C) In The Following Triangle, Calculate The XE Measure And Indicated In The Graph. 80 ° And 15 40 ° X

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Understanding the Problem

In the given triangle, we are asked to calculate the measure of angle XE. The triangle has two known angles: 80° and 15°, and the third angle is represented by X. Our goal is to find the measure of angle X.

The Angle Sum Property of a Triangle

The angle sum property of a triangle states that the sum of the measures of the three interior angles of a triangle is always 180°. This property can be expressed as:

m∠A + m∠B + m∠C = 180°

where m∠A, m∠B, and m∠C are the measures of the three interior angles of the triangle.

Applying the Angle Sum Property

We can apply the angle sum property to the given triangle to find the measure of angle X. Let's denote the measure of angle X as x. We know that the measures of the other two angles are 80° and 15°. Therefore, we can write:

m∠X + 80° + 15° = 180°

Simplifying the Equation

To solve for x, we need to simplify the equation by combining the constants on the left-hand side:

x + 95° = 180°

Subtracting 95° from Both Sides

Now, we can subtract 95° from both sides of the equation to isolate x:

x = 180° - 95°

Calculating the Value of x

Finally, we can calculate the value of x by evaluating the expression:

x = 85°

Conclusion

Therefore, the measure of angle XE is 85°. This is the solution to the problem.

Visualizing the Triangle

Here is a visual representation of the triangle with the calculated angle XE:

  A (80°)
 / \
/   \
B (15°)
 \ /
  XE (85°)

Key Takeaways

  • The angle sum property of a triangle states that the sum of the measures of the three interior angles of a triangle is always 180°.
  • We can apply the angle sum property to find the measure of an unknown angle in a triangle.
  • By simplifying the equation and isolating the unknown angle, we can calculate its measure.

Real-World Applications

The concept of the angle sum property of a triangle has numerous real-world applications in various fields, such as:

  • Architecture: In designing buildings, architects use the angle sum property to ensure that the angles of the roof and walls are correct.
  • Engineering: Engineers use the angle sum property to design and build structures, such as bridges and tunnels.
  • Navigation: Pilots and sailors use the angle sum property to navigate and determine their position.

Practice Problems

Here are some practice problems to help you reinforce your understanding of the angle sum property of a triangle:

  1. In a triangle, the measures of two angles are 30° and 60°. Find the measure of the third angle.
  2. In a triangle, the measures of two angles are 90° and 45°. Find the measure of the third angle.
  3. In a triangle, the measures of two angles are 120° and 30°. Find the measure of the third angle.

Answer Key

  1. 90°
  2. 45°
  3. 30°
    Frequently Asked Questions (FAQs) about the Angle Sum Property of a Triangle ====================================================================================

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 three interior angles of a triangle is always 180°.

Q: How do I apply the angle sum property to find the measure of an unknown angle?

A: To apply the angle sum property, you need to know the measures of two angles in the triangle. Let's denote the measure of the unknown angle as x. You can then write an equation using the angle sum property:

m∠A + m∠B + m∠C = 180°

where m∠A, m∠B, and m∠C are the measures of the three interior angles of the triangle.

Q: What if I have a triangle with two right angles? Can I still apply the angle sum property?

A: Yes, you can still apply the angle sum property to a triangle with two right angles. In this case, the sum of the measures of the two right angles is 180°, and the measure of the third angle is 0°.

Q: Can I apply the angle sum property to a triangle with obtuse angles?

A: Yes, you can apply the angle sum property to a triangle with obtuse angles. However, you need to be careful when working with obtuse angles, as they are greater than 90°.

Q: What if I have a triangle with two obtuse angles? Can I still apply the angle sum property?

A: Yes, you can still apply the angle sum property to a triangle with two obtuse angles. However, the sum of the measures of the two obtuse angles will be greater than 180°.

Q: Can I apply the angle sum property to a triangle with a straight angle?

A: Yes, you can apply the angle sum property to a triangle with a straight angle. In this case, the sum of the measures of the two angles is 180°, and the measure of the third angle is 0°.

Q: What if I have a triangle with a reflex angle? Can I still apply the angle sum property?

A: No, you cannot apply the angle sum property to a triangle with a reflex angle. A reflex angle is greater than 180°, and the angle sum property does not apply to triangles with reflex angles.

Q: Can I apply the angle sum property to a triangle with a zero angle?

A: No, you cannot apply the angle sum property to a triangle with a zero angle. A zero angle is not a valid angle, and the angle sum property does not apply to triangles with zero angles.

Q: What if I have a triangle with two congruent angles? Can I still apply the angle sum property?

A: Yes, you can still apply the angle sum property to a triangle with two congruent angles. In this case, the sum of the measures of the two congruent angles will be equal to the sum of the measures of the two other angles.

Q: Can I apply the angle sum property to a triangle with a right angle and a straight angle?

A: No, you cannot apply the angle sum property to a triangle with a right angle and a straight angle. A right angle is 90°, and a straight angle is 180°. The sum of the measures of the two angles is 270°, which is greater than 180°.

Q: What if I have a triangle with a right angle and an obtuse angle? Can I still apply the angle sum property?

A: Yes, you can still apply the angle sum property to a triangle with a right angle and an obtuse angle. In this case, the sum of the measures of the two angles will be greater than 180°.

Q: Can I apply the angle sum property to a triangle with a zero angle and a straight angle?

A: No, you cannot apply the angle sum property to a triangle with a zero angle and a straight angle. A zero angle is not a valid angle, and a straight angle is 180°. The sum of the measures of the two angles is 180°, which is not a valid equation.

Conclusion

The angle sum property of a triangle is a fundamental concept in geometry that states that the sum of the measures of the three interior angles of a triangle is always 180°. By applying this property, you can find the measure of an unknown angle in a triangle. However, there are some special cases where the angle sum property does not apply, such as triangles with reflex angles or zero angles.