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Geometry - Fundamental Ideas

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    Study Guide Geometry – Fundamental Ideas 1. Angles and Angle Pairs Figure 1 ∠ BAC What Is an Angle? An angle is formed by two rays that share the same endpoint . • The shared endpoint is called the vertex . • The two rays are called the sides of the angle . • Angles are measured in degrees (°) . • A degree measure is always between 0° and 180° . For example, if rays AB and AC meet at point A , then: • A is the vertex. • AB and AC are the sides of the angle. How Do We Name Angles? There are several correct ways to name the same angle. Let’s look at them.

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    Study Guide Figure 2 Different names for the same angle. By the Vertex If there is only one angle at point A, we can call it: By a Number or Letter Inside the Angle Sometimes angles are labeled with a number or lowercase letter. By Three Points This is the most common method. Important: The middle letter is always the vertex . Example 1 : Renaming Angles Figure 3 Different names for the same angle

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    Study Guide • Angle 3 can also be named ∠ IMJ or ∠ JMI . • ∠ KMJ is the same as ∠ 4 . Different names — but the same angle! 1.2 Measuring Angles: The Protractor Postulate Figure 4 Using the Protractor Postulate This rule helps us measure angles correctly. If point O is on a line, every ray starting at O can be matched with a number between 0° and 180° . To find the measure of an angle: 1. Find the numbers that match each ray. 2. Subtract the smaller number from the larger number. If ( a > b ) , then: Example 2 : Finding Angle Measures Using a protractor scale:

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    Study Guide Figure 5 Using the Protractor Postulate (a) Find m ∠ SON (b) Find m ∠ ROT (c) Find m ∠ MOE Always subtract the smaller reading from the larger one. Figure 6 Addition of angles.

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    Study Guide 1.3 Adding Angles: The Angle Addition Postulate If a ray lies between two other rays, you can add the smaller angles to get the larger angle. If ray OB lies between OA and OC, then: Think of it like putting two small pieces together to make one bigger piece. Example 3 : Adding Angle Measures If: Figure 7 Addition of angles. • ( m ∠ 1 = 32° ) • ( m ∠ 2 = 45° ) And one ray lies between the others, then: So:

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    Study Guide 1.4 Angle Bisector An angle bisector is a ray that divides an angle into two equal angles . Figure 8 Bisector of an angle If ray OY bisects ∠ XOZ, then: Important Fact: An angle that is not a straight angle has exactly one bisector . 1.5 Special Types of Angles Angles are grouped based on their measure. Right Angle

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    Study Guide Figure 9 A right angle. • Measures exactly 90° • Marked with a small square • All right angles are equal Acute Angle Figure 10 An acute angle.

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    Study Guide • Less than 90° • Small and sharp Obtuse Angle Figure 11 An obtuse angle. • More than 90° • Less than 180° Straight Angle Figure 12 A straight angle

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    Study Guide • Exactly 180° • Forms a straight line Some textbooks simply call this a straight angle . Example 4 : Classifying Angles Figure 13 Classification of angles Let’s identify each angle type. (a) ∠ BFD Right angle (b) ∠ AFE Straight angle (c) ∠ BFC Acute angle

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    Study Guide (d) ∠ DFA Obtuse angle Summary • An angle forms when two rays share a vertex. • Angles measure between 0 ° and 180 ° . • Always subtract smaller from larger when measuring. • You can rename angles in multiple ways. • Angles can be added together. • An angle bisector splits an angle into two equal parts. • Special angle types: acute, right, obtuse, straight. 2. Special Angles Some pairs of angles have special names. These names depend on: • Where the angles are located • Or what their measures add up to Let’s break each type down in a simple way. 2.1 Adjacent Angles Adjacent angles are two angles that:

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