9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
3 min read•june 25, 2024
Angles and triangles are fundamental to geometry, shaping our understanding of shapes and spaces. From complementary and to the properties of , these concepts form the building blocks for more complex geometric reasoning.
Triangles, with their unique properties and the powerful , 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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add up to 90° ()
If A measures 35°, its complement angle B measures 55° (90° - 35°)
Supplementary angles add up to 180° ()
If angle C measures 112°, its supplement angle D measures 68° (180° - 112°)
formed by intersecting lines are always (equal in measure)
If one angle in a pair of vertical angles measures 70°, the other also measures 70°
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 , and alternate exterior angles that are congruent
Corresponding angles formed by parallel lines and a transversal can be used in
Triangle Properties and Applications
Triangle properties and applications
Sum of the three interior angles in a 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
has all sides equal in length and all angles measure 60°
has two sides equal in length and the angles opposite those sides are congruent
has all sides different in length and all angles have different measures
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 , the square of the length (side opposite the right angle) equals the sum of the squares of the other two side lengths
Formula: a2+b2=c2, 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=32+42=25=5 units
Applications of the Pythagorean theorem
Calculate height of a tree or building using its shadow length and
Determine shortest distance between two points on a coordinate plane (x-y graph)
Find diagonal length of a rectangular prism (box) or cube