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Geometry Concepts: GED Test Prep (page 2)

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Updated on Mar 9, 2011

Complementary Angles

Two angles are complementary if the sum of their measures is equal to 90 degrees.

Complementary Angles

Supplementary Angles

Two angles are supplementary if the sum of their measures is equal to 180 degrees.

Supplementary Angles

Adjacent Angles

Adjacent angles have the same vertex, share a side, and do not overlap.

Adjacent Angles

The sum of all of the measures of adjacent angles around the same vertex is equal to 360 degrees.

Adjacent Angles

Angles of Intersecting Lines

When two lines intersect, two sets of nonadjacent angles called vertical angles are formed. Vertical angles have equal measures and are supplementary to adjacent angles.

Angles of Intersecting Lines

  • m1 = m3 and m2 = m4
  • m1 = m4 and m3 = m2
  • m1 + m2 = 180 and m2 + m3 = 180
  • m3 + m4 = 180 and m1 + m4 = 180

Bisecting Angles and Line Segments

Both angles and lines are said to be bisected when divided into two parts with equal measures.

Example

Bisecting Angles and Line Segments

Line segment AB is bisected at point C.

Bisecting Angles and Line Segments

According to the figure, A is bisected by ray AC.

Angles Formed by Parallel Lines

  • When two parallel lines are intersected by a third line, vertical angles are formed.
  • Of these vertical angles, four will be equal and acute, and four will be equal and obtuse.
  • Any combination of an acute and an obtuse angle will be supplementary.

Angles Formed by Parallel Lines

In the previous figure:

  • b, c, f, and g are all acute and equal.
  • a, d, e, and h are all obtuse and equal.
  • Also, any acute angle added to any obtuse angle will be supplementary.

Examples

mb + md = 180°

mc + me = 180°

mf + mh = 180°

mg + ma = 180°

Example

In the following figure, if m||n and a||b, what is the value of x?

Angles Formed by Parallel Lines

Solution

Because both sets of lines are parallel, you know that x° can be added to x + 10 to equal 180. The equation is thus x + x + 10 = 180.

Example

Solve for x:

Therefore, mx = 85 and the obtuse angle is equal to 180 – 85 = 95.

Angles of a Triangle

The measures of the three angles in a triangle always equal 180 degrees.

Angles of a Triangle

Exterior Angles

Exterior Angles

An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles:

  • An exterior angle and interior angle that share the same vertex are supplementary.
  • An exterior angle is equal to the sum of the nonadjacent interior angles.

Example

m1 = m3 + m5

m4 = m2 + m5

m6 = m3 + m2

The sum of the exterior angles of a triangle equal 360 degrees.

Triangles

It is possible to classify triangles into three categories based on the number of equal sides:

Triangles

It is also possible to classify triangles into three categories based on the measure of the greatest angle:

Triangles

Angle-Side Relationships

Knowing the angle-side relationships in isosceles, equilateral, and right triangles will be useful in taking the GED Exam.

  • In isosceles triangles, equal angles are opposite equal sides.
  • Angle-Side Relationships

  • In equilateral triangles, all sides are equal and all angles are equal.
  • Angle-Side Relationships

  • In a right triangle, the side opposite the right angle is called the hypotenuse. This will be the largest side of the right triangle.

Angle-Side Relationships

Pythagorean Theorem

The Pythagorean theorem is an important tool for working with right triangles. It states a2+ b2= c2, where a and b represent the legs and c represents the hypotenuse.

This theorem allows you to find the length of any side as long as you know the measure of the other two.

    Pythagorean theorem

45–45–90 Right Triangles

Pythagorean Theorem

A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle. In an isosceles right triangle:

  • The length of the hypotenuse is multiplied by the length of one of the legs of the triangle.
  • The length of each leg is multiplied by the length of the hypotenuse.

Pythagorean Theorem

30–60–90 Triangles

In a right triangle with the other angles measuring 30 and 60 degrees:

  • The leg opposite the 30-degree angle is half of the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.)
  • The leg opposite the 60-degree angle is times the length of the other leg.

Pythagorean Theorem

Example

    Pythagorean Theorem

Comparing Triangles

Triangles are said to be congruent (indicated by the symbol ≅) when they have exactly the same size and shape. Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is easy to tell if two triangles are congruent by looking. However, in geometry, you must be able to prove that the triangles are congruent.

If two triangles are congruent, one of the following three criteria must be satisfied.

    Side-Side-Side (SSS)—The side measures for both triangles are the same.
    Side-Angle-Side (SAS)—The sides and the angle between them are the same.
    Angle-Side-Angle (ASA)—Two angles and the side between them are the same

Example: Are ΔABC and ΔBCD congruent?

Given: ABD is congruent to CBD and ADB is congruent to CDB. Both triangles share side BD.

Pythagorean theorem

Step 1: Mark the given congruencies on the drawing.

Pythagorean theorem

Step 2: Determine whether this is enough information to prove the triangles are congruent.

Yes, two angles and the side between them are equal. Using the ASA rule, you can determine that triangle ABD is congruent to triangle CBD.

Polygons and Parallelograms

A polygon is a closed figure with three or more sides.

Polygons and Parallelograms

Terms Related to Polygons

Vertices are corner points, also called endpoints, of a polygon. The vertices in this polygon are A, B, C, D, E, and F.

A diagonal of a polygon is a line segment between two nonadjacent vertices. The two diagonals in this polygon are line segments BF and AE.

  • A regular polygon has sides and angles that are all equal.
  • An equiangular polygon has angles that are all equal.

Angles of a Quadrilateral

A quadrilateral is a four-sided polygon. Because a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360 degrees.

Angles of a Quadrilateral

Interior Angles

To find the sum of the interior angles of any polygon, use this formula:

S = 180(x – 2)°,with x being the number of polygon sides

Example

Find the sum of the angles in the following polygon:

Interior Angles

S = (5 – 2) × 180°

S = 3 × 180°

S = 540°

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