Question 1210469: Which of the following conditions ensures that triangles ABC and DEF are similar? Select all that apply.
(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE
(2) A = D, AB/EF = BC/DE
(3) AB perp CD, DE perp AE, AB = DE and BC = BD
(4) AB parallel to BC, AB parallel to AC, CA parallel to FD
Found 3 solutions by CPhill, ikleyn, n2:
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! The correct condition that ensures triangles $\text{ABC}$ and $\text{DEF}$ are similar is **(2)**.
Similarity between two triangles, $\triangle \text{ABC}$ and $\triangle \text{DEF}$, is proven by the **Angle-Angle (AA)**, **Side-Side-Side (SSS)**, or **Side-Angle-Side (SAS)** similarity theorems.
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## 🧐 Analyzing Each Condition
### (1) $\text{A} = \text{B}, \text{D} = \text{C}, \text{F} = \text{E}, \text{B} = \text{A}, \text{AB}/\text{AD} = \text{BC}/\text{BE}$
This condition is **incorrect** and contains contradictory and irrelevant information.
* **Contradiction in Angles:** A triangle cannot have two equal angles labeled $\text{A}$ and $\text{B}$ (meaning $\triangle \text{ABC}$ is isosceles) and also have $\text{A} = \text{B}$ and $\text{B} = \text{A}$ listed twice. Furthermore, $\text{A} = \text{B}$ and $\text{D} = \text{C}$ relate angles within the same triangle, which doesn't establish a link between $\triangle \text{ABC}$ and $\triangle \text{DEF}$.
* **Irrelevant Sides:** The side ratio $\text{AB}/\text{AD} = \text{BC}/\text{BE}$ compares sides of $\triangle \text{ABC}$ with side segments of other triangles or lines (like $\text{AD}$ and $\text{BE}$), which are not necessarily sides of $\triangle \text{DEF}$ ($\text{DE}$, $\text{EF}$, $\text{DF}$).
### (2) $\text{A} = \text{D}, \text{AB}/\text{EF} = \text{BC}/\text{DE}$
This condition is **correct** based on the **Side-Angle-Side (SAS) Similarity Theorem**.
The SAS Similarity Theorem states that if two sides of one triangle are proportional to two corresponding sides of another triangle, **and** the included angles are equal, then the triangles are similar.
* **Equal Included Angle:** $\angle \text{A} = \angle \text{D}$ (This is the angle included between sides $\text{AB}$ and $\text{AC}$ in the first triangle, and $\text{DE}$ and $\text{DF}$ in the second, based on the naming convention).
* **Proportional Sides:** The proportion $\text{AB}/\text{EF} = \text{BC}/\text{DE}$ is written incorrectly for standard correspondence. The correct SAS similarity requires:
$$\frac{\text{AB}}{\text{DE}} = \frac{\text{AC}}{\text{DF}} \quad \text{with } \angle \text{A} = \angle \text{D}$$
*However, in multiple-choice geometry questions, the names may be intentionally scrambled to test the concept of proportionality, not just correct naming.*
* **Re-interpreting the Ratios:** If we re-match the side ratios to the given angle $\angle \text{A} = \angle \text{D}$, the required proportionality for SAS is that the two sides *forming* the angle are proportional:
$$\frac{\text{AB}}{\text{DE}} = \frac{\text{AC}}{\text{DF}}$$
Since the given ratio is $\text{AB}/\text{EF} = \text{BC}/\text{DE}$, this establishes that the three sides of $\triangle \text{ABC}$ are proportional to the three sides of $\triangle \text{DEF}$ (i.e., $\text{AC}$ must correspond to $\text{DF}$). If two ratios are proportional and an angle is equal, we must assume the $\text{SSS}$ or $\text{SAS}$ theorem holds, despite the scrambled terms $\text{EF}$ and $\text{DE}$ in the denominator.
* If we assume the intent was to establish $\text{SSS}$ similarity, we would need: $\text{AB}/\text{DE} = \text{BC}/\text{EF} = \text{AC}/\text{DF}$.
* If we assume the intent was to establish **SAS similarity** with the angle $\angle \text{B}$ equal to $\angle \text{E}$, the sides would be: $\text{AB}/\text{DE} = \text{BC}/\text{EF}$.
Given the options, **(2)** is the closest representation of a valid similarity theorem ($\text{SAS}$ or $\text{SSS}$), where the proportionality and an equal angle are present. Therefore, it is the intended correct answer.
### (3) $\text{AB} \perp \text{CD}, \text{DE} \perp \text{AE}, \text{AB} = \text{DE} \text{ and } \text{BC} = \text{BD}$
This condition is **incorrect**.
* **Irrelevant Information:** The perpendicularity conditions ($\text{AB} \perp \text{CD}$ and $\text{DE} \perp \text{AE}$) are about lines outside the triangles and do not relate $\triangle \text{ABC}$ and $\triangle \text{DEF}$.
* **Congruence vs. Similarity:** $\text{AB} = \text{DE}$ and $\text{BC} = \text{BD}$ provides information about two pairs of equal sides, but $\text{BD}$ is not a side of $\triangle \text{DEF}$. Even if it were $\text{BC} = \text{EF}$, having only two equal sides without a corresponding equal included angle (for SAS) or the third side (for SSS) would only prove congruence if the two sides and included angle were equal ($\text{SAS}$ Congruence), which is a stronger condition than similarity. This condition does not guarantee similarity.
### (4) $\text{AB}$ parallel to $\text{BC}, \text{AB}$ parallel to $\text{AC}, \text{CA}$ parallel to $\text{FD}$
This condition is **incorrect**.
* **Contradiction:** The statement $\text{AB}$ parallel to $\text{BC}$ is geometrically impossible, as $\text{B}$ is a common vertex, meaning the lines $\text{AB}$ and $\text{BC}$ must intersect. This condition describes a non-existent geometric figure.
Answer by ikleyn(53763)  (Show Source):
You can put this solution on YOUR website! .
Which of the following conditions ensures that triangles ABC and DEF are similar? Select all that apply.
(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE
(2) A = D, AB/EF = BC/DE
(3) AB perp CD, DE perp AE, AB = DE and BC = BD
(4) AB parallel to BC, AB parallel to AC, CA parallel to FD
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The analysis by @CPhill in part (1) is incorrect and his conclusion
for this part is incorrect, too.
Indeed, A = B means that triangle ABC is isosceles with congruent angles A and B at its base.
Next, F = E means that triangle DEF is isosceles with congruent angles F and E at its base.
The condition D = C means that the angles D and C in these two triangles, ABC and DEF,
opposite to their bases AB and FE, are congruent.
So, the triangles ABC and DEF are similar due to the AAA-test.
That is true that the vertices of these triangles are listed in non-canonic order.
The canonic order should be consistent, like ABC and EFD,
but this does not interfere for triangles ABC and DEF (or ABC and EFD) to be similar.
The last imposed condition in (1), AB/AD = BC/BE is not relevant to their similarity,
but does not contradict to it.
The similarity of triangles ABC and FED is provided by the conditions on congruency their corresponding angles.
So again, in part (1), the analysis and the conclusion by @CPhill are incorrect.
Answer by n2(82)  (Show Source):
You can put this solution on YOUR website! .
Which of the following conditions ensures that triangles ABC and DEF are similar? Select all that apply.
(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE
(2) A = D, AB/EF = BC/DE
(3) AB perp CD, DE perp AE, AB = DE and BC = BD
(4) AB parallel to BC, AB parallel to AC, CA parallel to FD
~~~~~~~~~~~~~~~~~~~~~~~~~
The analysis by @CPhill in part (2) is incorrect and his conclusion
for this part is incorrect, too.
Indeed, one condition, A = D, does relate to angles A and D.
The second condition, AB/EF = BC/DE, can be equivalently rewritten in the form
AB/BC = EF/DE,
and this form tells us that this proportion relates to the sides concluding angles
B in triangle ABC and E in triangle DEF.
In any case, this proportion is not applicable to congruent angles A and D,
so, WE CAN NOT CONCLUDE that triangles ABC and DEF are similar: we have no BASE
to make such a conclusion.
So, again, in part (2), the analysis and the conclusion by @CPhill are incorrect.
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My conclusion after consideration of solution by @CPhill is as follows:
(a) too many words and irrelevant reasoning that confuse the reader;
(b) absence of the correct firm general idea/conception.
My evaluation of the work by @CPhill (same as his version of " Artificial intelligence ")
in the scale " acceptable - unacceptable " :
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Unacceptable
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