All Geometry Postulates And Theorems
The relation betwixt 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 every bit follows:
When two parallel lines are cutting by a transversal then resulting alternate exterior angles are congruent.
The alternate outside angles have the same caste measures because the lines are parallel to each other.
When 2 parallel lines are cut by a transversal then resulting alternate interior angles are congruent.
The alternate interior angles accept the aforementioned caste measures because the lines are parallel to each other.
One mode to find the alternate interior angles is to draw a zig-zag line on the diagram.
If 2 angles are complementary to the same angle or of congruent angles, then the 2 angles are congruent.
If ii angles are supplements to the same bending or of congruent angles, then the 2 angles are congruent.
If ii angles are both supplement and congruent then they are correct 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 coinciding.
Now permit united states of america move onto geometry theorems which apply on triangles.
Triangle Theorems
Nosotros know that there are different types of triangles based on the length of the sides like a scalene triangle, isosceles triangle, equilateral triangle and nosotros also take triangles based on the degree of the angles like the acute angle triangle, correct-angled triangle, obtuse bending triangle.
Though there are many Geometry Theorems on Triangles simply Let us see some basic geometry theorems.
Theorem i
In any triangle, the sum of the three interior angles is 180°.
Suppose XYZ are three sides of a Triangle, then every bit per this theorem; ∠Ten + ∠Y + ∠Z = 180°
Theorem two
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, ∠ane, ∠2, and ∠three are interior angles. And ∠iv, ∠v, 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 [Ii sides of the triangle are equal]
Hence,
∠Y = ∠Z
Where ∠Y and ∠Z are the base angles.
Now Let's acquire some advanced level Triangle Theorems.
Theorem 3: If a line is drawn parallel to i side of a triangle to intersect the midpoints of the other two sides, and then the 2 sides are divided in the same ratio.
XYZ is a triangle and L M is a line parallel to Y Z such that information technology intersects XY at fifty and XZ at M.
Hence, equally per the theorem:
40/LY = 10 M/1000 Z
Theorem four
If a line divides whatsoever 2 sides of a triangle in the same ratio, and 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 aforementioned ratio, such that;
XL/LY = X M/Thou Z
Theorem 5
If in two triangles, corresponding angles are equal, and so their corresponding sides are in the same ratio and hence the two triangles are similar.
Let ∆ABC and ∆PQR are ii triangles.
Then as per the theorem,
AB/PQ = BC/QR = Air-conditioning/PR (If ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R)
And ∆ABC ~ ∆PQR
Theorem half dozen
If in 2 triangles, the sides of one triangle are proportional to other sides of the triangle, and then their respective angles are equal and hence the ii triangles are similar.
Let us now proceed to discussing geometry theorems dealing with circles or circle theorems.
Circumvolve Theorems
Circumvolve 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 every bit circle theorems.
Now let's report different geometry theorems of the circle.
Circle Theorems 1
Angles in the same segment and on the aforementioned chord are ever equal.
Circle Theorems 2
A line fatigued from the center of a circumvolve to the mid-point of a chord is perpendicular to the chord at 90°.
Circumvolve Theorems 3
The angle at the center of a circle is twice the angle at the circumference.
Circle Theorems iv
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-circumvolve is always 90°.
Circle Theorems vi
Tangents from a common bespeak (A) to a circumvolve are ever equal in length. AB=BC
Circle Theorems seven
The angle between the tangent and the radius is ever 90°
Circle Theorems eight
In a cyclic quadrilateral, all vertices prevarication on the circumference of the circle. Opposites angles add together upward to 180°.
Go along 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'south now understand some of the parallelogram theorems.
Parallelogram Theorems 1
If both pairs of contrary sides of a quadrilateral are coinciding, and then the quadrilateral is a parallelogram.
Parallelogram Theorems two
If both pairs of opposite angles of a quadrilateral are coinciding, then the quadrilateral is a parallelogram.
Parallelogram Theorems 3
If the diagonals of a quadrilateral bifurcate each other, then the quadrilateral is a parallelogram.
Parallelogram Theorems 4
If one pair of contrary 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.
Yous must have heard your teacher maxim that Geometry Theorems are very important just accept you ever wondered why? We exit you with this idea hither to detect out more until y'all read more on proofs explaining these theorems. Proving the geometry theorems listing including all the angle theorems, triangle theorems, circumvolve theorems and parallelogram theorems can be washed with the help of proper figures.
Written by Rashi Murarka
FAQs
Which of the post-obit states the pythagorean theorem?
The Pythagorean theorem consists of a formula a^two+b^2=c^ii which is used to effigy out the value of (by and large) the hypotenuse in a right triangle. The a and b are the 2 "not-hypotenuse" sides of the triangle (Opposite and Next).
What is the vertical angles theorem?
Angles that are opposite to each other and are formed by two intersecting lines are coinciding.
All Geometry Postulates And Theorems,
Source: https://www.cuemath.com/learn/geometry-theorems/
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