The Interior Angles Of A Triangle Have Measures 102 ∘ , 16 ∘ 102^{\circ}, 16^{\circ} 10 2 ∘ , 1 6 ∘ , And M ∘ M^{\circ} M ∘ . What Is The Value Of M M M ?Enter Your Answer In The Box. M = □ ∘ M=\square^{\circ} M = □ ∘

The Interior Angles of a Triangle: Finding the Missing Measure

In geometry, the interior angles of a triangle are the angles formed by the intersection of the sides of the triangle. The sum of the interior angles of a triangle is always 180 degrees. In this article, we will explore how to find the missing measure of an interior angle of a triangle given the measures of the other two angles.

The Sum of the Interior Angles of a Triangle

The sum of the interior angles of a triangle is a fundamental property of geometry. It states that the sum of the measures of the interior angles of a triangle is always 180 degrees. This property can be expressed mathematically as:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

where $m_1$, $m_2$, and $m_3$ are the measures of the interior angles of the triangle.

Finding the Missing Measure

In this problem, we are given the measures of two interior angles of a triangle: $102^{\circ}$ and $16^{\circ}$. We are asked to find the measure of the third interior angle, denoted by $m$. To find the measure of $m$, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$102^{\circ} + 16^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$118^{\circ} + m = 180^{\circ}$$

Subtracting $118^{\circ}$ from both sides, we get:

$$m = 62^{\circ}$$

Therefore, the measure of the third interior angle is $62^{\circ}$.

In this article, we explored how to find the missing measure of an interior angle of a triangle given the measures of the other two angles. We used the formula for the sum of the interior angles of a triangle to find the measure of the third angle. The measure of the third angle was found to be $62^{\circ}$.

Here are a few example problems that illustrate how to find the missing measure of an interior angle of a triangle:

• A triangle has interior angles measuring $120^{\circ}$ and $30^{\circ}$. Find the measure of the third interior angle.
• A triangle has interior angles measuring $90^{\circ}$ and $60^{\circ}$. Find the measure of the third interior angle.
• A triangle has interior angles measuring $105^{\circ}$ and $25^{\circ}$. Find the measure of the third interior angle.

Here are the solutions to the example problems:

• A triangle has interior angles measuring $120^{\circ}$ and $30^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$120^{\circ} + 30^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$150^{\circ} + m = 180^{\circ}$$

Subtracting $150^{\circ}$ from both sides, we get:

$$m = 30^{\circ}$$

Therefore, the measure of the third interior angle is $30^{\circ}$.

• A triangle has interior angles measuring $90^{\circ}$ and $60^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$90^{\circ} + 60^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$150^{\circ} + m = 180^{\circ}$$

Subtracting $150^{\circ}$ from both sides, we get:

$$m = 30^{\circ}$$

Therefore, the measure of the third interior angle is $30^{\circ}$.

• A triangle has interior angles measuring $105^{\circ}$ and $25^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$105^{\circ} + 25^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$130^{\circ} + m = 180^{\circ}$$

Subtracting $130^{\circ}$ from both sides, we get:

$$m = 50^{\circ}$$

Therefore, the measure of the third interior angle is $50^{\circ}$.

The final answer is: $\boxed{62}$
The Interior Angles of a Triangle: Q&A

In our previous article, we explored how to find the missing measure of an interior angle of a triangle given the measures of the other two angles. In this article, we will answer some frequently asked questions about the interior angles of a triangle.

Q: What is the sum of the interior angles of a triangle?

A: The sum of the interior angles of a triangle is always 180 degrees.

Q: How do I find the missing measure of an interior angle of a triangle?

A: To find the missing measure of an interior angle of a triangle, you can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

where $m_1$, $m_2$, and $m_3$ are the measures of the interior angles of the triangle.

Q: What if I have two interior angles of a triangle and I want to find the third angle?

A: If you have two interior angles of a triangle and you want to find the third angle, you can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substitute the given values and solve for the missing angle.

Q: Can I use the formula for the sum of the interior angles of a triangle to find the measure of an exterior angle of a triangle?

A: No, the formula for the sum of the interior angles of a triangle is used to find the measure of an interior angle of a triangle, not an exterior angle.

Q: What is the relationship between the interior angles of a triangle and the exterior angles of a triangle?

A: The sum of an interior angle and its corresponding exterior angle is always 180 degrees.

Q: Can I use the formula for the sum of the interior angles of a triangle to find the measure of an angle in a quadrilateral?

A: No, the formula for the sum of the interior angles of a triangle is used to find the measure of an interior angle of a triangle, not an angle in a quadrilateral.

Q: What is the sum of the interior angles of a quadrilateral?

A: The sum of the interior angles of a quadrilateral is always 360 degrees.

Q: Can I use the formula for the sum of the interior angles of a triangle to find the measure of an angle in a polygon with more than four sides?

A: No, the formula for the sum of the interior angles of a triangle is used to find the measure of an interior angle of a triangle, not an angle in a polygon with more than four sides.

Q: What is the formula for the sum of the interior angles of a polygon with n sides?

A: The formula for the sum of the interior angles of a polygon with n sides is:

$$180(n-2)$$

In this article, we answered some frequently asked questions about the interior angles of a triangle. We hope that this article has been helpful in clarifying any confusion you may have had about the interior angles of a triangle.

Here are a few example problems that illustrate how to find the missing measure of an interior angle of a triangle:

• A triangle has interior angles measuring $120^{\circ}$ and $30^{\circ}$. Find the measure of the third interior angle.
• A triangle has interior angles measuring $90^{\circ}$ and $60^{\circ}$. Find the measure of the third interior angle.
• A triangle has interior angles measuring $105^{\circ}$ and $25^{\circ}$. Find the measure of the third interior angle.

Here are the solutions to the example problems:

• A triangle has interior angles measuring $120^{\circ}$ and $30^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$120^{\circ} + 30^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$150^{\circ} + m = 180^{\circ}$$

Subtracting $150^{\circ}$ from both sides, we get:

$$m = 30^{\circ}$$

Therefore, the measure of the third interior angle is $30^{\circ}$.

• A triangle has interior angles measuring $90^{\circ}$ and $60^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$90^{\circ} + 60^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$150^{\circ} + m = 180^{\circ}$$

Subtracting $150^{\circ}$ from both sides, we get:

$$m = 30^{\circ}$$

Therefore, the measure of the third interior angle is $30^{\circ}$.

• A triangle has interior angles measuring $105^{\circ}$ and $25^{\circ}$. To find the measure of the third interior angle, we can use the formula for the sum of the interior angles of a triangle:

$$m_1 + m_2 + m_3 = 180^{\circ}$$

Substituting the given values, we get:

$$105^{\circ} + 25^{\circ} + m = 180^{\circ}$$

Simplifying the equation, we get:

$$130^{\circ} + m = 180^{\circ}$$

Subtracting $130^{\circ}$ from both sides, we get:

$$m = 50^{\circ}$$

Therefore, the measure of the third interior angle is $50^{\circ}$.

The final answer is: $\boxed{62}$