{ \Delta JKL$}$ Has { J = 7, K = 11$}$, And { M \angle J = 18^{\circ}$}$. Complete The Statements To Determine All Possible Measures Of Angle K.Triangle JKL Meets The { \square$}$ Criteria, Which Means It Is The

Introduction

In geometry, triangles are fundamental shapes that have been studied extensively. One of the key aspects of triangles is the relationship between their angles. In this article, we will explore how to determine the measures of angles in a triangle, specifically when given some information about the triangle's sides and angles. We will use the given information about triangle JKL to complete the statements and determine all possible measures of angle K.

Understanding the Given Information

We are given that triangle JKL has the following properties:

• Side lengths: $j = 7$ and $k = 11$
• Angle measure: $m \angle J = 18^{\circ}$

The triangle also meets the $\square$ criteria, which means it is a right-angled triangle.

Determining Angle Measures

To determine the measures of angle K, we need to use the properties of right-angled triangles. In a right-angled triangle, the sum of the measures of the two acute angles is always $90^{\circ}$. We can use this property to find the measure of angle K.

Step 1: Find the Measure of Angle K

Since triangle JKL is a right-angled triangle, we know that the sum of the measures of the two acute angles is $90^{\circ}$. We can use this information to find the measure of angle K.

Let $m \angle K = x$. Then, we know that $m \angle J + m \angle K = 90^{\circ}$.

Substituting the given value of $m \angle J = 18^{\circ}$, we get:

$$18^{\circ} + x = 90^{\circ}$$

Solving for $x$, we get:

$$x = 72^{\circ}$$

Therefore, the measure of angle K is $72^{\circ}$.

Step 2: Find the Measure of Angle L

Now that we have found the measure of angle K, we can use the fact that the sum of the measures of the three angles in a triangle is always $180^{\circ}$ to find the measure of angle L.

Let $m \angle L = y$. Then, we know that $m \angle J + m \angle K + m \angle L = 180^{\circ}$.

Substituting the values of $m \angle J = 18^{\circ}$ and $m \angle K = 72^{\circ}$, we get:

$$18^{\circ} + 72^{\circ} + y = 180^{\circ}$$

Solving for $y$, we get:

$$y = 90^{\circ}$$

Therefore, the measure of angle L is $90^{\circ}$.

Conclusion

In this article, we have used the properties of right-angled triangles to determine the measures of angles in triangle JKL. We have found that the measure of angle K is $72^{\circ}$ and the measure of angle L is $90^{\circ}$. These results demonstrate the importance of understanding the properties of triangles and how to use them to solve problems.

Key Takeaways

• The sum of the measures of the two acute angles in a right-angled triangle is always $90^{\circ}$.
• The sum of the measures of the three angles in a triangle is always $180^{\circ}$.
• Using the properties of triangles, we can determine the measures of angles in a triangle.

Further Reading

For more information on triangles and their properties, see the following resources:

• Triangle Properties
• Right-Angled Triangles
• Angle Measures

Glossary

• Acute angle: An angle whose measure is less than $90^{\circ}$.
• Right-angled triangle: A triangle with one angle that measures $90^{\circ}$.
• Sum of angles: The sum of the measures of the three angles in a triangle.
  Frequently Asked Questions (FAQs) =====================================

Q: What is the relationship between the side lengths and angle measures in a triangle?

A: In a triangle, the side lengths and angle measures are related through the properties of the triangle. For example, in a right-angled triangle, the side lengths are related to the angle measures through the Pythagorean theorem.

Q: How do I determine the measure of an angle in a triangle?

A: To determine the measure of an angle in a triangle, you can use the properties of the triangle, such as the sum of the measures of the three angles being $180^{\circ}$, or the relationship between the side lengths and angle measures.

Q: What is the difference between an acute angle and a right angle?

A: An acute angle is an angle whose measure is less than $90^{\circ}$, while a right angle is an angle whose measure is exactly $90^{\circ}$.

Q: How do I find the measure of an angle in a right-angled triangle?

A: To find the measure of an angle in a right-angled triangle, you can use the fact that the sum of the measures of the two acute angles is $90^{\circ}$.

Q: What is the relationship between the measures of the three angles in a triangle?

A: The sum of the measures of the three angles in a triangle is always $180^{\circ}$.

Q: How do I determine the measure of an angle in a triangle when given the measures of the other two angles?

A: To determine the measure of an angle in a triangle when given the measures of the other two angles, you can use the fact that the sum of the measures of the three angles is $180^{\circ}$.

Q: What is the significance of the $\square$ criteria in a triangle?

A: The $\square$ criteria in a triangle refers to the fact that the triangle is a right-angled triangle, meaning that one of its angles measures $90^{\circ}$.

Q: How do I use the properties of triangles to solve problems?

A: To use the properties of triangles to solve problems, you can apply the properties of the triangle, such as the sum of the measures of the three angles being $180^{\circ}$, or the relationship between the side lengths and angle measures.

Q: What are some common properties of triangles that I should know?

A: Some common properties of triangles that you should know include:

• The sum of the measures of the three angles in a triangle is always $180^{\circ}$.
• The sum of the measures of the two acute angles in a right-angled triangle is always $90^{\circ}$.
• The relationship between the side lengths and angle measures in a triangle.

Q: How do I apply the properties of triangles to real-world problems?

A: To apply the properties of triangles to real-world problems, you can use the properties of the triangle to solve problems in fields such as architecture, engineering, and physics.

Q: What are some common applications of triangles in real-world problems?

A: Some common applications of triangles in real-world problems include:

• Building design and construction
• Bridge design and construction
• Aircraft design and construction
• Medical imaging and diagnostics

Q: How do I use triangles to solve problems in mathematics?

A: To use triangles to solve problems in mathematics, you can apply the properties of the triangle, such as the sum of the measures of the three angles being $180^{\circ}$, or the relationship between the side lengths and angle measures.

Q: What are some common mathematical concepts that involve triangles?

A: Some common mathematical concepts that involve triangles include:

• Geometry
• Trigonometry
• Algebra
• Calculus

Q: How do I use triangles to solve problems in science and engineering?

A: To use triangles to solve problems in science and engineering, you can apply the properties of the triangle, such as the sum of the measures of the three angles being $180^{\circ}$, or the relationship between the side lengths and angle measures.

Q: What are some common scientific and engineering applications of triangles?

A: Some common scientific and engineering applications of triangles include:

• Building design and construction
• Bridge design and construction
• Aircraft design and construction
• Medical imaging and diagnostics