Answered

Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

If isosceles triangle [tex]\(ABC\)[/tex] has a [tex]\(130^{\circ}\)[/tex] angle at vertex [tex]\(B\)[/tex], 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]

Sagot :

GinnyAnswer
To determine which statement is true for the given isosceles triangle [tex]\(ABC\)[/tex] with a [tex]\(130^\circ\)[/tex] angle at vertex [tex]\(B\)[/tex], follow these steps:

1. Understand the properties of an isosceles triangle:
In an isosceles triangle, two sides are equal, and the angles opposite those sides are also equal.

2. Determine the sum of angles in a triangle:
The sum of the interior angles in any triangle is always [tex]\(180^\circ\)[/tex].

3. Assign given information:
Given that [tex]\(m \angle B = 130^\circ\)[/tex], and this is an isosceles triangle with the equal angles at vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex].

4. Calculate the sum of the other two angles (Angle A and Angle C):
Since the sum of angles in any triangle is [tex]\(180^\circ\)[/tex],
[tex]\[ m \angle A + m \angle B + m \angle C = 180^\circ \][/tex]
Substituting [tex]\(m \angle B = 130^\circ\)[/tex],
[tex]\[ m \angle A + m \angle C = 180^\circ - 130^\circ = 50^\circ \][/tex]

5. Determine the individual measures of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are equal in an isosceles triangle,
[tex]\[ m \angle A = m \angle C = \frac{50^\circ}{2} = 25^\circ \][/tex]

Now that we have determined [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\(m \angle C = 25^\circ\)[/tex], we can evaluate the given statements:

- [tex]\( m \angle A=15^\circ \)[/tex] and [tex]\( m \angle C=35^\circ \)[/tex]: False. We determined that both [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex] each measure [tex]\(25^\circ\)[/tex].

- [tex]\( m \angle A + m \angle B = 155^\circ\)[/tex]: True. [tex]\(m \angle A = 25^\circ\)[/tex] and [tex]\(m \angle B = 130^\circ\)[/tex], thus,
[tex]\[ 25^\circ + 130^\circ = 155^\circ \][/tex]

- [tex]\( m \angle A + m \angle C = 60^\circ\)[/tex]: False. We determined that [tex]\(m \angle A + m \angle C = 50^\circ\)[/tex].

- [tex]\( m \angle A=20^\circ \)[/tex] and [tex]\( m \angle C=30^\circ \)[/tex]: False. We determined that both [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex] each measure [tex]\(25^\circ\)[/tex].

Therefore, the correct statement is:
[tex]\( m \angle A + m \angle B = 155^\circ \)[/tex].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.