The Sum of Angles in a Triangle

Oct 13, 2025 by miodeya.com 76 views

Understanding the Problem

In this problem, we are given the measures of two angles in a triangle and asked to find the measure of the third angle. The measures of angles 2 and 3 are given as $(5x + 14)^\circ$ and $(7x - 14)^\circ$ , respectively. We need to find the measure of angle 1 in degrees.

The Sum of Angles in a Triangle

The sum of the measures of the angles in a triangle is always $180^\circ$ . This is a fundamental property of triangles that we can use to solve for the measure of angle 1.

Setting Up the Equation

Let's set up an equation using the fact that the sum of the measures of the angles in a triangle is $180^\circ$ . We know that the measure of angle 1 is $x^\circ$ , the measure of angle 2 is $(5x + 14)^\circ$ , and the measure of angle 3 is $(7x - 14)^\circ$ . Therefore, we can write the equation:

x + (5x + 14) + (7x - 14) = 180

Simplifying the Equation

Now, let's simplify the equation by combining like terms:

x + 5x + 14 + 7x - 14 = 180

13x = 180

Solving for x

Now, let's solve for x by dividing both sides of the equation by 13:

x = \frac{180}{13}

x = 13.84615...

Finding the Measure of Angle 1

Now that we have found the value of x, we can find the measure of angle 1 by substituting x into the expression for the measure of angle 1:

\text{Measure of angle 1} = x^\circ

\text{Measure of angle 1} = 13.84615...^\circ

However, we are given four options for the measure of angle 1: $88^\circ$ , $89^\circ$ , $90^\circ$ , and $91^\circ$ . We need to find the option that is closest to the value of x that we found.

Rounding the Value of x

Let's round the value of x to the nearest whole number:

x \approx 14

Finding the Measure of Angle 1

Now that we have rounded the value of x, we can find the measure of angle 1 by substituting x into the expression for the measure of angle 1:

\text{Measure of angle 1} = x^\circ

\text{Measure of angle 1} = 14^\circ

However, we are given four options for the measure of angle 1: $88^\circ$ , $89^\circ$ , $90^\circ$ , and $91^\circ$ . We need to find the option that is closest to the value of x that we found.

Comparing the Options

Let's compare the options to the value of x that we found:

$88^\circ$ is less than the value of x that we found.
$89^\circ$ is less than the value of x that we found.
$90^\circ$ is less than the value of x that we found.
$91^\circ$ is greater than the value of x that we found.

Conclusion

Based on our calculations, we can see that the measure of angle 1 is closest to $89^\circ$ . Therefore, the correct answer is:

The final answer is $89^\circ$ .

Additional Information

In this problem, we used the fact that the sum of the measures of the angles in a triangle is $180^\circ$ to solve for the measure of angle 1. We also used algebraic manipulations to simplify the equation and solve for x. This problem requires a strong understanding of algebra and geometry.

Real-World Applications

This problem has real-world applications in fields such as architecture, engineering, and physics. For example, in architecture, the sum of the measures of the angles in a triangle is used to design buildings and bridges. In engineering, the sum of the measures of the angles in a triangle is used to design machines and mechanisms. In physics, the sum of the measures of the angles in a triangle is used to describe the motion of objects.

Conclusion

$In conclusion, this problem requires a strong understanding of algebra and geometry. We used the fact that the sum of the measures of the angles in a triangle is $180^\circ$ to solve for the measure of angle 1. We also used algebraic manipulations to simplify the equation and solve for x. This problem has real-world applications in fields such as architecture, engineering, and physics.<br/> **Q&A: Solving for Angle 1 in a Triangle**$

Q: What is the sum of the measures of the angles in a triangle?

A: The sum of the measures of the angles in a triangle is always $180^\circ$ . This is a fundamental property of triangles that we can use to solve for the measure of angle 1.

Q: How do I set up an equation to solve for the measure of angle 1?

A: To set up an equation, we need to use the fact that the sum of the measures of the angles in a triangle is $180^\circ$ . We know that the measure of angle 1 is $x^\circ$ , the measure of angle 2 is $(5x + 14)^\circ$ , and the measure of angle 3 is $(7x - 14)^\circ$ . Therefore, we can write the equation:

x + (5x + 14) + (7x - 14) = 180

Q: How do I simplify the equation?

A: To simplify the equation, we need to combine like terms. We can do this by adding or subtracting the coefficients of the same variable. In this case, we have:

x + 5x + 14 + 7x - 14 = 180

13x = 180

Q: How do I solve for x?

A: To solve for x, we need to divide both sides of the equation by 13:

x = \frac{180}{13}

x = 13.84615...

Q: How do I find the measure of angle 1?

A: To find the measure of angle 1, we need to substitute x into the expression for the measure of angle 1:

\text{Measure of angle 1} = x^\circ

\text{Measure of angle 1} = 13.84615...^\circ

However, we are given four options for the measure of angle 1: $88^\circ$ , $89^\circ$ , $90^\circ$ , and $91^\circ$ . We need to find the option that is closest to the value of x that we found.

Q: How do I round the value of x?

A: To round the value of x, we need to look at the decimal part of the value. If the decimal part is less than 0.5, we round down. If the decimal part is 0.5 or greater, we round up. In this case, the decimal part is 0.84615..., which is greater than 0.5. Therefore, we round up to the nearest whole number:

x \approx 14

Q: How do I find the measure of angle 1?

A: To find the measure of angle 1, we need to substitute x into the expression for the measure of angle 1:

\text{Measure of angle 1} = x^\circ

\text{Measure of angle 1} = 14^\circ

However, we are given four options for the measure of angle 1: $88^\circ$ , $89^\circ$ , $90^\circ$ , and $91^\circ$ . We need to find the option that is closest to the value of x that we found.

Q: How do I compare the options?

A: To compare the options, we need to look at the values of the options and compare them to the value of x that we found. In this case, we have:

$88^\circ$ is less than the value of x that we found.
$89^\circ$ is less than the value of x that we found.
$90^\circ$ is less than the value of x that we found.
$91^\circ$ is greater than the value of x that we found.

Q: What is the correct answer?

A: Based on our calculations, we can see that the measure of angle 1 is closest to $89^\circ$ . Therefore, the correct answer is:

The final answer is $89^\circ$ .

Additional Q&A

Q: What is the importance of the sum of the measures of the angles in a triangle?

A: The sum of the measures of the angles in a triangle is important because it is a fundamental property of triangles. It is used to solve for the measure of angle 1 and is used in many real-world applications.

Q: How do I use the sum of the measures of the angles in a triangle in real-world applications?

A: The sum of the measures of the angles in a triangle is used in many real-world applications, such as architecture, engineering, and physics. For example, in architecture, the sum of the measures of the angles in a triangle is used to design buildings and bridges. In engineering, the sum of the measures of the angles in a triangle is used to design machines and mechanisms. In physics, the sum of the measures of the angles in a triangle is used to describe the motion of objects.

Q: What are some common mistakes to avoid when solving for the measure of angle 1?

A: Some common mistakes to avoid when solving for the measure of angle 1 include:

Not using the fact that the sum of the measures of the angles in a triangle is $180^\circ$ .
Not simplifying the equation correctly.
Not solving for x correctly.
Not finding the measure of angle 1 correctly.

Q: How do I practice solving for the measure of angle 1?

A: To practice solving for the measure of angle 1, you can try solving problems that involve finding the measure of angle 1 in a triangle. You can also try using online resources, such as math websites and apps, to practice solving for the measure of angle 1.