question 1
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draw isosceles triangle abc with base $\overline{ac}$ and vertex b. reflect triangle abc across line ac. next, consider the larger shape which is formed by combining triangle abc along with its image. (it may help to shade in both triangle abc and its image with the same color, and ignore the line of reflection.) this new shape is a ____________. (choose all names that apply.)
i. rectangle
ii. parallelogram
iii. rhombus
iv. square.
a) $\circ$ i, ii, iii, and iv
b) $\circ$ iii only
c) $\circ$ ii, iii, and iv only
d) $\circ$ ii and iii only
e) $\circ$ ii only
f) $\circ$ none of the above.
question 2
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draw isosceles triangle abc with base $\overline{ac}$ and vertex b. rotate triangle abc $180^\circ$ using the midpoint of ab as the center of rotation. next, consider the larger shape which is formed by combining triangle abc along with its image. (it may help to shade in both triangle abc and its image with the same color, and ignore the segment which contains the center of rotation.) this new shape is a ____________. (choose all names that apply.)
i. rectangle
ii. parallelogram
iii. rhombus
iv. square.
a) $\circ$ ii, iii, and iv only
b) $\circ$ ii and iii only

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### Question 1
# Brief Explanations:
1. **Understand the Reflection**: When we reflect isosceles triangle $ ABC $ (with base $ \overline{AC} $) across line $ AC $, the image of $ B $ (let's call it $ B' $) will form a new triangle $ AB'C $. Now, combining $ \triangle ABC $ and $ \triangle AB'C $:
   - In $ \triangle ABC $, $ AB = BC $ (since it's isosceles with base $ AC $). After reflection, $ AB = AB' $ and $ BC = B'C $. So $ AB = BC = AB' = B'C $.
   - A quadrilateral with all sides equal is a rhombus. Also, a rhombus is a type of parallelogram (since opposite sides are equal and parallel).
   - It is not necessarily a rectangle or a square because we don't know if the angles are right angles (the original triangle is just isosceles, not necessarily right - angled).
2. **Evaluate the Options**:
   - Option I (rectangle): A rectangle has right angles. We can't be sure the angles here are right angles, so it's not a rectangle.
   - Option II (parallelogram): A rhombus is a parallelogram (opposite sides are parallel and equal), so this applies.
   - Option III (rhombus): Since all four sides $ AB, BC, B'C, AB' $ are equal (because $ AB = BC $ from the isosceles triangle and $ AB = AB', BC = B'C $ from reflection), it is a rhombus.
   - Option IV (square): A square is a rhombus with right angles. We don't have information about right angles, so it's not a square.

# Answer:
d) II and III only

### Question 2
# Brief Explanations:
1. **Understand the Rotation**: We rotate isosceles triangle $ ABC $ (with base $ \overline{AC} $) $ 180^{\circ} $ about the mid - point of $ AB $ (let's call the mid - point $ M $). Let the image of $ C $ be $ C' $.
   - In a $ 180^{\circ} $ rotation, $ AM = MB $ and $ CM = MC' $. Also, since $ \triangle ABC $ is isosceles with $ AB = BC $, after rotation, we can show that opposite sides of the quadrilateral $ ACC'B $ (the combined shape) are equal and parallel.
   - A quadrilateral with opposite sides equal and parallel is a parallelogram. Also, since $ AB = BC $ and the rotation preserves side lengths, we can show that all sides are equal (so it is a rhombus).
   - We don't have information to confirm right angles, so it's not a rectangle or a square.
2. **Evaluate the Options**:
   - Option I (rectangle): No right angles confirmed, so not a rectangle.
   - Option II (parallelogram): Opposite sides are equal and parallel, so it is a parallelogram.
   - Option III (rhombus): All sides are equal (because of the isosceles triangle and the $ 180^{\circ} $ rotation), so it is a rhombus.
   - Option IV (square): No right angles confirmed, so not a square.

# Answer:
b) II and III only

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