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PLEASE HELP 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

Respuesta :

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: [tex] \frac{AB}{BC}=\frac{AD}{DC} [/tex]

Proof:

Since Lines EA and BD are parallel,

[tex] \angle 1=\angle 4 [/tex] (Corresponding angles)

So, [tex] \angle 2=\angle 3 [/tex] (Alternate angles)

[tex] \angle 1=\angle 3 [/tex] (because it is given that BD bisects ∠ABC )

So, by the above three equations, we get

[tex] \angle 2=\angle 4 [/tex]

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

[tex] \frac{AD}{DC}=\frac{BE}{BC} [/tex]

By using equation 1, we get

[tex] \frac{AD}{DC}=\frac{AB}{BC} [/tex]

Hence, proved.

Ver imagen pinquancaro

We can prove AD/DC = AB/BC by using the concept of the triangle and congruent theorem.

What is the triangle?

In terms of geometry, the triangle is a three-sided polygon with three edges and three vertices. The triangle's interior angles add up to 180°.

The question is incomplete.

The complete question is in the picture, please refer to the attached picture.

We have a triangle shown in the picture and line BD bisects ∠ABC. Auxiliary Line EA is drawn such that Line AE Line BD | |.

Statement 2:

Angle DBC ≅ Angle ABD    (BD bisects ∠ABC)

Statement 3:

AE || BD  (given)

Statement 4:

Angle AEB ≅ Angle DBC (Corresponding angles)

Statement 5:

Angle AEB ≅ Angle ABD   (Angle DBC ≅ Angle ABD)

Statement 6:

Angle ABD ≅ Angle BAE   (Alternative angles)

Statement 7:

Angle AEB ≅ Angle BAE   (Transitive property of equality)

Statement 8:

EB ≅ AB (triangle ABE is an isosceles triangle)

Statement 9:

EB = AB (concept of isosceles triangle)

Statement 10:

AD/DC = EB/BC  (triangle proportionality theorem)

Statement 11:

AD/DC = AB/BC   (substitution property of equality)

Thus, we can prove AD/DC = AB/BC by using the concept of the triangle and congruent theorem.

Learn more about the triangle here:

brainly.com/question/25813512

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Ver imagen maheshpatelvVT

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