### Complementary Angles

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

### Supplementary Angles

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

### Adjacent Angles

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

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

### 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.

- 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**

Line segment *AB* is bisected at point *C*.

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.

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.

Examplesm

b+ md= 180°m

c+ me= 180°m

f+ mh= 180°m

g+ ma= 180°

ExampleIn the following figure, if

m||nanda||b, what is the value ofx?

SolutionBecause both sets of lines are parallel, you know that

x° can be added to x + 10 to equal 180. The equation is thusx+x+ 10 = 180.

ExampleSolve for

x:

Therefore, m

x= 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.

### 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.

Examplem1 = 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:

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

### 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.
- In equilateral triangles, all sides are equal and all angles are equal.
- In a right triangle, the side opposite the right angle is called the
**hypotenuse**. This will be the largest side of the right triangle.

### Pythagorean Theorem

The **Pythagorean theorem** is an important tool for working with right triangles. It states *a*^{2}+ *b*^{2}= *c*^{2}, 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.

### 45–45–90 Right Triangles

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.

### 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.

**Example**

### 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 ΔABCand ΔBCDcongruent?Given:

ABDis congruent toCBDandADBis congruent toCDB. Both triangles share sideBD.

Step 1:Mark the given congruencies on the drawing.

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

ABDis congruent to triangleCBD.

### Polygons and Parallelograms

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

### 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.

### 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

ExampleFind the sum of the angles in the following polygon:

S= (5 – 2) × 180°

S= 3 × 180°

S= 540°

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