Given: circle C is constructed so that CD = DE = AD;— CA is the radius of circle C

Prove: AE is tangent to c

Step-by-step explanation:

16. m<CAD + m<DAE =  -  Definition of complementary angles

18.  m<CAD + m<DAE = m<CAE =     -  Substitution property

20. m<CAE =                        -  Definition of perpendicular

21. AE ⊥ AC  -  AE is tangent to circle C

To prove that AE is tangent to C. Given that CA is the radius of circle C,

Then,

m<CAD + m<DAE =

So that,

AE ⊥ AC

Thus, AE is tangent to C.

(Statement > Reason)

1 )  Circle C is constructed so that CD = DE = AD; Line segment CA is a radius of circle C. > Given

2) Line segment CD is congruent to line segment DE; line segment DE is congruent to line segment AD > Definition of congruence

3) Triangle ACD is an isosceles triangle; Triangle ADE is an isosceles triangle. > Isosceles triangle theorem

4) m<CAD + m<DCA + m<ADC = 180 degrees; m<DAE = m<AED + m<EDA = 180 degrees > Substitution property

5) CD = DE > Isosceles triangle theorem

6) m<CAD = m<DCA; m<CAE = m<AED > Definition of congruence

7) m<CAD + m<CAD + m<ADC = 180 degrees; m<DAE = m<DAE + m<EDA = 180 degrees > Substitution property

8) 2(m<CAD) + m<ADC = 180 degrees; 2(m<DAE) + m<EDA = 180 degrees > Addition

9) m<ADC = 180 degrees - 2(m<CAD); m<EDA = 180 degrees - 2(m<DAE) > Addition subtraction property

10) Angle ADC and Angle EDA are a linear pair. > Linear pair postulate

11) Angle ADC + Angle EDA > Linear pair postulate

12) m<ADC + m<EDA = 180 degrees > Definition of supplementary angles

13) 180 degrees - 2(m<CAD) + 180 degrees - 2(m<DAE) = 180 degrees > Substitution property

14) 360 degrees - 2(m<CAD) - 2(m<DAE) = 180 degrees > Addition

15) -2(m<CAD) - 2(m<DAE) = -180 degrees > Subtraction property

16) m<CAD + m<DAE = 90 degrees > Definition of complementary angles

17) m<CAD + m<DAE = m<CAE > Angle addition postulate

18) m<CAD + m<DAE = m<CAE = 90 degrees > Substitution property

19) Angle CAE is a right angle > Definition of right angle

20) m<CAE = 90 degrees > Definition of perpendicular

21) Line segment AE is tangent to circle C. > AE ⊥ AC

Step-by-step explanation:

I'm not sure if the reason for #9 is correct, but that's what someone else in a different answer said, so I went ahead and put it.