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Line and Angle Relationships: Definitions, Examples, and Applications

Learn about acute angles, right angles, obtuse angles, and straight angles, as well as vertical angles, adjacent angles, complementary angles, and supplementary angles. Understand the concepts of parallel lines, perpendicular lines, and transversals, and discover how to classify and calculate various angles.

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Line and Angle Relationships: Definitions, Examples, and Applications

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  1. Lesson 6-1 Line and Angle Relationships

  2. Definitions • Acute Angles – Angles with measures less than 90°. • Right Angles - Angles with a measure of 90. • Obtuse Angles - Angles with measures between 90° and 180°. • Straight Angles – Angles with measures equal to 180.

  3. Vertical Angles are opposite angles formed by intersecting lines. They are congruent. • Adjacent Angles have the same vertex, share a common side, and do not overlap. • The sum of the measures of complementary angles is 90°. • The sum of the measures of supplementary angles is 180°

  4. Examples 1 and 2 Classify each angle or angle pair using all names that apply. 1 m ∠1 is greater than 90°. So, ∠1 is an obtuse angle. Ex. 1 ∠1 and ∠2 are adjacent angles since they have the same vertex, share a common side, and do not overlap. Ex. 2 1 2 Together they form a straight angle measuring 180°. So, ∠1 and ∠2 are also supplementary angles.

  5. Classify each angle or angle pair using all names that apply. a. b. 30° 60° c. 3 4

  6. Example 3 In the figure m∠ABC = 90°. Find the value of x. A B x° 65° m∠ABD + m∠DBC = 90° x + 65 = 90 - 65= -65 x = 25 D C

  7. Find the value of x in each figure. d. x° 38° e. x° 150°

  8. Definitions • Lines that intersect at right angles are called perpendicular lines. • Two lines in a plane that never intersect or cross are called parallel lines. q p m n Symbol: p q Symbol: m ⟘ n

  9. Definitions • A line that intersects two or more other lines is called a transversal. When a transversal intersects two lines, eight angles are formed that have special names. • If two lines cut by a transversal are parallel, then these special pairs of angles are congruent. 1 2 4 3 6 5 7 8 transversal

  10. Definitions • Alernate Inerior Angles – Those on opposite sides of the transversal and inside the other two lines are congruent. Ex. ∠2 ≅ ∠8 • Alternate Exterior Angles – Those on opposite sides of the transversal and outside the other two lines, are congruent. Ex. ∠4 ≅ ∠6 • Corresponding Angles - Those in the same position on the two lines in relation to the transversal, are congruent. Ex. ∠3 ≅ ∠7 6 5 2 1 7 8 3 4

  11. Example 4 You are building a bench for a picnic table. The top of the bench will be parallel to the ground. If m∠1 = 148°, find m∠2 and m∠3. 3 2 Since ∠1 and ∠2 are alternate interior angles, they are congruent. So, m∠2 = 148°. 1 Since ∠2 and ∠3 are supplementary, the sum of their measures is 180°. Therefore, m∠3 = 180° - 148° or 32°.

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