Question

BD−→− bisects ∠ABC such that m∠ABD = (4x−5)∘ and m∠DBC = (3x+2)∘.

Find m∠ABC.

m∠ABC = ∘

Answer 1

Answer:

$\angle ABC=(46)^{\circ}$

Step-by-step explanation:

Given: BD bisects $\angle ABC$

$\angle ABD=(4x-5)^{\circ}\\\angle DBC=(3x+2)^{\circ}$

To find: $\angle ABC$

Solution:

A bisector of an angle divides it into angles of equal measures.

As BD bisects $\angle ABC$, $\angle ABD=\angle DBC$

Put $\angle ABD=(4x-5)^{\circ}\,,\,\angle DBC=(3x+2)^{\circ}$

$\angle ABD=\angle DBC\\(4x-5)^{\circ}=(3x+2)^{\circ}\\4x-3x=2+5\\x=7^{\circ}$

So,

$\angle ABD=(4x-5)^{\circ}=(4(7)-5)^{\circ}=(28-5)^{\circ}=(23)^{\circ}\\\angle DBC=(3x+2)^{\circ}=(3(7)+2)^{\circ}=(21+2)^{\circ}=(23)^{\circ}$

$\angle ABC=\angle ABD+\angle DBC\\=(23)^{\circ}+(23)^{\circ}\\=(46)^{\circ}$

Answer 2

A bisector divides lines, angles and shapes into equal segments

The measure of angle ABC is 46 degrees

The given parameters are:

• m∠ABD = (4x−5)∘
• m∠DBC = (3x+2)∘

Given that line BD is the bisector of angle ABC, it means that:

$\angle ABD = \angle DBC$

and

$\angle ABC = 2\times \angle ABD$

So, we have:

$4x - 5 = 3x +2$

Collect like terms

$4x - 3x = 5 +2$

Evaluate like terms

$x = 7$

Recall that

$\angle ABC = 2\times \angle ABD$

So, we have:

$\angle ABC = 2 \times (4x - 5)$

Substitute 7 for x

$\angle ABC = 2 \times (4\times 7 - 5)$

$\angle ABC = 2 \times (28 - 5)$

$\angle ABC = 2 \times 23$

$\angle ABC = 46$

Hence, the measure of angle ABC is 46 degrees

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