Question 1210461: In the diagram, AD bisects BAC.
By adding one more condition, we can prove that triangles ABD and ACD are congruent. Which one of the following could be that condition? Select all that apply.
(a) BC = AD
(b) AB = PQ
(c) angle ABC = angle BPQ
(d) angle BCD + angle PAQ = 90
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! The goal is to find an additional condition that proves $\triangle ABD \cong \triangle ACD$.
## 📐 Congruence Criteria
We are given two facts about $\triangle ABD$ and $\triangle ACD$:
1. **Shared Side:** $AD$ is common to both triangles, so **$AD = AD$ (Side)**.
2. **Angle Bisector:** $AD$ bisects $\angle BAC$, which means the angle at the vertex $A$ is divided into two equal angles. Therefore, **$\angle BAD = \angle CAD$ (Angle)**.
So far, we have one **Side** and one **Angle** ($S$ and $A$). To prove congruence, we need one of the following criteria:
* **SAS** (Side-Angle-Side): We need the other side adjacent to the known angle: $AB = AC$.
* **AAS** (Angle-Angle-Side): We need another angle and a non-included side: $\angle ABD = \angle ACD$ (and $AD=AD$) or $\angle ADB = \angle ADC$ (and $AD=AD$).
* **ASA** (Angle-Side-Angle): We need another angle, and the side must be the included side: $\angle ADB = \angle ADC$.
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## 🔎 Evaluating the Options
The options introduce segments and angles from other potential figures ($PQ$, $BPQ$, $PAQ$), which are not necessarily part of $\triangle ABD$ or $\triangle ACD$. We must assume that the non-$A, B, C, D$ elements in the options are meant to relate to the required conditions through substitution or equivalence.
### (a) $BC = AD$
**False.** This compares the side $BC$ of the entire triangle to the segment $AD$. This is not one of the required congruence conditions.
### (b) $AB = PQ$
**True (Potentially).** The required condition for **SAS** congruence is **$AB = AC$**.
If the condition given is **$AB = PQ$**, and we assume the intended required condition **$AB = AC$** can be satisfied by assuming $PQ$ is equal to $AC$ (i.e., $PQ = AC$), then $AB = AC$.
More simply, if we assume the statement means $AB = AC$, then **SAS** is satisfied: $\mathbf{AB = AC}$ (Side), $\mathbf{\angle BAD = \angle CAD}$ (Angle), $\mathbf{AD = AD}$ (Side).
### (c) $\angle ABC = \angle BPQ$
**True (Potentially).** The required condition for **AAS** congruence is $\angle ABD = \angle ACD$, which is the same as **$\angle ABC = \angle ACB$**.
If the condition given is $\mathbf{\angle ABC = \angle ACB}$, then **AAS** is satisfied: $\mathbf{\angle ABD = \angle ACD}$ (Angle), $\mathbf{\angle BAD = \angle CAD}$ (Angle), $\mathbf{AD = AD}$ (Non-included Side).
Assuming the statement intends to mean $\angle ABC = \angle ACB$.
### (d) $\angle BCD + \angle PAQ = 90^\circ$
**False.** This introduces a numerical constraint on two angles and does not provide an equality of sides or angles between $\triangle ABD$ and $\triangle ACD$.
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## ⭐️ Conclusion (Standard Geometry Test)
In a typical geometry problem of this nature, the options are testing whether the student knows the necessary congruence criteria, even if the distracting labels ($PQ$, $BPQ$) are used.
The two easiest ways to prove congruence, given the existing $\mathbf{S}$ and $\mathbf{A}$, are by establishing the **SAS** or **AAS** criteria.
1. **SAS requires $AB = AC$**
2. **AAS requires $\angle ABC = \angle ACB$**
Since $\triangle ABD$ and $\triangle ACD$ are the focus, we must select the options that are equivalent to $AB=AC$ or $\angle ABC = \angle ACB$.
* (b) is equivalent to $AB = AC$.
* (c) is equivalent to $\angle ABC = \angle ACB$.
Therefore, both (b) and (c) could be the necessary additional condition, assuming the external angles/segments ($PQ, BPQ$) are equivalent to the required internal ones ($AC, ACB$).
The statements that could be that condition are **(b)** and **(c)**.
Answer by ikleyn(53277)  (Show Source):
You can put this solution on YOUR website! .
In the diagram, AD bisects BAC.
By adding one more condition, we can prove that triangles ABD and ACD are congruent.
Which one of the following could be that condition? Select all that apply.
(a) BC = AD
(b) AB = PQ
(c) angle ABC = angle BPQ
(d) angle BCD + angle PAQ = 90
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Neither the diagram nor the link to it are provided, so a reader does not have
adequate information about the problem. Thus the input is meaningless.
As the problem is worded and presented in the post, the objects P, Q, and PQ are undefined,
and, therefore, can not be used without explanations of their meaning.
So and therefore, as presented in the post, the problem is FATALLY DEFECTIVE
and the post itself can not be considered as a Math problem.
In general, recent parcels to this forum (of yesterday) produce very bad impression
about the ability of their creator to compose adequate Math problems.
The fact that the Artificial Intelligence was employed to solve meaningless problem,
produce very bad impression about intentions of its authors.
Another attempt to deceive the reader.
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