The relation between the angles that are formed by two lines is illustrated by the geometry theorems called “Angle theorems”. Some of the important angle theorems involved in angles are as follows:
When two parallel lines are cut by a transversal then resulting alternate exterior angles are congruent.
The alternate exterior angles have the same degree measures because the lines are parallel to each other.
When two parallel lines are cut by a transversal then resulting alternate interior angles are congruent.
The alternate interior angles have the same degree measures because the lines are parallel to each other.
One way to find the alternate interior angles is to draw a zig-zag line on the diagram.
If two angles are complementary to the same angle or of congruent angles, then the two angles are congruent.
If two angles are supplements to the same angle or of congruent angles, then the two angles are congruent.
If two angles are both supplement and congruent then they are right angles.
If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary.
Angles that are opposite to each other and are formed by two intersecting lines are congruent.
Now let us move onto geometry theorems which apply on triangles.
Triangle Theorems
We know that there are different types of triangles based on the length of the sides like a scalene triangle, isosceles triangle, equilateral triangle and we also have triangles based on the degree of the angles like the acute angle triangle, right-angled triangle, obtuse angle triangle.
Though there are many Geometry Theorems on Triangles but Let us see some basic geometry theorems.
Theorem 1
In any triangle, the sum of the three interior angles is 180°.
Suppose XYZ are three sides of a Triangle, then as per this theorem; ∠X + ∠Y + ∠Z = 180°
Theorem 2
If a side of the triangle is produced, the exterior angle so formed is equal to the sum of corresponding interior opposite angles.
For a triangle, XYZ, ∠1, ∠2, and ∠3 are interior angles. And ∠4, ∠5, and ∠6 are the three exterior angles.
Theorem 3
The base angles of an isosceles triangle are congruent.
Suppose a triangle XYZ is an isosceles triangle, such that;
XY = XZ [Two sides of the triangle are equal]
Hence,
∠Y = ∠Z
Where ∠Y and ∠Z are the base angles.
Now Let’s learn some advanced level Triangle Theorems.
Theorem 3: If a line is drawn parallel to one side of a triangle to intersect the midpoints of the other two sides, then the two sides are divided in the same ratio.
XYZ is a triangle and L M is a line parallel to Y Z such that it intersects XY at l and XZ at M.
Hence, as per the theorem:
XL/LY = X M/M Z
Theorem 4
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Suppose XYZ is a triangle and a line L M divides the two sides of triangle XY and XZ in the same ratio, such that;
XL/LY = X M/M Z
Theorem 5
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.
Let ∆ABC and ∆PQR are two triangles.
Then as per the theorem,
AB/PQ = BC/QR = AC/PR (If ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R)
And ∆ABC ~ ∆PQR
Theorem 6
If in two triangles, the sides of one triangle are proportional to other sides of the triangle, then their corresponding angles are equal and hence the two triangles are similar.
Let us now proceed to discussing geometry theorems dealing with circles or circle theorems.
Circle Theorems
Circle theorems helps to prove the relation of different elements of the circle like tangents, angles, chord, radius, and sectors. Or we can say circles have a number of different angle properties, these are described as circle theorems.
Now let’s study different geometry theorems of the circle.
Circle Theorems 1
Angles in the same segment and on the same chord are always equal.
Circle Theorems 2
A line drawn from the center of a circle to the mid-point of a chord is perpendicular to the chord at 90°.
Circle Theorems 3
The angle at the center of a circle is twice the angle at the circumference.
Circle Theorems 4
The angle between the tangent and the side of the triangle is equal to the interior opposite angle.
Circle Theorems 5
The angle in a semi-circle is always 90°.
Circle Theorems 6
Tangents from a common point (A) to a circle are always equal in length. AB=BC
Circle Theorems 7
The angle between the tangent and the radius is always 90°
Circle Theorems 8
In a cyclic quadrilateral, all vertices lie on the circumference of the circle. Opposites angles add up to 180°.
Proceed to the discussion on geometry theorems dealing with paralellograms or parallelogram theorems.
Parallelogram Theorems
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Let’s now understand some of the parallelogram theorems.
Parallelogram Theorems 1
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Parallelogram Theorems 2
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Parallelogram Theorems 3
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Parallelogram Theorems 4
If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.
Summary
In Geometry, you learn many theorems which are concerned with points, lines, triangles, circles, parallelograms, and other figures. Geometry Theorems are important because they introduce new proof techniques.
You must have heard your teacher saying that Geometry Theorems are very important but have you ever wondered why? We leave you with this thought here to find out more until you read more on proofs explaining these theorems. Proving the geometry theorems list including all the angle theorems, triangle theorems, circle theorems and parallelogram theorems can be done with the help of proper figures.
Written by Rashi Murarka
FAQs
Which of the following states the pythagorean theorem?
The Pythagorean theorem consists of a formula a^2+b^2=c^2 which is used to figure out the value of (mostly) the hypotenuse in a right triangle. The a and b are the 2 "non-hypotenuse" sides of the triangle (Opposite and Adjacent).
What is the vertical angles theorem?
Angles that are opposite to each other and are formed by two intersecting lines are congruent.
