Example Question #1 : Geometry
The image provided contains a set of parallel lines, and , and a transversal line, . If angle is equal to , then which of the other angles is equal to
First, we need to define some key terms:
Parallel Lines: Parallel lines are lines that will never intersect with each other.
Transversal Line: A transversal line is a line that crosses two parallel lines.
In the the image provided, lines and are parallel lines and line is a transversal line because it crosses the two parallel lines.
It is important to know that transversal lines create angle relationships:
- Vertical angles are congruent
- Corresponding angles are congruent
- Alternate interior angles are congruent
- Alternate exterior angles are congruent
Let's look at angle in the image provided below to demonstrate our relationships.
Angle and are vertical angles.
Angle and are corresponding angles.
Angle and are exterior angles.
Angle is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs and as well as angle and .
For this problem, we want to find the angle that is congruent to angle . Based on our answer choices, angle and are vertical angles; thus, both angle and are congruent and equal
Example Question #2 : Geometry
Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.
The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or angle. When a triangle includes a right angle, the triangle is said to be a "right triangle."
We can use the Pythagorean Theorem to help us solve this problem.
The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:
We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:
Example Question #3 : Geometry
Calculate the volume of the cone provided. Round the answer to the nearest hundredth.
In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:
Now that we have this formula, we can substitute in the given values and solve: