triangle GAD$ are congruent. Then, by CPCTC, their corresponding sides $\overline{GL}$ and $\overline{GD}$ must be congruent.
#### Step-by-step reasoning:
1.
Statement 1 & 2: Given that GA bisects angle G, so the two smaller angles at G are congruent.
2.
Statement 3: Given that GA is perpendicular to LD — this is crucial because it gives us right angles at A.
3.
Statement 4: Because perpendicular lines form right angles, both $\angle GAL$ and $\angle GAD$ are 90°.
4.
Statement 5: Since all right angles are congruent, these two angles are congruent.
5.
Statement 6: Side GA is common to both triangles — so it’s congruent to itself (reflexive property).
6.
Statement 7: Now we have:
- $\angle LGA \cong \angle DGA$ (from angle bisector),
- $\overline{GA} \cong \overline{GA}$ (common side),
- $\angle GAL \cong \angle GAD$ (both right angles).
→ This satisfies the
ASA (Angle-Side-Angle) congruence criterion. So the two triangles are congruent.
7.
Statement 8: Since the triangles are congruent, their corresponding parts are congruent. Therefore, side GL corresponds to side GD → $\overline{GL} \cong \overline{GD}$.
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✔ Final Answer:
The missing entries are filled as shown above. The key idea is using ASA to prove triangle congruence, then applying CPCTC to get the desired side congruence.
This also shows that triangle LGD is isosceles with GL = GD — which makes sense since the altitude from the apex (G) to the base (LD) is also the angle bisector — a property of isosceles triangles!
Parent Tip: Review the logic above to help your child master the concept of geometry statements and reasons worksheet.
