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Unit 6

MM-150 SURVEY OF MATHEMATICS – Jody Harris

Geometry

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Basic Terms

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Angles An angle is the union of two rays with a

common endpoint; denoted The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle:

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Angles The measure of an angle is the amount of

rotation from its initial to its terminal side. Angles are classified by their degree

measurement. Right Angle is 90o

Acute Angle is less than 90o

Obtuse Angle is greater than 90o but less than 180o

Straight Angle is 180o

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Types of Angles

____ Straight Angle____ Right Angle____ Obtuse Angle ____ Acute Angle

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Types of Angles Adjacent Angles - angles that have a common

vertex and a common side but no common interior points.

Complementary Angles - two angles whose sum of their measures is 90 degrees.

Supplementary Angles - two angles whose sum of their measures is 180 degrees.

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

D

ABC and CBD

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

D

ABC and CBD

ABC + CBD = 180

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

D

ABC and CBD

ABC + CBD = 180

x

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

D

ABC and CBD

ABC + CBD = 180

x4x

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

Dx4x

ABC and CBD

ABC + CBD = 180

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

Dx4x

ABC and CBD

4x + x = 180 ABC + CBD = 180

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

Dx4x

ABC and CBD

4x + x = 1805x = 180

ABC + CBD = 180

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

Dx4x

ABC and CBD

4x + x = 1805x = 180x = 36

ABC + CBD = 180

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Example

If are supplementary and the measure of ABC is 4 times larger than CBD, determine the measure of each angle.

A B

C

Dx4x

ABC and CBD

4x + x = 1805x = 180x = 36

ABC + CBD = 180

ABC = 4x = 4(36) = 144

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ExampleFind x. Assume that angle 1 and angle 2 are complementary angles.

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ExampleFind x. Assume that angle 1 and angle 2 are complementary angles.

x + 5x + 3 = 90

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ExampleFind x. Assume that angle 1 and angle 2 are complementary angles.

x + 5x + 3 = 906x + 3 = 90

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ExampleFind x. Assume that angle 1 and angle 2 are complementary angles.

x + 5x + 3 = 906x + 3 = 90

6x = 87

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ExampleFind x. Assume that angle 1 and angle 2 are complementary angles.

x + 5x + 3 = 906x + 3 = 90

6x = 87x = 14.5

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More definitions Vertical angles are the nonadjacent angles

formed by two intersecting straight lines. Vertical angles have the same measure. A line that intersects two different lines, at two

different points is called a transversal.

Special angles are given to the angles formed by a transversal crossing two parallel lines.

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Special Names

5 6

1 24

87

3

One interior and one exterior angle on the same side of the transversal–have the same measure

Corresponding angles

Exterior angles on the opposite sides of the transversal–have the same measure

Alternate exterior angles

Interior angles on the opposite side of the transversal–have the same measure

Alternate interior angles

5 6

1 24

87

3

5 6

1 24

87

3

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Types of Triangles

Acute TriangleAll angles are acute.

Obtuse TriangleOne angle is obtuse.

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Types of Triangles continuedRight TriangleOne angle is a right angle.

Isosceles TriangleTwo equal sides.Two equal angles.

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Types of Triangles continuedEquilateral TriangleThree equal sides. Three equal angles (60º) each.

Scalene TriangleNo two sides are equal in length.

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Similar Figures Two polygons are similar if their corresponding

angles have the same measure and the lengths of their corresponding sides are in proportion.

4

3

4

6

6 6

9

4.5

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Similar Figures Two polygons are similar if their corresponding

angles have the same measure and the lengths of their corresponding sides are in proportion.

4

3

4

6

6 6

9

4.5

Put brown inside green

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Example Catherine Johnson wants to measure the height of

a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse?

x

75 125

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Quadrilaterals

Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360o.

Quadrilaterals may be classified according to their characteristics.

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Classifications Trapezoid

Two sides are parallel.

Parallelogram

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

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Classifications continued Rhombus

Both pairs of opposite sides are parallel. The four sides are equal in length.

Rectangle

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.

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Classifications continued Square

Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

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Formulas

P = s1 + s2 + b1 + b2

P = s1 + s2 + s3

P = 2b + 2w

P = 4s

P = 2l + 2w

Perimeter

Trapezoid

Triangle

A = bhParallelogram

A = s2Square

A = lwRectangle

AreaFigure

12A bh

11 22 ( )A h b b