Question

1. triangle hef is the image of triangle fgh after a 180 degree rotation about point k. 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 khe is congruent to angle kfg.
d. angle ghk is congruent to angle khe.
e. segment eh is congruent to segment fg.
f. segment gh is congruent to segment ef.

1. when triangle abc is reflected across line ab, the image is triangle abd. why are segment ad and segment ac congruent?

a. congruent parts of congruent figures are corresponding.
b. corresponding parts of congruent figures are congruent.
c. an isosceles triangle has a pair of congruent sides.
d. segment ab is a perpendicular bisector of segment dc.
(from unit 2, lesson 1.)

Explanation:

Question 3

• Option A: A \(180^{\circ}\) rotation is a rigid transformation. Rigid transformations preserve congruence. But \(\triangle FGH\) and \(\triangle FEH\) are not congruent.
• Option B: Since \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation about point \(K\), \(\triangle EFH\cong\triangle GFH\) (rotation is a rigid - motion, which preserves side - lengths and angle - measures).
• Option C: Because of the \(180^{\circ}\) rotation, \(\angle KHE\) and \(\angle KFG\) are congruent (rotation preserves angle measures).
• Option D: \(\angle GHK\) and \(\angle KHE\) are not congruent.
• Option E: Since \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation about point \(K\), \(EH = FG\) (rotation preserves side - lengths).
• Option F: Since \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation about point \(K\), \(GH=EF\) (rotation preserves side - lengths).

When \(\triangle ABC\) is reflected across line \(AB\) to get \(\triangle ABD\), \(\triangle ABC\cong\triangle ABD\) (reflection is a rigid transformation). For congruent figures, corresponding parts are congruent. \(AD\) and \(AC\) are corresponding parts of congruent triangles \(\triangle ABD\) and \(\triangle ABC\) respectively.

Answer:

B. Triangle \(EFH\) is congruent to triangle \(GFH\); C. Angle \(KHE\) is congruent to angle \(KFG\); E. Segment \(EH\) is congruent to segment \(FG\); F. Segment \(GH\) is congruent to segment \(EF\)

Question 4