Chapter 1 Summary and Review

Key Concepts

1. The sum of the angles in a triangle is [latex]180°{.}[/latex]
2. A right triangle has one angle of [latex]90°{.}[/latex]
3. All of the angles of an equilateral triangle are equal.
4. The base angles of an isosceles triangle are equal.
5. Vertical angles are equal.
6. If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal.
7. Two triangles are congruent if they have exactly the same size and shape.
8. The altitude of an equilateral triangle divides it into two congruent right triangles.
9. In a [latex]30°-60°-90°[/latex] right triangle, the leg opposite the [latex]30°[/latex] angle is half the length of the hypotenuse.
10. Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional.
11. Similar Triangles.
    Two triangles are similar if either
    • their corresponding angles are equal, or
    • their corresponding sides are proportional.
12. If two right triangles have one pair of corresponding acute angles with the same measure, then the triangles are similar.
13. Distance Formula.
    The distance [latex]d[/latex] between two points [latex]P_{1}(x_1, y_1)[/latex] and [latex]P_{2}(x_2, y_2)[/latex] is
    [latex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/latex]
14. Any number that can be written as a quotient of two integers [latex]\dfrac{a}{b},~~ b\not=0,~~{,}[/latex] is called a rational number. The decimal form of a rational number is either a terminating decimal or a repeating decimal.
15. An irrational number is one that cannot be written as a quotient of two integers [latex]\dfrac{a}{b},~~ b\not=0,~~{.}[/latex] We cannot write down an exact decimal equivalent for an irrational number.
16. A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.
17. Circle.
    The equation for a circle of radius [latex]r[/latex] centered at the origin is
    [latex]x^2+y^2=r^2[/latex]
18. The circle [latex]x^2 + y^2 = 1{,}[/latex] which is centered at the origin and has radius [latex]1[/latex] unit, is called the unit circle.
19. Circumference of a Circle.
    The circumference of a circle of radius [latex]r[/latex] is given by
    [latex]C=2\pi r[/latex]
20. Area of a Circle.
    The area of a circle of radius [latex]r[/latex] is given by
    [latex]A=\pi r^2[/latex]