9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem

Angles and triangles are fundamental to geometry, shaping our understanding of shapes and spaces. From complementary and supplementary angles to the properties of parallel lines, these concepts form the building blocks for more complex geometric reasoning.

Triangles, with their unique properties and the powerful Pythagorean theorem, offer practical applications in real-world problem-solving. Understanding these basics opens doors to advanced geometric concepts and their use in various fields.

Angle Properties and Relationships

Angle properties for problem-solving

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• Complementary angles add up to 90° (right angle)
  • If angle A measures 35°, its complement angle B measures 55° (90° - 35°)
• Supplementary angles add up to 180° (straight angle)
  • If angle C measures 112°, its supplement angle D measures 68° (180° - 112°)
• Vertical angles formed by intersecting lines are always congruent (equal in measure)
  • If one angle in a pair of vertical angles measures 70°, the other also measures 70°
• Adjacent angles share a common vertex and side but do not overlap
  • Adjacent angles forming a straight line are supplementary (add up to 180°)
  • Adjacent angles forming a right angle are complementary (add up to 90°)

Parallel Lines and Transversals

• Parallel lines are lines in the same plane that never intersect
• A transversal is a line that intersects two or more lines
• When a transversal crosses parallel lines, it creates corresponding angles, alternate interior angles, and alternate exterior angles that are congruent
• Corresponding angles formed by parallel lines and a transversal can be used in geometric proofs

Triangle Properties and Applications

Triangle properties and applications

• Sum of the three interior angles in a triangle always equals 180°
  • If two angles measure 45° and 60°, the third angle measures 75° (180° - 45° - 60°)
• Relationship between sides and angles in a triangle
  • Longest side is opposite the largest angle, shortest side is opposite the smallest angle
  • Triangle with side lengths 3, 4, and 6 units has the largest angle opposite the 6-unit side and smallest angle opposite the 3-unit side
• Types of triangles based on side lengths
  • Equilateral triangle has all sides equal in length and all angles measure 60°
  • Isosceles triangle has two sides equal in length and the angles opposite those sides are congruent
  • Scalene triangle has all sides different in length and all angles have different measures
• Similarity in triangles occurs when they have the same shape but different sizes, with corresponding angles equal and corresponding sides proportional

Pythagorean theorem in context

• In a right triangle, the square of the hypotenuse length (side opposite the right angle) equals the sum of the squares of the other two side lengths
  • Formula: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse length and $a$ and $b$ are the other side lengths
  • Right triangle with side lengths 3 and 4 units has hypotenuse length $c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ units
• Applications of the Pythagorean theorem
  • Calculate height of a tree or building using its shadow length and angle of elevation
  • Determine shortest distance between two points on a coordinate plane (x-y graph)
  • Find diagonal length of a rectangular prism (box) or cube
  • Solve trigonometry problems involving right triangles