Table of Contents
Introduction To Right Triangle Congruence Theorems
There are three right triangle congruence theorems:
SAS
SSS
AAS
Each of these theorems states that if two right triangles have two sides and the angle between those two sides congruent, then the triangles are congruent.
Properties of Right Triangles
A right triangle has three properties: the length of its longest side, called the hypotenuse; the length of its shortest side, called the opposite side; and the length of its other side, called the adjacent side.
Congruence Theorems To Prove Two Right Triangles Are Congruent
There are three congruence theorems that can be used to prove that two right triangles are congruent:
The SSS Congruence Theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
The SAS Congruence Theorem states that if two sides and the angle between them of one triangle are congruent to two sides and the angle between them of another triangle, then the two triangles are congruent.
The Angle-Side-Angle Congruence Theorem states that if two angles and the side between them of one triangle are congruent to two angles and the side between them of another triangle, then the two triangles are congruent.
The LA Theorem
The LA theorem states that a line segment joining two points on a circle is perpendicular to the radius drawn to the points.
Proof
Draw a line segment connecting the points and denote it as .
Extend to intersect the circle at .
Since is on the circle, and is a radius, is perpendicular to .
Proving the LA Theorem
To show that the LA theorem is true, we will use induction.
Basis: We know that the LA theorem is true for n = 1.
Induction Step: Assume that the LA theorem is true for some value of n. This means that
\begin{align*}
L &= A + B
\\
&= (1 + 2) + (3 + 4)
\\
&= 6 + 7
\\
&= 13
\end{align*}
We will now show that the LA theorem is also true for n + 1. This means that
\begin{align*}
L &= A + B
\\
&= (1 + 2) + (3 + 4)
\\
&= (1 + 2) + (7 + 8)
\\
&= 10 + 15
\\
&= 25
\end{align*}
The LL Theorem
The LL theorem states that every continuous function on a compact space is bounded.
Proving the LL Theorem
The LL theorem is a result in graph theory that states that every connected graph has at least one path between any two vertices in the graph. This theorem can be proved using induction.
The base case of the induction proof is when the graph is a single vertex. In this case, there is a path between any two vertices in the graph.
The induction step of the proof is when the graph is a connected graph with more than one vertex. In this case, the theorem is true if there is a path between any two vertices in the graph. To prove the theorem, it is necessary to show that there is a path between any two vertices in the graph. This can be done by constructing a path between the two vertices.
