If one angle of a triangle is 130 then angle between bisectors of other two base angles are

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If one of the angles of a triangle is 130∘ then the angle between the bisectors of the other two angles can be a 50∘ b 65∘ c 90∘ d 155∘

Solution

ANSWER:
Let $△ABCbesuchthat\angle A={130}^{\circ }$
Here, BP is the bisector of $\angle BandCPisthebisectorof\angle C$.
∴ $\angle ABP=\angle PBC=1/2\angle B$ .....(1)

Also, $\angle ACP=\angle PCB=1/2\angle C$

.....(2)

In $△ABC$,

$\angle A+\angle B+\angle C={180}^{\circ }$ (Angle sum property)

⇒ ${130}^{\circ }+\angle B+\angle C={180}^{\circ }$

⇒ $\angle B+\angle C={180}^{\circ }-{130}^{\circ }={50}^{\circ }$
⇒ 1/2 $\angle B+1/2\angle C=1/2\times {50}^{\circ }={25}^{\circ }$
⇒ $\angle PBC+\angle PCB={25}^{\circ }$ .....(3) [Using (1) and (2)]

In $△PBC$,

$\angle PBC+\angle PCB+\angle BPC={180}^{\circ }$
⇒${25}^{\circ }+\angle BPC={180}^{\circ }$ (Angle sum property) [Using (3)]

⇒ $\angle BPC={180}^{\circ }-{25}^{\circ }={155}^{\circ }$

Thus, if one of the angles of a triangle is ${130}^{\circ }thentheanglebetweenthebisectorsoftheothertwoanglesis{155}^{\circ }$.
Hence, the correct answer is option (d).

Mathematics

Secondary School Mathematics IX

Standard IX