c Try this the next time you are in a room with a mirror!
A
a See solution
B
b See solution
C
c No
Practice makes perfect
a Let's draw â–³ ABD and â–³ CBD including the given information.
Examining the diagram, we see that â–³ ABD and â–³ CBD share BD as a side. By the Reflexive Property of Congruence, we know this side is congruent in the triangles. Additionally, because
DB ⊥ AC,
we can by the definition of perpendicular lines say that ∠CBD is a right angle as well.
Now we have enough information to prove congruence by the ASA Congruence Theorem. Let's write this as a two-column proof as well.
Statement
Reason
1.
∠CDB ≅ ∠ADB, DB⊥ AC
1.
Given
2.
∠ABD and ∠CBD are right angles
2.
Definition of perpendicular lines
3.
∠ABD≅ ∠CBD
3.
Right Angles Congruence Theorem
4.
BD≅ BD
4.
Reflexive Property of Congruence
5.
△ ABD ≅ △ CBD
5.
ASA Congruence Theorem
b From Part A, we know that △ ABD ≅ △ CBD. Let's identify the remaining pair of congruent angles in these triangles.
Finally, we highlight â–³ ACD and relevant angles for this triangle.
Since two angles in △ ADC are congruent, we can by the Converse of the Base Angles Theorem claim that AD≅ CD, making it an isosceles triangle.
c No. Where she currently stands, she sees her toes at the bottom of the mirror. If she moves away from the mirror, â–³ ACD will remain isosceles reflecting her entire body including her feet, in the mirror.
