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A triangle with an angle measure of 40° was cut into 6 triangles by its angle bisectors. Some of these new triangles are right triangles. What could be the measure of the largest angle of this triangle?

Sagot :

livracette

Answer: :)

Step-by-step explanation:

By the Angle Bisector Theorem,

BDDC=ABAC

Proof:

Draw BE←→∥AD←→

.

Extend CA⎯⎯⎯⎯⎯

to meet BE←→ at point E

.

By the Side-Splitter Theorem,

CDDB=CAAE

---------( 1

)

The angles ∠4and∠1

are corresponding angles.

So, ∠4≅∠1

.

Since AD⎯⎯⎯⎯⎯

is a angle bisector of the angle ∠CAB,∠1≅∠2

.

By the Alternate Interior Angle Theorem , ∠2≅∠3

.

Therefore, by transitive property, ∠4≅∠3

.

Since the angles ∠3and∠4

are congruent , the triangle ΔABE is an isosceles triangle with AE=AB

.

Replacing AE

by AB in equation ( 1

),

CDDB=CAAB

Example:

Find the value of x

.

By Triangle-Angle-Bisector Theorem,

ABBC=ADDC

.

Substitute.

512=3.5x

Cross multiply.

5x=42

Divide both sides by 5

.

5x5=425x=8.4

The value of x

is 8.4 .

The largest angle is 8.4.

what is Angle-Bisector theorem?

The Angle-Bisector theorem in geometry is concerned with the proportions of the two segments that a line that bisects the opposite angle divides a triangle's side into. It compares their proportional lengths to the proportional lengths of the triangle's other two sides.

Given

By the Angle Bisector Theorem,

BDDC=ABAC

Proof:

Draw BE←→∥AD←→

Extend CA��⎯⎯⎯⎯to meet BE←→ at point E

By the Side-Splitter Theorem,

CDDB=CAAE---------( 1)

The angles ∠4and∠1

are corresponding angles.

So, ∠4≅∠1

Since AD⎯⎯⎯⎯⎯

is a angle bisector of the angle ∠CAB,∠1≅∠2

By the Alternate Interior Angle Theorem , ∠2≅∠3

Therefore, by transitive property, ∠4≅∠3

Since the angles ∠3and∠4

are congruent , the triangle ΔABE is an isosceles triangle with AE=AB

Replacing AE

by AB in equation ( 1),

CDDB=CAAB

Example:

Find the value of x

By Triangle-Angle-Bisector Theorem,

ABBC=ADDC

Substitute.

512=3.5x

Cross multiply.

5x=42

Divide both sides by 5

5x/5=42/5=8.4

The value of x is 8.4 .

To learn more about the angle-bisector theorem refer to:

https://brainly.com/question/2478436

#SPJ2

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