basic definitions of geometry - ms. talhami ... 1 basic definitions of geometry labeling: now, let's
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Basic Definitions of Geometry
Labeling: Now, let's take a look at how figures are labeled in geometry, and the meanings of these labels. This information should sound familiar. Lines:
Lines are traditionally labeled by expressing two points through which the line passes.
Lines may also be labeled with a single scripted letter, and referred to by that name.
When drawing rectangle ABCD: the letters must follow, in order, around the outside of the figure. You may start at any vertex point.
When drawing rectangle ABCD: you may label in either a clockwise or counterclockwise direction around the outside of the figure.
When drawing rectangle ABCD: you may NOT label "across" the figure as shown here. This is not rectangle ABCD. (It is rectangle ACBD.)
∠ABC or ∠CBA
Angles are labeled by specifying 3 points, with the center point being the vertex of the angle. This angle is NOT ∠BAC.
Angles may be labeled with a single letter at the vertex, as long as it is perfectly clear that there is only one angle at this vertex.
∠A or ∠
Angles may be represented by a single lower case letter or by a Greek letter, as long as it is clear which angle is being referenced.
∠1 and ∠2
Angles may also be represented by numbers, as long as it is clear to which angle the number applies.
Congruent angles are angles that have the same measure.
∠ABC ≅ ∠DEF the angles are congruent
m∠ABC = m∠DEF the measures of the angles are equal Bisector of an angle is a ray whose endpoint is the vertex of the angles and that divides that angle into two congruent angles. Classifying Angles According to Their Measures
§ An Acute angle is an angle whose degree measure is greater than 0 and less than 90o § A Right angle is an angle whose degree measure is 90o § An Obtuse angle is an angle whose degree measure is greater than 90 and less than 180o. § A Straight angle measures 180o.
or any other three letter combination of A,B and C will apply to this triangle.
When using letters to refer to the sides of a triangle, it is customary to label the sides as small case letters. Across from the vertex labeled capital A will be the side labeled small case a, and so on.
A right triangle is designated with a "box" drawn in the location of the right angle.
Never assume that diagrams are drawn “to scale” Notations(The figure vs its measures)
There are notations that refer to the actual geometric figures, and there are notations that refer to the measures (sizes, lengths) of the figures.
The bar over the letters means you are referring to the segment itself (the actual physical segment).
The letters, without a bar on top, refer to the length of the segment from point A to point B.
The angle symbols,
, mean you are referring to the angle itself (the actual physical angle).
The m in front of the angle notation refers to the measure of the angle labeled A, B and C (with vertex at B).
By definition, the term congruent means "having equal length or measure".
Segments are congruent. Angles are congruent. Triangles are congruent. The congruent symbol is used when referring to the actual physical entities (diagrams).
When referring to a length or measure, the equal sign should be used. You speak of numbers as being equal (or not equal).
(Triangles are not referred to as being "equal", since they are not numbers, and we do not speak of the measure of a Δ.)
Undefined Terms: There are three words in geometry that are not formally defined. These words are point, line and plane. While these words are "undefined" in the formal sense, we can still "describe" these words.
POINT • a point indicates a location (or position) in space. • a point has no dimension (actual size). • a point has no length, no width, and no height (thickness). • a point is usually named with a capital letter. • in the coordinate plane, a point is named by an ordered pair, (x,y). While we represent a point with a dot, the dot can be very tiny or very large. Remember, a point has no size.
LINE (straight line) • a line has no thickness. • a line's length extends in one dimension. • a line goes on forever in both directions. • a line has infinite length, zero width, and zero height. • a line is assumed to be straight. • a line is drawn with arrowheads on both ends. • a line is named by a single lowercase script letter, or by any two (or more) points which lie on the line.
PLANE • a plane has two dimensions. • a plane forms a flat surface extending indefinitely in all directions. • a plane has infinite length, infinite width and zero height (thickness). • a plane is drawn as a four-sided figure resembling a tabletop or a parallelogram. • a plane is named by a single letter (plane m) or by three coplanar, but non-collinear,* points (plane ABC).
Collinear points are points that lie on the same straight line. Coplanar points are points that line in the same plane.
Complementary Angles are two angles whose sum is 90 degrees. Supplementary Angles are two angles whose sum is 180 degrees. A Linear Pair are two adjacent angles whose sum is 180 degrees. Vertical Angles are two angles in which the two nonadjacent angles (opposite) formed when two lines intersect. They are always congruent.
Vertical angles are located across from one another in the corners of the "X" formed by two straight lines. In the diagram at the right, lines m and n are straight: ∠1 and ∠2 are vertical angles. ∠3 and ∠4 are vertical angles.
1 – 3 Solve the following 1. x = ______ y = ______
2. x = ______ y = ______
3. x = ______ y = ______
4 – 8 Are the following statements true or false? 4. ∠5 and ∠3 are vertical angles T or F 5. ∠1 and ∠5 are a linear pair T or F 6. ∠4 and ∠3 are adjacent angles T or F 7. ∠4 and ∠1 are vertical angles T or F 8. ∠3 and ∠4 are a linear pair T or F 9. If ∠A and ∠B are supplementary and m∠A = 150o, what is m∠B?
10. If ∠A and ∠B are complements and m∠A = 27o, what is m∠B?
11. If ∠A and ∠B are vertical angles and m∠A = 36o, what is m∠B?
12. If ∠A and ∠B are a linear pair and m∠A = 2x + 8 and m∠B = 3x + 2, what is the value of x?
13. If ∠A and ∠B are vertical angles and m∠A = 7x - 5 and m∠B = 4x + 10, what is the value of x?
14 – 23 Circle True or False 14. ∠1 = ∠4 T or F 15. ∠6 = ∠16 T or F 16. ∠3 = ∠5 T or F 17. ∠4 = ∠5 T or F 18. ∠2 = ∠10 T or F 19. ∠9 = ∠15 T or F 20. ∠12 = ∠14 T or F 21. ∠9 = ∠11 T or F 22. m∠11 + ∠15 = 180o T or F 23. m∠1 + ∠8 = 180o T or F
Angles formed by Parallel Lines Cut by a Transversal
m II n and l is the transversal Congruent Angles
• Vertical Angles ∠1 and ∠4, ∠2 and ∠3, ∠5 and ∠8, ∠6 and ∠7 • Corresponding Angles ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 • Alternating Interior Angles ∠3 and ∠6, ∠4 and ∠5 • Alternation Exterior Angles ∠1 and ∠8, ∠2 and ∠7
• ∠3 and ∠5, ∠4 and ∠6, ∠1 and ∠7, ∠2 and ∠8
1 – 6 Solve for the unknown values: 1. x = ________
2. x = ________
3. x = ________
4. x = ________
5. x = ________
6. x = ________
7 – 12 Solve the following: 7. m∠1 = _______ 8. m∠2 = _______ 9. m∠3 = _______ 10. m∠4 = _______ 11. m∠5 = _______ 12. m∠6 = _______ 13. Solve the following: x = _________ y = ________
14. Solve the following: x = _________ y = ________
Find the value of x and name the type of pairs of angles they are: 1.
A Triangle is a polygon that has exactly three sides. Classifying Triangles According to Sides
§ No Sides equal - Scalene § 2 sides equal - Isosceles § 3 sides equal - Equilateral.
Classifying Triangles According to Angles
§ Has a right angle - Right § Has an obtuse angle - Obtuse § All acute angles - Acute § All equal angles - Equiangular
The sum of the measures of the interior angles of a