Question

6 from unit 2, lesson 2
line ef is a line of symmetry for figure abecdf. clare says that quadrilateral abef
is congruent to quadrilateral cdfe because sides ab and cd are corresponding.

a. why is clares congruence statement incorrect?

b. write a correct congruence statement for the quadrilaterals.
7 from unit 2, lesson 2
triangle hef is the image of triangle hgf after a reflection across line fh.

select all statements that must be true.
a triangle fgh is congruent to triangle feh.
b triangle efh is congruent to triangle gfh.
c angle hfe is congruent to angle fhg.
d angle efg is congruent to angle ehg.
e segment eh is congruent to segment fg.
f segment gh is congruent to segment eh.

Explanation:

Question 6

a.

When a figure is reflected over a line of symmetry, corresponding parts are congruent. In a congruence statement, the order of the vertices must correspond to the order of the corresponding parts. Clare only considered one pair of corresponding sides (\(AB\) and \(CD\)) but did not account for the order of all vertices. The correct correspondence should be based on the reflection over line \(EF\).

Since line \(EF\) is the line of symmetry for figure \(ABECDF\), when we reflect quadrilateral \(ABEF\) over line \(EF\), point \(A\) corresponds to point \(D\), point \(B\) corresponds to point \(C\), and the quadrilateral \(ABEF\) is congruent to quadrilateral \(DCEF\).

• Option A: Since \(\triangle HEF\) is the image of \(\triangle HGF\) after a reflection across line \(FH\), then \(\triangle FGH\cong\triangle FEH\) (by the definition of reflection, which is a rigid transformation that preserves congruence).
• Option B: \(\triangle EFH\) and \(\triangle GFH\) are not congruent. The reflection is of \(\triangle HGF\) to \(\triangle HEF\), not a direct congruence between \(\triangle EFH\) and \(\triangle GFH\).
• Option C: \(\angle HFE\) and \(\angle FHG\) are not congruent. The congruent angles are those that are images of each other under the reflection. For example, \(\angle EHF\cong\angle GHF\).
• Option D: \(\angle EFG\) and \(\angle EHG\) are not congruent. There is no reflection or other congruence - preserving transformation that would make these angles congruent.
• Option E: Segment \(EH\) is congruent to segment \(GH\) (by reflection, not \(FG\)).
• Option F: Since \(\triangle HEF\) is the image of \(\triangle HGF\) after a reflection across line \(FH\), segment \(GH\) is congruent to segment \(EH\) (reflection preserves segment lengths).

Answer:

Clare's statement is incorrect because in a congruence statement for figures related by a line of symmetry (reflection), the order of vertices must match the reflection correspondence. She did not consider the full vertex - to - vertex correspondence.

b.