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:

Step1: Analyze question 6

• A 180 - degree rotation about point \(K\) is a rigid transformation. Rigid transformations (rotations, reflections, translations) preserve the shape and size of the figure, so corresponding parts of the pre - image (\(\triangle FGH\)) and the image (\(\triangle HEF\)) are congruent.
• For triangles, if \(\triangle HEF\) is the image of \(\triangle FGH\) after a 180 - degree rotation about point \(K\):
• By the property of rotation, \(\triangle FGH\cong\triangle FEH\) (b). The vertices of \(\triangle FGH\) and \(\triangle FEH\) are related by the rotation rule. If we consider the rotation of each vertex around \(K\) by 180 degrees, \(F\) maps to \(F\) (since rotating a point 180 degrees around itself leaves it unchanged), \(G\) maps to \(E\) and \(H\) maps to \(H\) (again, rotating a point 180 degrees around itself leaves it unchanged). Also, \(\angle EFH\cong\angle GFH\) (c). When we rotate \(\triangle FGH\) 180 degrees about \(K\) to get \(\triangle HEF\), the angles at \(F\) (for the two triangles \(\triangle FGH\) and \(\triangle EFH\)) are congruent because of the rotation (a rigid transformation that preserves angle measures). And \(\angle KHE\cong\angle KFG\) (d). The rotation preserves the angle measures of the angles formed at the intersection points (by the property of rotation as a rigid transformation).

Step2: Analyze question 7

• When \(\triangle ABC\) is reflected across line \(AB\) to get \(\triangle ABD\), reflection is a rigid transformation.
• By the definition of congruent figures (figures that can be mapped onto each other using rigid transformations), \(\triangle ABC\cong\triangle ABD\). Corresponding parts of congruent figures (CPCTC - Corresponding Parts of Congruent Triangles are Congruent) are congruent. So \(AC\cong AD\) because of the reflection (a rigid transformation that maps \(C\) to \(D\) across the line of reflection \(AB\)). The reason is that congruent parts of congruent figures are corresponding (a).

Answer:

1. b. Triangle \(EFH\) is congruent to triangle \(GFH\), c. Angle \(EFH\) is congruent to angle \(GFH\), d. Angle \(KHE\) is congruent to angle \(KFG\)
2. a. Congruent parts of congruent figures are corresponding