Notes 38

1. Notes 38 Geometry Overview

2. Vocabulary Point-an exact location. It is usually represented as a dot, but it has no size at all. Line-a straight path that extends without end in opposite directions. Plane-a flat surface that has no thickness and extends forever. Ray-a part of a line. It has one endpoint and extends forever one direction. Line segment- part of a line or a ray that extends from one endpoint to another. Congruent- figures that have the same shape and size. Line segments are congruent if they have the same length.

3. Vocabulary Angle- formed by two rays with a common endpoint. Vertex- the common endpoint of an angle where the two rays meet. Right angle- an angle that measures exactly 90°. Acute angle- an angle that measures less than 90°. Obtuse angle- an angle that measures more than 90° but less than 180°. Straight angle- an angle that measures exactly 180°. Complementary angles- sum of the measures of two angles is 90° Supplementary angles- sum of the measures of two angles is 180°

4. XY, or YX, or l X Y Use two points on the line or a lowercase letter to name a line. Helpful Hint A number line is an example of a line. A point is an exact location. It is usually represented as a dot, but it has no size at all. point A Use a capital letter to name a point. • A l A lineis a straight path that extends without end in opposite directions.

5. Q S R Helpful Hint A coordinate plane is an example of a plane. A plane is a Flat surface that Has no thickness and extends forever. plane QRS Use three points in any order, not on the same line, to name a plane.

6. Additional Example 1: Identifying Points, Lines, and Planes Identify the figures in the diagram. D E F A. three points D, E, and F Choose any two points on a line to name the line. B. two lines DE, DF Choose any three points, not on the same line, in any order. C. a plane plane DEF

7. Check It Out: Example 1 Identify the figures in the diagram. G H I F A. four points B. two lines C. a plane

8. GH Name the endpoint first when naming a ray. G H LM, or ML Use the endpoints to name a line segment. L M A ray is a part of a line. It has one endpoint and extends forever one direction. A line segmentis part of a line or a ray that extends from one endpoint to another.

9. Additional Example 2: Identifying Line Segments and Rays Identify the figures in the diagram. M N O A. three rays Name the endpoint of a ray first. MN, NM, MO B. two line segments Use the endpoints in any order to name a segment. MN, MO

10. Check It Out: Example 2 Identify the figures in the diagram. D C A. three rays B A B. three line segments

11. Figures are congruent if they have the same shape and size. Line segments are congruent if they have the same length. You can use tick marks to indicate congruent line segments. In the triangle below, line segments AB and BC are congruent.

12. AB CD ACBD BF DF EC AE Reading Math The symbol means “is congruent to.” Additional Example 3: Identifying Congruent Line Segments Identify the line segments that are congruent in the figure. One tick mark Two tick marks Three tick marks

13. A B C E D Check It Out: Example 3 Identify the line segments that are congruent in the figure.

14. A Vertex 1 B C An angleis formed by two rays with a common endpoint. The two rays are the sides of the angle. The common endpoint is the vertex. Angles are measured in degrees (°).

15. An angle’s measure determines the type of angle it is. A right angle is an angle that that measures exactly 90°. The symbol indicates a right angle. An acute angle is an angle that measures less than 90°. Anobtuse angle is an angle that measures more than 90° but less than 180°. A straightangle is an angle that measures exactly 180°.

16. Additional Example 1: Classifying Angles Tell whether each angle is acute, right, obtuse or straight. A. B. acute angle obtuse angle

17. Reading Math You can name this angle ABC, CBA, B, or 1. A • 1 B• •C

18. Check It Out: Example 1 Tell whether each angle is acute, right, obtuse, or straight. B. A.

19. If the sum of the measures of two angles is 90°, then the angles are complementary angles. If the sum of the measures of two angles is 180°, then the angles are supplementary angles.

20. To find mPMQ start with the measure that QM crosses, 105°, and subtract the measure that MP crosses, 75°. mPMQ = 105° -75° = 30°. mOMP = 60°. P Q O N R M Additional Example 2A: Identifying Complementary and Supplementary Angles Use the diagram to tell whether the angles are complementary, supplementary, or neither. OMP and PMQ Since 60° + 30° = 90°, PMQ andOMP are complementary.

21. P Q Reading Math O Read mNMO as “the measure of angle NMO.” N R M Additional Example 2B: Identifying Complementary and Supplementary Angles Use the diagram to tell whether the angles are complementary, supplementary, or neither. NMO and OMR mNMO = 15° and mOMR = 165° Since 15° + 165° = 180°, NMO andOMR are supplementary.

22. To find mPMQ start with the measure that QM crosses, 105°, and subtract the measure that MP crosses, 75°. mPMQ = 105° -75° = 30°. mQMR = 75°. P Q O N R M Additional Example 2C: Identifying Complementary and Supplementary Angles Use the diagram to tell whether the angles are complementary, supplementary, or neither. PMQ and QMR Since 30° + 75° = 105°, PMQ andQMR are neither complementary nor supplementary.

23. D E C F B A Check It Out: Example 2A Use the diagram to tell whether the angles are complementary, supplementary, or neither. BAC and CAF

24. Additional Example 3: Finding Angle Measures Angles A and B are complementary. If mA is 56°, what is the mB? Since A and B are complementary, mA + mB = 90°. mA + mB = 90° 56° + mB = 90° Substitute 56° for mA. Subtract 56° from both sides. – 56° – 56° mB = 34° The measure of B = 34°.

25. Check It Out: Example 3 Angles P and Q are supplementary. If mP is 32°, what is the mQ?