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Geometry for Nursing School Entrance Exam Study Guide (page 4)

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Triangle Rules

The following rules tend to appear more frequently on standardized tests than other rules. A typical test question follows each rule.

Example: One base angle of an isosceles triangle is 30°. Find the vertex angle.

1. Draw a picture of an isosceles triangle. Drawing it to scale helps: Since it is an isosceles triangle, draw both base angles the same size (as close to 30° as you can) and make sure the sides opposite them are the same length. Label one base angle as 30°.
2. Since the base angles are congruent, label the other base angle as 30°.
3. There are two steps needed to find the vertex angle:
• Add the two base angles together: 30° + 30° = 60°
• The sum of all three angles is 180°. To find the vertex angle, subtract the sum of the two base angles (60°) from 180°: 180° – 60° = 120°

Thus, the vertex angle is 120°.

Check: Add all three angles together to make sure their sum is 180°:

30° + 30° + 120° = 180°

Example: In the triangle shown at the right, which side is the shortest?

1. Determine the size of A, the missing angle, by adding the two known angles and then subtracting their sum from 180°: Thus, A is 44°.
2. Since A is the smallest angle, side BC, which is opposite A, is the shortest side.

Right Triangles

Example: What is the perimeter of the triangle shown at the right?

1. Since the perimeter is the sum of the lengths of the sides, we must first find the missing side. Use the Pythagorean theorem since you know this is a right triangle.
2. Substitute the given sides for two of the letters. Remember: Side c is always the hypotenuse.
3. To solve this equation, subtract 9 from both sides:
4. Then, take the square root of both sides. Thus, the missing side has a length of 4 units:
5. Adding the three sides yields a perimeter of 12:

A radical is simplified if there is no perfect square factor of the radicand. For example, is simplified because 10 has no perfect square factors. But, is not simplified because 20 has a perfect square factor of 4.

In order to simplify a radical, rewrite the radical as the product of two radicals, one of which is the largest perfect square factor of the radicand. The square root of a perfect square always simplifies to a rational number. Simplify the perfect square radical to get your final answer.

Example: Simplify .

A quadrilateral is a four-sided polygon. Following are examples of quadrilaterals that are most likely to appear on standardized tests (and in everyday life):

These quadrilaterals have something in common beside having four sides:

• Opposite sides are the same size and parallel.
• Opposite angles are the same size.

However, each quadrilateral has its own distinguishing characteristics:

The naming conventions for quadrilaterals are similar to those for triangles:

• The figure is named by the letters at its four consecutive corners, usually in alphabetic order: rectangle ABCD.
• A side is named by the letters at its ends: side AB.
• An angle is named by its vertex letter: A.

The sum of the angles of a quadrilateral is 360°: A + B + C + D = 360°

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