Question
Transformations and Similarity Online Practice
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Question
Use the image to answer the question.
An illustration shows two quadrilaterals. Quadrilateral A C B D is smaller than quadrilateral G H F E. In quadrilateral A C B D, angle A measures 90 degrees, angle C measures 108 degrees, angle B measures 90 degrees, and angle D measures 72 degrees. In quadrilateral G H F E, angle G measures 90 degrees, angle H measures 108 degrees, angle F measures 90 degrees, and angle E measures 72 degrees.
Which statement correctly describes the similarity between the two quadrilaterals?
(1 point)
Responses
Quadrilateral BDAC
∼
Quadrilateral GHFE
Quadrilateral upper B upper D upper A upper C tilde Quadrilateral upper G upper H upper F upper E
Quadrilateral ABCD
∼
Quadrilateral EFGH
Quadrilateral upper A upper B upper C upper D tilde Quadrilateral upper E upper F upper G upper H
Quadrilateral ADBC
∼
Quadrilateral FHGE
Quadrilateral upper A upper D upper B upper C tilde Quadrilateral upper F upper H upper G upper E
Quadrilateral ACBD
∼
Quadrilateral GHFE
Answers
GPT-4o mini
To determine the correct statement about the similarity between the two quadrilaterals, we first consider the given information.
Both quadrilaterals \(ABCD\) (smaller) and \(GHFE\) (larger) have the same angles:
- Angle \(A\) measures 90 degrees
- Angle \(B\) measures 90 degrees
- Angle \(C\) measures 108 degrees
- Angle \(D\) measures 72 degrees
Since both quadrilaterals have the same corresponding angles, we can conclude that they are similar by the Angle-Angle (AA) postulate for similarity in quadrilaterals.
Now let's analyze the response options:
1. **Quadrilateral BDAC ∼ Quadrilateral GHFE** - This option flips the order of the vertices.
2. **Quadrilateral ABCD ∼ Quadrilateral EFGH** - This option incorrectly states the larger quadrilateral as \(EFGH\).
3. **Quadrilateral ADBC ∼ Quadrilateral FHGE** - This option also improperly refers to the quadrilaterals.
4. **Quadrilateral ACBD ∼ Quadrilateral GHFE** - This option correctly maintains the order of the smaller quadrilateral \(ABCD\) and properly references the larger quadrilateral \(GHFE\).
The correct response is:
**Quadrilateral ACBD ∼ Quadrilateral GHFE**
Both quadrilaterals \(ABCD\) (smaller) and \(GHFE\) (larger) have the same angles:
- Angle \(A\) measures 90 degrees
- Angle \(B\) measures 90 degrees
- Angle \(C\) measures 108 degrees
- Angle \(D\) measures 72 degrees
Since both quadrilaterals have the same corresponding angles, we can conclude that they are similar by the Angle-Angle (AA) postulate for similarity in quadrilaterals.
Now let's analyze the response options:
1. **Quadrilateral BDAC ∼ Quadrilateral GHFE** - This option flips the order of the vertices.
2. **Quadrilateral ABCD ∼ Quadrilateral EFGH** - This option incorrectly states the larger quadrilateral as \(EFGH\).
3. **Quadrilateral ADBC ∼ Quadrilateral FHGE** - This option also improperly refers to the quadrilaterals.
4. **Quadrilateral ACBD ∼ Quadrilateral GHFE** - This option correctly maintains the order of the smaller quadrilateral \(ABCD\) and properly references the larger quadrilateral \(GHFE\).
The correct response is:
**Quadrilateral ACBD ∼ Quadrilateral GHFE**