If isosceles triangle ABC has a [tex]$130^{\circ}$[/tex] angle at vertex B, which statement must be true?

A. [tex]$m \angle A=15^{\circ}$[/tex] and [tex]$m \angle C=35^{\circ}$[/tex]
B. [tex]$m \angle A+m \angle B=155^{\circ}$[/tex]
C. [tex]$m \angle A+m \angle C=60^{\circ}$[/tex]
D. [tex]$m \angle A=20^{\circ}$[/tex] and [tex]$m \angle C=30^{\circ}$[/tex]



Answer :

To solve the problem, consider the properties of an isosceles triangle and the fact that the sum of the interior angles of any triangle is always [tex]\(180^\circ\)[/tex].

Given:
- Triangle ABC is isosceles with [tex]\( \angle B = 130^\circ \)[/tex].
- Since ABC is isosceles, the remaining two angles, [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex], are equal.

Let's denote the measures of [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] as [tex]\( x \)[/tex]:

1. The sum of all angles in a triangle is [tex]\(180^\circ\)[/tex]:
[tex]\[ \angle A + \angle B + \angle C = 180^\circ \][/tex]

2. Substitute the known values into the equation:
[tex]\[ x + 130^\circ + x = 180^\circ \][/tex]

3. Combine like terms:
[tex]\[ 2x + 130^\circ = 180^\circ \][/tex]

4. Subtract [tex]\(130^\circ\)[/tex] from both sides:
[tex]\[ 2x = 50^\circ \][/tex]

5. Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 25^\circ \][/tex]

Thus, both [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are [tex]\( 25^\circ \)[/tex].

Now, let's check which statement is true based on our findings:

1. [tex]\( \angle A = 15^\circ \)[/tex] and [tex]\( \angle C = 35^\circ \)[/tex]:
- This is false because [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are both [tex]\(25^\circ\)[/tex].

2. [tex]\( \angle A + \angle B = 155^\circ \)[/tex]:
- Calculate [tex]\( \angle A + \angle B \)[/tex]:
[tex]\[ 25^\circ + 130^\circ = 155^\circ \][/tex]
- This is true.

3. [tex]\( \angle A + \angle C = 60^\circ \)[/tex]:
- Calculate [tex]\( \angle A + \angle C \)[/tex]:
[tex]\[ 25^\circ + 25^\circ = 50^\circ \][/tex]
- This is false.

4. [tex]\( \angle A = 20^\circ \)[/tex] and [tex]\( \angle C = 30^\circ \)[/tex]:
- This is false because [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are both [tex]\(25^\circ\)[/tex].

Therefore, the correct statement is:

[tex]\[ \boxed{ \angle A + \angle B = 155^\circ } \][/tex]

So, the true statement is:

[tex]\[ m \angle A + m \angle B = 155^\circ \][/tex]