Amazing Angles and Shapes - Intermediate

Geometry is the study of shapes and sizes. A geometer studies the length, area and volume of various objects, and comes up with ways of understanding them better. He starts off by learning about angles and shapes, and figuring out the relationship between these ideas. This allows him to make predictions and draw conclusions about the physical world. For example, the inclination of a hill, the maximum tilt of a space shuttle before launch, and the path of a pool ball when it bounces off the side can all be expressed in terms of an angle.

Here are some tips for you to get started:

1. Draw out the geometry diagram, as it helps you get familiar with it.
2. Use known relationships about angles, especially in regular polygons.
3. Recognize special shapes like isosceles triangles, equilateral triangles or parallelograms, and use their properties.
4. Add in various lines or points that you think could be helpful.
5. If you're stuck at one part, zoom out and look at the big picture.

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What is the vertex angle (in degrees) of a 5 pointed star?

The given image doesn't allow us to get started, since it's hard to see how one can progress. Let's draw in various lines or points that could be helpful:
Draw the inner regular pentagon.

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If you're stuck at one part, zoom out and look at the big picture:
We still can't work with the vertex angle as yet, so let's look elsewhere.

Use known relationships about angles, especially in regular polygons:
The regular pentagon has a total angle of \( (5-2) \times 180 ^ \circ = 540 ^ \circ \).
Each interior angle would be \( \frac{ 540^\circ}{5} = 108^ \circ \).
The exterior angle would be \( 180 ^ \circ - 108 ^ \circ = 72 ^ \circ \).

Now, we can work with the tip of the vertex star. Recognize special shapes:
Since both sides of the star have the same length, we have an isosceles triangle. The base angle is calculated to be \(72 ^ \circ \) above.
Hence the vertex angle would be \( 180 ^ \circ - (2 \times 72 ^ \circ) = 36 ^ \circ \).

Back to Quiz: Amazing Angles and Shapes - Intermediate

Geometry problems involving angles and shapes can be classified as

• Basic information about angles: Picking up the terminology allows you to quickly understand what is being talked about. When in doubt, do a search for the term.
• Parallel Lines and Perpendicular Lines: Recognizing pairs of parallel and perpendicular lines allows you to exploit their properties. This is extremely useful when finding angles.
• Properties of Triangles: These comprise the basic building blocks of shapes. A strong grasp of this fundamental concept will allow you to develop much further.
• Identify Similar Triangles: Spotting pairs of similar triangles allows you to use the area and perimeter relations that tie them together.
• Classify Quadrilaterals: Spotting a quadrilateral and classifying it correctly allows us to utilize its special properties.
• Regular Polygons: Knowing that a polygon is regular gives us a lot of additional information for free! We know what the angles, side lengths and area have to be.