**Introduction**

Angles are fundamental to our understanding of shapes and how they fit together. While you might have come across acute, right, and obtuse angles, today we will delve deep into the world of reflex angles. They’re not as commonly discussed as the others, but they play a crucial role in various mathematical concepts and everyday situations.

**Grade Appropriateness**

Reflex angles are typically introduced between grades 4 and 7, depending on the curriculum. However, their applications and complexities grow as students progress, making them relevant through middle school and beyond.

**Math Domain**

Reflex angles fall under the domain of Geometry. Geometry deals with the sizes, shapes, properties, and dimensions of objects and structures.

**Applicable Common Core Standards**

*CCSS.MATH.CONTENT.4.MD.C.6:* Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

*CCSS.MATH.CONTENT.7.G.B.5:* Use facts about supplementary, complementary, vertical, and adjacent angles to write and solve simple equations for a given unknown angle in a figure.

**Definition of the Topic**

A reflex angle is any angle greater than 180° but less than 360°. For example, 240°, 270°, and 330° are reflex angles.

There is a reflex angle for every acute or obtuse angle.

**Key Concepts**

** Degree Measurement:** Angles are measured in degrees.

** Relationship with Interior Angle:** For any given line segment, the reflex angle and its corresponding interior angle (whether acute, right, or obtuse) will always add up to 360°.

**Discussion with Illustrative Examples**

A reflex angle is the type of angle that measures exactly more than 180^{o} but less than 360^{o}.

**Measuring a Reflex Angle Using a Protractor**

The following are the easy steps to measure a reflex angle using a protractor.

**Step 1:** Place the vertex of the angle in the middle of the protractor and measure the inner angle.

The inner angle is 140^{o}. When subtracted to 360^{o}, this reflex angle is 220^{o}.

**Step 2: **Subtract the measurement of the inner angle from 360°.

Here, we have 360°-140° to calculate the reflex angle. Hence, the reflex angle of 140° is 220°.

**Calculating Reflex Angles**

In calculating the measure of a reflex angle given the interior angle (θ), we follow the formula,

Reflex Angle of θ=360°-θ, where 0<θ<180°

Hence, if an interior angle is 60∘, then the reflex angle would be 360°-60°=300°.

**Examples with Solution**

**Example 1**

Find the reflex angle of 95°.

**Solution**

To find the measure of the reflex angle, we must subtract 95° from 360°.

Reflex Angle of 95°=360°-95°

Reflex Angle of 95°=265°

Thus, the reflex angle of 95° is 265°.

**Example 2**

Given an interior angle of 140 °, find the reflex angle.

**Solution**

Reflex Angle of 140°=360°-140°

Reflex Angle of 140°=220°

Therefore, the reflex angle of 140° is 220°.

**Example 3 **

If a reflex angle measures 290°, what is the measure of its corresponding interior angle?

**Solution **

Interior Angle of 290°=360°-290°

Interior Angle of 290°=70°

Hence, a reflex angle measuring 290° has a corresponding interior angle that measures 70°.

**Real-life Application with Solution**

The following are a few examples of reflex angles around us in real life.

1. Suppose you open a book slightly; the smaller angle between the pages is acute or obtuse. Now, think of the larger angle on the other side of the pages. That larger angle is the reflex angle!

2. Imagine you are on a merry-go-round. You start at a horse and decide to move to the horse directly opposite you. You have covered half the circle or 180°. Now, if you were to go just one more horse past the opposite one, you’d cover a reflex angle!

3. Understanding reflex angles is helpful in navigation, engineering, and even in understanding the movement of planets.

**Practice Test**

1. Is an angle that measures 120° a reflex angle?

2. Given an interior angle of 160°, find the reflex angle.

3. What is the reflex angle for an interior angle measuring 90°?

4. If a reflex angle measures 200°, what is the measure of its corresponding interior angle?

5. When you know a reflex angle, how can you find its corresponding interior angle?

6. A clock’s hands are at 4 and 12. What reflex angle do they make?

7. If a pizza slice has an interior angle of 80∘, what is the reflex angle outside that slice?

*Answers:*

1. No. An angle measuring 120° is not a reflex angle but an obtuse one.

2. Reflex Angle of 160°=360°-160°=200°

3. Reflex Angle of 90°=360°-90°=270°

4. Interior Angle of 200°=360°-200°=160°

5. To find the corresponding interior angle of a reflex angle, subtract the given reflex angle from 360°.

6. Since the clock has 12 sectors, the angle between two numbers is 30°. Thus, the angle between the hour and minute hands at 4:00 is 30°×4=120°. Calculating its reflex angle, we have,

Reflex Angle of 120°=360°-120°

Reflex Angle of 120°=240°

7. Reflex Angle of 80°=360°-80°=280°

**Frequently Asked Questions (FAQs) **

**Is ****180°****considered a reflex angle?**

No, 180° is known as a straight angle.

**Can a triangle have a reflex angle?**

No. The sum of the inner angles in a triangle is always 180 degrees, and a reflex angle has a measurement greater than 180° but less than 360°. Reflex angles are thus impossible for a triangle to have.

**Why is it called a reflex angle?**

The term “reflex” comes from Latin, meaning “bent back.” It refers to the angle bending back on itself.

**Do circles have reflex angles?**

Circles do not have angles, but sectors of circles can form reflex angles when the arc is more than half the circle.

**How is a reflex angle different from an obtuse angle?**

An obtuse angle has a measurement greater than 90° but less than 180°, while a reflex angle has a measurement greater than

180° but less than 360°.

Remember, understanding angles, including reflex angles, will significantly help you in geometric calculations and understanding the world around you. Happy learning!

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