2 Assertion (A): AABC APQR Such That ZA = 65°, ZC = 60°. Hence ZQ= 55°. Reason (R): Sum Of All Angles Of A Triangle Is 180°.​

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Introduction

In the world of mathematics, particularly in geometry, understanding the properties of triangles is crucial. One of the fundamental properties of triangles is the sum of all angles, which is a constant value. In this article, we will explore this property and use it to solve a problem involving a triangle with specific angle measurements.

The Problem

We are given a triangle AABC with angle ZA measuring 65° and angle ZC measuring 60°. We are also given a triangle APQR with angle ZQ measuring 55°. Our task is to use the property of the sum of all angles in a triangle to find the measure of angle ZP.

The Property of Sum of Angles in a Triangle

The property states that the sum of all angles in a triangle is always 180°. This is a fundamental concept in geometry and is used to solve a wide range of problems involving triangles.

Applying the Property to the Given Triangles

Let's start by applying the property to triangle AABC. We know that the sum of all angles in a triangle is 180°, so we can write the equation:

ZA + ZB + ZC = 180°

We are given that ZA = 65° and ZC = 60°, so we can substitute these values into the equation:

65° + ZB + 60° = 180°

Simplifying the equation, we get:

ZB = 55°

Now, let's apply the property to triangle APQR. We know that the sum of all angles in a triangle is 180°, so we can write the equation:

ZA + ZB + ZC = 180°

We are given that ZQ = 55°, so we can substitute this value into the equation:

ZA + ZB + 55° = 180°

Simplifying the equation, we get:

ZA + ZB = 125°

Finding the Measure of Angle ZP

Now that we have found the measure of angle ZB, we can use it to find the measure of angle ZP. We know that the sum of all angles in a triangle is 180°, so we can write the equation:

ZA + ZB + ZP = 180°

We are given that ZA = 65° and ZB = 55°, so we can substitute these values into the equation:

65° + 55° + ZP = 180°

Simplifying the equation, we get:

ZP = 60°

Conclusion

In this article, we used the property of the sum of all angles in a triangle to solve a problem involving two triangles with specific angle measurements. We found the measure of angle ZB in triangle AABC and used it to find the measure of angle ZP in triangle APQR. This problem demonstrates the importance of understanding the properties of triangles in geometry.

Key Takeaways

  • The sum of all angles in a triangle is always 180°.
  • We can use this property to solve problems involving triangles with specific angle measurements.
  • Understanding the properties of triangles is crucial in geometry.

Further Reading

If you want to learn more about the properties of triangles, I recommend checking out the following resources:

References

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

A: The sum of all angles in a triangle is always 180°. This is a fundamental property of triangles and is used to solve a wide range of problems involving triangles.

Q: How do I use the property of sum of angles in a triangle to solve a problem?

A: To use the property of sum of angles in a triangle, you need to know the measure of at least two angles in the triangle. You can then use the equation:

ZA + ZB + ZC = 180°

to find the measure of the third angle.

Q: What if I don't know the measure of any angles in the triangle?

A: If you don't know the measure of any angles in the triangle, you can use the property of sum of angles in a triangle to find the measure of one angle. For example, if you know that the sum of all angles in a triangle is 180°, you can use the equation:

ZA + ZB + ZC = 180°

to find the measure of one angle.

Q: Can I use the property of sum of angles in a triangle to find the measure of an angle in a right triangle?

A: Yes, you can use the property of sum of angles in a triangle to find the measure of an angle in a right triangle. In a right triangle, one angle is 90°, so you can use the equation:

ZA + ZB + 90° = 180°

to find the measure of the other two angles.

Q: What if I have a triangle with three right angles?

A: If you have a triangle with three right angles, it is not a valid triangle. A triangle must have at least one angle that is less than 180°.

Q: Can I use the property of sum of angles in a triangle to find the measure of an angle in an obtuse triangle?

A: Yes, you can use the property of sum of angles in a triangle to find the measure of an angle in an obtuse triangle. In an obtuse triangle, one angle is greater than 90°, so you can use the equation:

ZA + ZB + ZC = 180°

to find the measure of the other two angles.

Q: What if I have a triangle with two right angles and one obtuse angle?

A: If you have a triangle with two right angles and one obtuse angle, it is not a valid triangle. A triangle must have at least one angle that is less than 180°.

Q: Can I use the property of sum of angles in a triangle to find the measure of an angle in an equilateral triangle?

A: Yes, you can use the property of sum of angles in a triangle to find the measure of an angle in an equilateral triangle. In an equilateral triangle, all three angles are equal, so you can use the equation:

ZA + ZB + ZC = 180°

to find the measure of one angle.

Q: What if I have a triangle with three equal angles?

A: If you have a triangle with three equal angles, it is an equilateral triangle.

Q: Can I use the property of sum of angles in a triangle to find the measure of an angle in an isosceles triangle?

A: Yes, you can use the property of sum of angles in a triangle to find the measure of an angle in an isosceles triangle. In an isosceles triangle, two angles are equal, so you can use the equation:

ZA + ZB + ZC = 180°

to find the measure of one angle.

Q: What if I have a triangle with two equal angles and one angle that is not equal to the other two?

A: If you have a triangle with two equal angles and one angle that is not equal to the other two, it is an isosceles triangle.

Conclusion

In this article, we have answered some frequently asked questions about triangle angles. We have discussed the property of sum of angles in a triangle and how it can be used to solve problems involving triangles. We have also discussed some special types of triangles, such as right triangles, obtuse triangles, equilateral triangles, and isosceles triangles.