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 .
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