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**Angles of Elevation and Depression**

Geometry Angles of Elevation and Depression CONFIDENTIAL

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Warm up Find each length: X Y z CONFIDENTIAL

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**Angles of Elevation and Depression**

An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram./1 is the angle of elevation from the tower T to the plane P. Next page CONFIDENTIAL

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An angle of depression is the angle formed by a horizontal line and a line of a sight to a point below the line./2 is the angle of depression from the plane to the tower. CONFIDENTIAL

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Since horizontal lines are parallel, /1 ≅ /2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. CONFIDENTIAL

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**Classifying Angle of Elevation and Depression**

Classify each angle as an angle of elevation or angle of depression. A ) /3 /3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. B) /4 /4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. CONFIDENTIAL

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Now you try! 1) Use the diagram to classify each angle as an angle of elevation or angle of depression. 1a. / b. /6 CONFIDENTIAL

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**Finding Distance by Using Angle of Elevation**

An air traffic controller at an airport sights a plane at an angle of elevation of 41˚. The pilot reports that plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Let x be the horizontal distance between the plane and the airport. Let A represent the airport and P represent the plane P 4000 ft 41˚ A Next page x CONFIDENTIAL

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**Multiply both sides by x and divide both sides by tan41˚.**

4000 ft 41˚ A x You are given the side opposite /A, and x is the side adjacent to /A. So write a tangent ratio. Multiply both sides by x and divide both sides by tan41˚. Simplify the expression. CONFIDENTIAL

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Now you try! 2) What if….? Suppose the plane is at an altitude of 3500ft and the angle of elevation from the airport to the plane is 29˚. What is the horizontal distance between the plane and the airport? Round to the nearest foot. CONFIDENTIAL

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**Finding Distance by Using Angle of Depression**

A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7˚. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire. 90 ft T F 7˚ S x Next page CONFIDENTIAL

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**By the Alternate Interior Theorem, m/F = 7˚.**

90 ft T S x 7˚ By the Alternate Interior Theorem, m/F = 7˚. write a tangent ratio. Multiply both sides by x and divide both sides by tan7˚. Simplify the expression. CONFIDENTIAL

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Now you try! 3) Suppose the ranger sees another fire and the angle of depression to the fire is 3˚. What is the horizontal distance to this fire? Round to the nearest foot. 90 ft T S x 3˚ CONFIDENTIAL

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Aviation Application A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the base of one tower is 37˚. The angle of depression to the base of the other tower is 58 ˚. What is the distance between the two towers? Round to the nearest tenth of a kilometer. Step: 1 Let P represent the plane and let A and B represent the two towers. Let x be the distance between the towers. Next page CONFIDENTIAL

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**By the Alternate Interior Angle Theorem, m/ CAP = 58˚**

Step 2 Find y. By the Alternate Interior Angle Theorem, m/ CAP = 58˚ Next page CONFIDENTIAL

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**By the Alternate Interior Angle Theorem, m/ CBP = 37˚**

Step 3 Find Z. By the Alternate Interior Angle Theorem, m/ CBP = 37˚ Next page CONFIDENTIAL

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**So the two towers are about 1.9 km. apart.**

Step 4 Find x. x = z – y x = – = 1.9 km. So the two towers are about 1.9 km. apart. CONFIDENTIAL

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Now you try! 4) A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78˚, and the angle of depression to the second airport is 19˚. What is the distance between the airport? Round to the nearest foot. CONFIDENTIAL

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**Now some practice problems for you!**

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**Classify each angle as an angle of elevation or angle of depression.**

Assessment Classify each angle as an angle of elevation or angle of depression. 1 ) /1 /2 /3 4) /4 CONFIDENTIAL

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5) When the angle of elevation to the sun is 37˚, a flagpole casts a shadow that is 24.2 ft long. What is the height of the flagpole to the nearest foot? CONFIDENTIAL

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6) The pilot of a traffic helicopter sights an accident at an angle of depression of 18˚. The helicopter’s altitude is 1560 ft. What is the horizontal distance from the helicopter to the accident? Round to the nearest foot. CONFIDENTIAL

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7) From the top of a canyon, the angle of depression to the far side of the river is 58˚, and the angle of depression to the near side of the river is 74˚.The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter. CONFIDENTIAL

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**Angles of Elevation and Depression**

Let’s review Angles of Elevation and Depression An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram./1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of a sight to a point below the line./2 is the angle of depression from the plane to the tower. CONFIDENTIAL

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Since horizontal lines are parallel, /1 ≅ /2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. CONFIDENTIAL

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**Classifying Angle of Elevation and Depression**

Classify each angle as an angle of elevation or angle of depression. A /3 /3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. B /4 /4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. CONFIDENTIAL

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**Finding Distance by Using Angle of Elevation**

An air traffic controller at an airport sights a plane at an angle of elevation of 41˚. The pilot reports that plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Let x be the horizontal distance between the plane and the airport. Let A represent the airport and P represent the plane P 4000 ft 41˚ A Next page x CONFIDENTIAL

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**Multiply both sides by x and divide both sides by tan41˚.**

4000 ft 41˚ A x You are given the side opposite /A, and x is the side adjacent to /A. So write a tangent ratio. Multiply both sides by x and divide both sides by tan41˚. Simplify the expression. CONFIDENTIAL

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**Finding Distance by Using Angle of Depression**

A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7˚. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire. 7˚ 90 ft x Next page CONFIDENTIAL

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**By the Alternate Interior Theorem, m/F = 7˚.**

90 ft S x By the Alternate Interior Theorem, m/F = 7˚. write a tangent ratio. Multiply both sides by x and divide both sides by tan7˚. Simplify the expression. CONFIDENTIAL

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**You did a great job today!**

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