Vertical Angles Theorem
Vertical Angles Theorem states that when two lines intersect, the angles formed across from each other (called vertical angles) are congruent. In other words, if two lines cross at a point, the angles opposite each other are equal.
Proof of Vertical Angle Theorem
The proof of vertical angle theorem is mentioned below:
Let’s consider two intersecting lines, line AB and line CD, as shown below:
When lines AB and CD intersect, they form four angles: ∠1, ∠2, ∠3, and ∠4.
Now, let’s examine the relationships between these angles:
- ∠1 and ∠3 are opposite each other and are formed by intersecting lines AB and CD.
- ∠2 and ∠4 are opposite each other and are also formed by intersecting lines AB and CD.
- ∠1 and ∠2 are adjacent angles.
- ∠3 and ∠4 are adjacent angles.
Given that lines AB and CD are straight lines, they form a straight angle. This means that ∠1 and ∠2 together form a straight angle, and ∠3 and ∠4 together form a straight angle.
Since a straight angle measures 180 degrees, we can write:
∠1 + ∠2 = 180° …..(i)
∠3 + ∠4 = 180° …..(ii)
Now, let’s rearrange equation (i) to solve for ∠1:
∠1 = 180° – ∠2 …..(iii)
Similarly, rearrange equation (ii) to solve for ∠3:
∠3 = 180° – ∠4 …..(iv)
Now, we can see that equations (iii) and (iv) show that ∠1 and ∠3 are equal in measure:
∠1 = ∠3
Hence, vertical angles ∠1 and ∠3 are equal
Congruent Angles
Congruent angles are angles that have equal measure. Thus, all the angles in the geometry that have sam measure are called congruent angles.
In this article, we will understand the meaning of congruent angles, their properties, the congruent angles theorem, the vertical angles theorem, the corresponding angles theorem, and the alternate angles theorem.
Table of Content
- What are Congruent Angles?
- Congruent Angles Theorem
- Vertical Angles Theorem
- Corresponding Angles Theorem
- Alternate Angles Theorem
- Congruent Supplements Theorem
- Congruent Complements Theorem
- How to Find Congruent Angles
- Constructing Congruent Angles
- Construction of a Congruent Angle to the Given Angle
- Construction of Two Congruent Angles
- Congruent Angles Properties
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