Question

select the correct answer from each drop down menu.
why are triangles adc and adb congruent?

triangle adc is congruent to triangle adb because triangle adc maps it onto triangle adb.
reflecting
rotating
translating
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Explanation:

Step1: Analyze the figure

Looking at the figure, we can see that line \(AD\) is the common side. \(AC = AB\) (marked with the same tick - marks) and \(CD=BD\) (marked with the same tick - marks). Also, the line \(AD\) acts as a perpendicular bisector (assuming the triangle is isosceles as \(AC = AB\) and \(CD = BD\)).

Step2: Recall transformation properties

• Reflection: A reflection over a line (in this case, line \(AD\)) flips a figure over the line. If we consider line \(AD\) as the line of reflection, every point of \(\triangle ADC\) has a corresponding point in \(\triangle ADB\) such that the line \(AD\) is the perpendicular bisector of the segment joining the corresponding points.
• Rotation: A rotation would require a center of rotation and an angle of rotation. There is no indication of a center (other than a point on \(AD\)) and an angle that would map \(\triangle ADC\) to \(\triangle ADB\) in a non - trivial way (since the "flip" nature is more prominent).
• Translation: A translation slides a figure. But \(\triangle ADC\) and \(\triangle ADB\) are not in a position where a simple slide (translation) would map one to the other as they are on opposite sides of line \(AD\).

Answer:

Triangle \(ADC\) is congruent to triangle \(ADB\) because reflecting triangle \(ADC\) over line \(AD\) maps it onto triangle \(ADB\).