Question

У трикутника АВС пр оведена бісектриса кута А який дорівнює 54 градусів через вершину В проведена пряма паралельна до цієї бісектриси.Точка О точка перетину проведеної прямої із продовженням сторони АС .Знайти кути трикутника АОВ

Answer 1

Ответ:

Розглянемо трикутник АМС. Сумка кутів трикутника дорівнює 180°, тоді ∠МАС+∠МСА+∠АМС=180°.

Сума суміжних кутів дорівнює 180°. Кути АMВ i AMC суміжні. Відомо, що ∠АМВ=117°, отже ∠АМС=180°-117°=63°

Бісектриса ділить кут навпіл отже ∠ВАС= ∠ВАМ+ ∠МАС=2∠МАС.

Трикутник АВС рівнобедрений тому кути при основі рівні тобто ∠ВАС=∠ВСА, отже оскільки ∠ВАС=2∠МАС, то і ∠ВСА=2∠МАС

Звідси ∠МАС+2∠МАС+63°.=180°.

3∠МАС=180°-63°

3∠МАС=117°

∠МАС=39°

∠ВАС=∠ВСА= ∠ВАМ+ ∠МАС=2∠МАС=2*39°=78°

∠АВС=180°-78°-78°=24°- за т. про суму кутів трикутника.

Відпповідь: ∠АВС=24°, ∠ВАС=∠ВСА=78°

Answer 2

Problem Analysis

We are given a triangle ABC with an angle A bisected by a line through vertex B, which measures 54 degrees. A line parallel to this bisector is drawn through vertex B, intersecting the extension of side AC at point O. We need to find the angles of triangle AOV.

Solution

To find the angles of triangle AOV, we can use the properties of angles formed by intersecting lines and the fact that the sum of the angles in a triangle is 180 degrees.

Let's denote the angles of triangle AOV as ∠AOV, ∠OVA, and ∠AVO.

Since the line through vertex B is parallel to the bisector of angle A, we know that ∠AOV and ∠OVA are alternate interior angles and therefore congruent. This means that ∠AOV = ∠OVA.

We also know that the sum of the angles in a triangle is 180 degrees. Therefore, we can write the equation:

∠AOV + ∠OVA + ∠AVO = 180 degrees

Substituting ∠AOV = ∠OVA, we get:

∠AOV + ∠AOV + ∠AVO = 180 degrees

Simplifying the equation, we have:

2∠AOV + ∠AVO = 180 degrees

Since we know that ∠AOV = ∠OVA, we can rewrite the equation as:

2∠OVA + ∠AVO = 180 degrees

Now, let's substitute the given angle measure of ∠OVA = 54 degrees:

2(54 degrees) + ∠AVO = 180 degrees

Simplifying the equation, we have:

108 degrees + ∠AVO = 180 degrees

Subtracting 108 degrees from both sides, we get:

∠AVO = 180 degrees - 108 degrees

∠AVO = 72 degrees

Therefore, the angle ∠AVO in triangle AOV measures 72 degrees.

To find the remaining angle ∠AOV, we can use the fact that the sum of the angles in a triangle is 180 degrees:

∠AOV + ∠OVA + ∠AVO = 180 degrees

Substituting ∠OVA = ∠AOV = 72 degrees, we have:

∠AOV + 72 degrees + 72 degrees = 180 degrees

Simplifying the equation, we get:

∠AOV = 180 degrees - 72 degrees - 72 degrees

∠AOV = 36 degrees

Therefore, the angle ∠AOV in triangle AOV measures 36 degrees.

Answer

The angles of triangle AOV are as follows: - ∠AOV = 36 degrees - ∠OVA = 72 degrees - ∠AVO = 72 degrees

Please let me know if I can help you with anything else.