Back To Course

High School Geometry: Help and Review13 chapters | 162 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will explore angles of depression with reference to angles of elevation. Using real world examples, we will seek to understand the similarities and differences of these angles.

Imagine that you are standing about 3 ft. away from a spot on your kitchen floor. As you look down at the spot, imagine a diagonal line is formed from your eyes to the spot. This line is called your **line of sight**, or the imaginary line that forms between your eye and an object. Now, what if you look out your kitchen window in a straight horizontal line and see someone coming towards you? Would your line of sight change? Yes, it would. However, most of the time we are looking out, not down, therefore that horizontal line is always present. The only thing that does change is whether you're looking down or up. When you were looking at that spot on your kitchen floor, you were looking down. The angle that formed between this diagonal line and the straight horizontal line you used to look out your kitchen window, is called the **angle of depression**.

The **angle of depression** is the angle between the horizontal and your line of sight (when looking down).

Now, pretend that the spot on your kitchen floor could look back up at you. Would it be using the same angle? It certainly would. This angle is called **the angle of elevation**. This angle is formed when looking up. Since both the angle of depression and angle of elevation are the same, angles 'x' and 'y' in this diagram have the same measure. And since both the top and bottom lines in the diagram are parallel lines, *x* and *y* form **alternate interior angles**, or congruent angles located on different parallel lines cut by a transversal line. Which means *x* = *y*.

Therefore, the angle of depression = angle of elevation.

The angle of depression is a very important angle to certain professionals, not to mention the fact that we form these angles every time we look at something, and without even realizing it. For example, pilots land planes at certain angles of depression. People in watch towers look down at certain angles of depression. A bird swooping down at a certain angle of depression to catch the worm is another common mathematical example.

If you go outside and look up at a bird nest on your roof, you are looking at a certain angle of elevation. If you do not have a good view of the nest, you would probably step back so that you can view at a better angle. This angle depends on the height and the horizontal distance.

In order to find the angle of depression, we must have some prior understanding of **trigonometric ratios**, *sine, cosine and tangent*. Remember, these ratios only apply to right triangles.

The main ratio that we use to find the angle of depression is tangent.

The **angle of depression** may be found by using this formula: tan y = opposite/adjacent. The opposite side in this case is usually the height of the observer or height in terms of location, for example, the height of a plane in the air. The adjacent is usually the horizontal distance between the object and the observer.

**Example 1**

Pretend that you are about 5 ft. tall and, as we said earlier, you are standing 3 ft. away from a spot on your kitchen floor. What is the angle of depression? Remember tan y = opposite (your height) divided by adjacent (the distance between you and the spot).

Tangent *y* = 5/3. 'Y' = the inverse of tangent x 5/3.

As the angle of depression and the angle of elevation are equal, or angles 'x' and 'y,' we need to use a scientific calculator to find the inverse function of the tangent, which gives us the angle of depression:

59.04 degrees.

**Example 2**

Two buildings are 70 ft. apart. One building is 800 ft. tall and the other is 500 ft. tall. What is the angle of elevation from the top of the shorter building to the top of the other building?

We can see that the horizontal distance (adjacent) is 70 ft. The height however is not very obvious. Since we are only considering the tops of the buildings, we have to subtract their heights. This gives 800ft. - 500ft. = 300ft. Tangent 'y' = 300/70. Again, we need to use a scientific calculator to find the inverse of the tangent, which gives us 76.87 degrees.

The **angle of depression** is the angle formed between the horizontal and our line of sight when we look down. The **angle of elevation** is the angle formed between the horizontal and our line of sight when we look up. The **angle of depression** equals the angle of elevation because they from alternate interior angles or congruent angles located on different parallel lines cut by a transversal line. We use these angles every day when we look at objects.

The formula for calculating the **angle of depression** is:

tan Y = opposite/adjacent (or) tan Y = height/ horizontal

Sometimes we may have to subtract the height of the objects in order to get the appropriate height that we need.

**Line of sight** - the imaginary line that forms between your eye and an object

**Angle of depression** - the angle between a diagonal line (moving 'downward') and a straight horizontal line; ex. angle between line of sight when looking at spot on floor and line of sight when looking straight ahead (horizontal line)

**Angle of elevation** - the angle between a diagonal line (moving 'upward') and a straight horizontal line; differs from angle of depression in that the angle formed is above the horizontal line rather than below it

**Alternate interior angles** - congruent angles located on different parallel lines cut by a transversal line

**Trigonometric ratios** - three ratios which apply to right triangles; sine, cosine, and tangent

After this lesson, see if you can do the following:

- Describe the terms: angle of depression and angle of elevation.
- Explain how the angle of depression and angle of elevation relate to one another.
- Identify examples of the angle of depression or the angle of elevation in everyday instances.
- Calculate an angle of depression or angle of elevation.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
12 in chapter 5 of the course:

Back To Course

High School Geometry: Help and Review13 chapters | 162 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Angle of Depression: Definition & Formula 5:20
- Midpoint Theorem: Definition & Application 4:23
- Perpendicular Bisector: Definition, Theorem & Equation 5:23
- Proof by Contradiction: Definition & Examples 6:07
- Pythagorean Identities in Trigonometry: Definition & Examples
- Square Matrix: Definition & Concept 4:41
- Tetrahedral in Molecular Geometry: Definition, Structure & Examples
- Proof by Induction: Steps & Examples
- Ceva's Theorem: Applications & Examples
- Go to Triangles, Theorems and Proofs: Help and Review

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Progressive Politics & American Imperialism
- Reconstruction, Westward Expansion, Industrialization & Urbanization
- North America & the 13 Colonies
- The Renaissance & The Age of Exploration
- Algorithmic Analysis, Sorting & Searching
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Convergent Sequence: Definition, Formula & Examples
- Mauryan Empire Art & Culture
- Multi-Dimensional Arrays in C Programming: Definition & Example
- Tests for Identifying Common Gases
- Singing Lesson Plan
- Arrays & Strings in JavaScript: Conversion & String Methods
- Heuristic Methods in AI: Definition, Uses & Examples
- Quiz & Worksheet - Average & Instantaneous Rates of Change
- Quiz & Worksheet - Speed, Velocity & Acceleration
- Quiz & Worksheet - Functions & Parameters Overview
- Quiz & Worksheet - Incremental & Radical Change
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Analytical Essay Topics for Teachers
- What is Project-Based Learning? | PBL Ideas & Lesson Plans

- Corporate Finance: Help & Review
- Practicing Ethical Behavior in the Workplace
- High School Geometry Textbook
- Building Team Trust in the Workplace
- Major Events in World History Study Guide
- Plant Biology: Homework Help
- Unemployment & the Economy
- Quiz & Worksheet - Characteristics of Performance Assessments
- Quiz & Worksheet - Promotion as a Marketing Strategy
- Quiz & Worksheet - Lewis Dot Structures of Polyatomic Ions
- Quiz & Worksheet - Foreign Market Entry in Marketing
- Quiz & Worksheet - Convert Fractional Notation to Percent Notation

- Personalizing a Word Problem to Increase Understanding
- The Blue Hotel by Stephen Crane: Summary & Analysis
- Arizona Science Standards for 6th Grade
- How to Earn Kanban Certification
- International Baccalaureate vs. Advanced Placement Tests
- Fun Math Games for Kids
- ESL Content Standards in Illinois
- High School Math Games
- Jabberwocky Lesson Plan
- Difference Between Engineering Management & Project Management
- Next Generation Science Standards for Kindergarten
- South Dakota Science Standards for Kindergarten

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject