Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![**Finding the Value of x, y, and z in a Cyclic Quadrilateral**
In the provided diagram, there is a cyclic quadrilateral (a quadrilateral inscribed in a circle). The given angles within the cyclic quadrilateral are:
- One angle is 95°.
- Another angle is 75°.
- One exterior angle adjacent to one of the sides of the quadrilateral is 60°.
- One exterior angle adjacent to another side of the quadrilateral is z°.
We are asked to find the values of the angles x°, y°, and z°.
### Explanation of the Problem
1. **Opposite Angles in a Cyclic Quadrilateral:**
- Opposite angles of a cyclic quadrilateral add up to 180°.
- From the given angles, 95° and the opposite angle x°:
\[
95° + x° = 180°
\]
Solving for x:
\[
x = 180° - 95° = 85°
\]
2. **External Angle in Cyclic Quadrilateral:**
- An external angle (z°) of a cyclic quadrilateral is equal to the interior opposite angle (y°).
- Since adjacent angle inside the quadrilateral to angle z° is given as 75°, and opposite the z° external angle is y°:
\[
y = 75°
\]
3. **Finding the Value of Remaining Angles:**
- The angle at the other opposing corner to 60° is also needed in cyclic quadrilateral.
- Angle at this vertex beside 60° is:
\[
y° = 60°
\]
### End Results:
To summarize:
- \( x = 85° \)
- \( y = 75° \)
- \( z = 75° \)
### Summary Table:
```
x = 85°
y = 75°
z = 75°
```
This demonstrates how the properties of a cyclic quadrilateral are utilized to determine unknown angles using given angles and basic geometric principles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7c29d2f-5cf3-43f5-aa90-abc5a822140a%2F5a4b6cbd-8b70-46bc-aedd-6ee79323e92a%2Fk018b4_processed.jpeg&w=3840&q=75)
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