Geometry Overview

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.

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°

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

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. A coordinate plane is an example of a plane. Helpful Hint Q R S

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

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

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

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

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

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.

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

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

An angle is 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 (°). A C B 1 Vertex

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°. An obtuse angle is an angle that measures more than 90° but less than 180°. A straight angle is an angle that measures exactly 180°.

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

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

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

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.

Use the diagram to tell whether the angles are complementary, supplementary, or neither. Additional Example 2A: Identifying Complementary and Supplementary Angles OMP and PMQ Since 60° + 30° = 90°, PMQ and OMP are complementary. O N P Q R M 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°.

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

Use the diagram to tell whether the angles are complementary, supplementary, or neither. Additional Example 2C: Identifying Complementary and Supplementary Angles PMQ and QMR O N P Q R M Since 30° + 75° = 105°, PMQ and QMR are neither complementary nor supplementary. 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°.

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

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

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