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the bisector of an angle of a triangle separates the opposite side into segments whose lengths are_________ to the lengths of the other two sides​

Sagot :

Answer:

proportional

Step-by-step explanation:

Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides
Hope it help

Answer:

PROPORTIONAL

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Step-by-step explanation:

Triangle Angle Bisector Theorem

An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

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 ∠4  and  ∠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 ∠3  and  ∠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 .