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

Angle Properties and Relationships

Angles and triangles are the building blocks of geometry. This section covers how angles relate to each other, how triangles behave, and how the Pythagorean theorem lets you find missing side lengths in right triangles.

Angle Properties for Problem-Solving

Two angle relationships come up constantly in geometry problems: complementary and supplementary angles.

• Complementary angles add up to 90° (a right angle)
  • If angle A measures 35°, its complement measures 55° because 90° - 35° = 55°
• Supplementary angles add up to 180° (a straight angle)
  • If angle C measures 112°, its supplement measures 68° because 180° - 112° = 68°

When two lines cross each other, they create two more useful angle relationships:

• Vertical angles are the pairs of opposite angles formed at the intersection. They're always equal. So if one angle measures 70°, the angle directly across from it also measures 70°.
• Adjacent angles share a common vertex and a common side but don't overlap. If two adjacent angles form a straight line, they're supplementary. If they form a right angle, they're complementary.

Parallel Lines and Transversals

Parallel lines are lines in the same plane that never intersect. A transversal is a line that cuts across two or more lines.

When a transversal crosses parallel lines, it creates several pairs of equal angles:

• Corresponding angles sit in the same position at each intersection (both upper-left, for example). They're congruent.
• Alternate interior angles are on opposite sides of the transversal, between the parallel lines. They're congruent.
• Alternate exterior angles are on opposite sides of the transversal, outside the parallel lines. They're also congruent.

These relationships let you find unknown angle measures. If you know just one angle where a transversal crosses parallel lines, you can figure out all eight angles at both intersections.

Triangle Properties and Applications

Triangle Properties

The three interior angles of any triangle always add up to 180°. This is one of the most-used facts in geometry.

• If two angles measure 45° and 60°, the third angle is 180° - 45° - 60° = 75°.

There's also a consistent relationship between sides and angles: the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle. For example, a triangle with sides of 3, 4, and 6 units has its largest angle opposite the 6-unit side.

Types of triangles by side length:

• Equilateral — all three sides are equal, and every angle measures 60°
• Isosceles — two sides are equal, and the angles opposite those equal sides are also equal
• Scalene — all three sides are different lengths, and all three angles are different measures

Similar triangles have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. For instance, if one triangle has sides of 3, 4, and 5, a similar triangle might have sides of 6, 8, and 10 (each side doubled).

The Pythagorean Theorem

The Pythagorean theorem applies only to right triangles (triangles with one 90° angle). It states that the square of the hypotenuse equals the sum of the squares of the other two sides:

$a^2 + b^2 = c^2$

Here, $c$ is the hypotenuse (the side opposite the right angle, and always the longest side), while $a$ and $b$ are the two shorter sides, called legs.

Finding a missing side, step by step:

1. Identify which side is the hypotenuse (opposite the right angle).
2. Plug the known values into $a^2 + b^2 = c^2$.
3. Solve for the unknown side.

Example: A right triangle has legs of 3 and 4 units. Find the hypotenuse.

1. You're solving for $c$, so: $3^2 + 4^2 = c^2$
2. $9 + 16 = c^2$
3. $c^2 = 25$, so $c = \sqrt{25} = 5$ units

You can also use the theorem to find a leg if you know the hypotenuse and the other leg. Just rearrange: $a^2 = c^2 - b^2$.

Common applications:

• Finding the distance between two points on a coordinate plane (the horizontal and vertical distances form the two legs)
• Calculating the diagonal of a rectangle or the space diagonal of a box
• Determining heights indirectly, like the height of a building when you know the distance from the base and the length of a line from that point to the top