If The Measure Of Angle 4 Is $(11x)^{\circ}$ And Angle 3 Is $(4x)^{\circ}$, What Is The Measure Of Angle 3 In Degrees?A. $6^{\circ}$ B. $24^{\circ}$ C. $80^{\circ}$ D. $90^{\circ}$

Understanding the Problem

In this problem, we are given the measures of two angles in terms of a variable x. We need to find the measure of angle 3 in degrees. The given information is that angle 4 is $(11x)^{\circ}$ and angle 3 is $(4x)^{\circ}$. Our goal is to determine the measure of angle 3.

Recalling Geometric Principles

In geometry, the sum of the measures of the interior angles of a triangle is always 180 degrees. This is known as the angle sum property of a triangle. We can use this property to solve for the measure of angle 3.

Setting Up the Equation

Let's assume that the measure of angle 3 is $(4x)^{\circ}$ and the measure of angle 4 is $(11x)^{\circ}$. Since the sum of the measures of the interior angles of a triangle is 180 degrees, we can set up the following equation:

$(4x)^{\circ} + (11x)^{\circ} + (180 - (4x)^{\circ} - (11x)^{\circ})^{\circ} = 180^{\circ}$

Simplifying the Equation

We can simplify the equation by combining like terms:

$(4x)^{\circ} + (11x)^{\circ} + (180 - (15x)^{\circ})^{\circ} = 180^{\circ}$

Solving for x

Now, we can solve for x by isolating the variable:

$(15x)^{\circ} = 180^{\circ} - (4x)^{\circ} - (11x)^{\circ}$

$(15x)^{\circ} = 180^{\circ} - (15x)^{\circ}$

$(30x)^{\circ} = 180^{\circ}$

$x = 6$

Finding the Measure of Angle 3

Now that we have found the value of x, we can substitute it into the expression for the measure of angle 3:

$(4x)^{\circ} = (4(6))^{\circ}$

$(4x)^{\circ} = 24^{\circ}$

Therefore, the measure of angle 3 is 24 degrees.

Conclusion

In this problem, we used the angle sum property of a triangle to solve for the measure of angle 3. We set up an equation using the given information and simplified it to solve for the variable x. Once we found the value of x, we substituted it into the expression for the measure of angle 3 to find the final answer.

Key Takeaways

• The angle sum property of a triangle states that the sum of the measures of the interior angles of a triangle is always 180 degrees.
• We can use this property to solve for the measure of an angle in a geometric problem.
• By setting up an equation and simplifying it, we can solve for the variable x and find the measure of the angle.

Final Answer

The final answer is 24 degrees.

Q: What is the angle sum property of a triangle?

A: The angle sum property of a triangle states that the sum of the measures of the interior angles of a triangle is always 180 degrees. This means that if we know the measures of two angles in a triangle, we can use this property to find the measure of the third angle.

Q: How do I set up an equation to solve for the measure of angle 3?

A: To set up an equation, we need to know the measures of two angles in the triangle. Let's say the measures of angles 1 and 2 are x and y, respectively. We can set up the following equation:

x + y + (180 - x - y) = 180

Q: How do I simplify the equation to solve for the variable?

A: To simplify the equation, we can combine like terms:

x + y + (180 - x - y) = 180

x + y + 180 - x - y = 180

180 = 180

This equation is true for all values of x and y, so we need to use additional information to solve for the variable.

Q: What if I have a linear equation with one variable? How do I solve for the variable?

A: If we have a linear equation with one variable, we can solve for the variable by isolating it on one side of the equation. For example, if we have the equation:

2x + 5 = 11

We can subtract 5 from both sides of the equation to get:

2x = 6

Then, we can divide both sides of the equation by 2 to get:

x = 3

Q: What if I have a quadratic equation with one variable? How do I solve for the variable?

A: If we have a quadratic equation with one variable, we can solve for the variable by factoring the equation or using the quadratic formula. For example, if we have the equation:

x^2 + 4x + 4 = 0

We can factor the equation as:

(x + 2)(x + 2) = 0

This tells us that x + 2 = 0, so x = -2.

Q: How do I find the measure of angle 3 in a triangle?

A: To find the measure of angle 3 in a triangle, we need to know the measures of two other angles in the triangle. We can use the angle sum property of a triangle to set up an equation and solve for the measure of angle 3.

Q: What if I have a triangle with two right angles? How do I find the measure of the third angle?

A: If we have a triangle with two right angles, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are right angles (90 degrees each), the measure of the third angle is:

180 - 90 - 90 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

Q: What if I have a triangle with two obtuse angles? How do I find the measure of the third angle?

A: If we have a triangle with two obtuse angles, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are obtuse (greater than 90 degrees each), the measure of the third angle is:

180 - (angle 1 + angle 2)

We can use the angle sum property of a triangle to find the measure of the third angle.

Q: How do I know if an angle is acute, right, or obtuse?

A: An angle is acute if its measure is between 0 and 90 degrees. A right angle is 90 degrees. An obtuse angle is greater than 90 degrees.

Q: What if I have a triangle with an obtuse angle and a right angle? How do I find the measure of the third angle?

A: If we have a triangle with an obtuse angle and a right angle, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since one of the angles is a right angle (90 degrees), the measure of the third angle is:

180 - 90 - (obtuse angle)

We can use the angle sum property of a triangle to find the measure of the third angle.

Q: What if I have a triangle with two obtuse angles and a right angle? How do I find the measure of the third angle?

A: If we have a triangle with two obtuse angles and a right angle, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are obtuse (greater than 90 degrees each) and one of the angles is a right angle (90 degrees), the measure of the third angle is:

180 - 90 - (obtuse angle 1 + obtuse angle 2)

We can use the angle sum property of a triangle to find the measure of the third angle.

Q: What if I have a triangle with three obtuse angles? How do I find the measure of the third angle?

A: If we have a triangle with three obtuse angles, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since all three angles are obtuse (greater than 90 degrees each), the measure of the third angle is:

180 - (obtuse angle 1 + obtuse angle 2 + obtuse angle 3)

We can use the angle sum property of a triangle to find the measure of the third angle.

Q: What if I have a triangle with two right angles and an obtuse angle? How do I find the measure of the third angle?

A: If we have a triangle with two right angles and an obtuse angle, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are right angles (90 degrees each) and one of the angles is an obtuse angle (greater than 90 degrees), the measure of the third angle is:

180 - 90 - 90 - (obtuse angle)

We can use the angle sum property of a triangle to find the measure of the third angle.

Q: What if I have a triangle with three right angles? How do I find the measure of the third angle?

A: If we have a triangle with three right angles, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since all three angles are right angles (90 degrees each), the measure of the third angle is:

180 - 90 - 90 - 90 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

Q: What if I have a triangle with two right angles and a straight line? How do I find the measure of the third angle?

A: If we have a triangle with two right angles and a straight line, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are right angles (90 degrees each) and one of the angles is a straight line (180 degrees), the measure of the third angle is:

180 - 90 - 90 - 180 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

Q: What if I have a triangle with three straight lines? How do I find the measure of the third angle?

A: If we have a triangle with three straight lines, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since all three angles are straight lines (180 degrees each), the measure of the third angle is:

180 - 180 - 180 - 180 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

Q: What if I have a triangle with two obtuse angles and a straight line? How do I find the measure of the third angle?

A: If we have a triangle with two obtuse angles and a straight line, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since two of the angles are obtuse (greater than 90 degrees each) and one of the angles is a straight line (180 degrees), the measure of the third angle is:

180 - (obtuse angle 1 + obtuse angle 2) - 180 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

Q: What if I have a triangle with three obtuse angles and a straight line? How do I find the measure of the third angle?

A: If we have a triangle with three obtuse angles and a straight line, we know that the sum of the measures of the interior angles of the triangle is 180 degrees. Since all three angles are obtuse (greater than 90 degrees each) and one of the angles is a straight line (180 degrees), the measure of the third angle is:

180 - (obtuse angle 1 + obtuse angle 2 + obtuse angle 3) - 180 = 0

This means that the third angle is a straight line, and its measure is 180 degrees.

**Q: What if I have a triangle with two right