Me! provide statements and reasons for the proof of the triangle angle bisector theorem. given: line bd bisects ∠abc. auxiliary line ea is drawn such that line ae line bd | |. auxiliary line be is an extension of line bc. prove: ad/dc = ab/bc

Refer to the attached image.

Given: Line BD bisects ∠ABC.

Construction: Auxiliary Line EA is drawn such that Line AE is parallel to Line BD . Auxiliary Line BE is an extension of Line BC.

To prove:

Proof:

Since Lines EA and BD are parallel,

(Corresponding angles)

So, (Alternate angles)

(because it is given that BD bisects ∠ABC )

So, by the above three equations, we get

So, BE=AB (Opposite sides equal to opposite angles are equal) (Equation 1)

Now, consider triangle ACE,

Since AE is parallel to BD.

By Basic Proportionality theorem, which states

" If a line is drawn parallel to one side of a triangle intersecting other two sides, then it divides the two sides in the same ratio."

So,we get

By using equation 1, we get

Hence, proved.

let soccer = x and golf = y

part a

1st equation: x + y = 125

2nd equation: x = y +45

part b

replace x in the 1st equation with the 2nd one:

y + 45 + y = 125

simplify:

2y + 45 = 125

subtract 45 from each side:

2y = 80

divide both sides by 2:

y = 80/2

y = 40

she plays golf for 40 minutes.

part c

no because she plays soccer for 45 minutes longer than golf 40 +45 =   85 minutes, not 80 minutes.