Triangle Proofs Worksheets
What Are Triangle Proofs?
Recall that triangles have three sides and are a construct of three points or vertices. Where any three points can come together to form a triangle, any three lines cannot come together to form a triangle. But in order to prove this, we need some logic and theorems.
Let us consider the fact that the sum of all the interior angles of a triangle is 180 degrees. We all know that this fact is true, but let us understand how this is proven to be true.
Triangle Sum Theorem - Consider a triangle ABC:
Take this triangle and draw a parallel line opposite to the side AC through vertex B.
We know that alternate interior angles are congruent when lines are parallel. Considering this fact, we know that ∢A is congruent to ∢DBA, hence ∢A=∢DBA. Similarly, ∢C and ∢EBC are alternate interior angles showing congruence with each other, hence ∢C=∢EBC. If we observe the diagram shown above, we can see that ∢EBC, ∢ABC, ∢DBA form a straight line. Therefore, ∢EBC+ ∢ABC+ ∢DBA = 180 degrees. Hence, the statement "the sum of the measures of the interior angles of a triangle is 180 degrees" is proven. You can use this theorem to solve other geometrical problems.
What is the Similarity of Triangles?
Two concepts are always associated with the study of shapes. Those two concepts are congruence and similarity. Both of these concepts differ a little from each other. In congruence, both shapes should look the same, which means that their angles, length of the sides, and their size must be equal. In short, the other shape should be the replica of the first one. Whereas, if we talk about similarity, the size of both shapes could be different. In similarity, the size and lengths of the sides do not need to be equal. However, the shape and their internal angle measures must be the same and equal to each other. Besides having an equal measure of internal angles, two similar triangles differ in their sizes according to a scale factor. A scale factor is a ratio in which the lengths of the sides of both similar triangles differ from each other. So, it can be said that one of the two similar triangles is an enlargement of the other.
Great Geometry Quotes
Where there is matter, there is geometry.--Johannes Kepler