Question

lesson 1 practice problems

1. when rectangle abcd is reflected across line ef, the image is dcba. how do you know that segment ab is congruent to segment dc?

a. a rectangle has 2 pairs of parallel sides.
b. any 2 sides of a rectangle are congruent.
c. congruent parts of congruent figures are corresponding.
d. corresponding parts of congruent figures are congruent.

1. triangle fgh is the image of isosceles triangle feh after a reflection across line hf. select all the statements that are a result of corresponding parts of congruent triangles being congruent.

a. efgh is a rectangle.
b. efgh has 4 congruent sides.
c. diagonal fh bisects angles efg and ehg.
d. diagonal fh is perpendicular to side fe.
e. angle feh is congruent to angle fgh.

Explanation:

1.

Step1: Analyze each option for question 1

• Option A: A rectangle having 2 pairs of parallel sides does not directly prove \(AB\cong DC\).
• Option B: In a rectangle, only opposite sides are congruent, not any 2 sides.
• Option C: The statement is mis - phrased. It should be corresponding parts of congruent figures are congruent.
• Option D: When rectangle \(ABCD\) is reflected across line \(EF\) to get \(DCBA\), the two rectangles are congruent. By the property that corresponding parts of congruent figures are congruent, since \(AB\) and \(DC\) are corresponding sides of congruent rectangles \(ABCD\) and \(DCBA\), \(AB\cong DC\).

2.

Step1: Analyze each option for question 2

• Option A: \(EFGH\) is not a rectangle. A rectangle has four right angles. There is no information to suggest \(EFGH\) has four right angles.
• Option B: \(EFGH\) does not have 4 congruent sides. We know \(FE\cong HE\) (given for \(\triangle FEH\) being isosceles), but after reflection \(\triangle FEH\cong\triangle FGH\), so \(FE = FG\) and \(HE=HG\), but not all four sides are equal.
• Option C: Since \(\triangle FEH\cong\triangle FGH\) (by reflection), \(\angle EFH=\angle GFH\) and \(\angle EHF=\angle GHF\). So diagonal \(FH\) bisects angles \(EFG\) and \(EHG\) (by the property of congruent triangles, corresponding angles are congruent).
• Option D: There is no information from the congruence of \(\triangle FEH\) and \(\triangle FGH\) to suggest that \(FH\perp FE\).
• Option E: Since \(\triangle FEH\cong\triangle FGH\) (by reflection), \(\angle FEH\cong\angle FGH\) (corresponding parts of congruent triangles are congruent).

Answer:

1. D. Corresponding parts of congruent figures are congruent.
2. C. Diagonal \(FH\) bisects angles \(EFG\) and \(EHG\); E. Angle \(FEH\) is congruent to angle \(FGH\).