Geometry Theorems - Triangles

Important Geometry theorems


Theorem 3  : Angle sum property of a triangle.

The sum of the angles of a triangle is 1800.

Triangles

In the picture above, PQR is a triangle with angles 1, 2 and 3

Then according to the theorem

Angle 1+Angle 2 +Angle 3 =1800

Theorem 4

If a side of a triangle is produced then the exterior angle so 

formed is equal to the sum of two interior opposite angles.

triangle side

In the picture above XYZ is a triangle whose side YZ is extended 

to R. 1, 2, and 3 are the interior angles of a triangle. Angle 1 and 

Angle 2 are the interior angles opposite to the exterior angle 4

Then according to the theorem

Angle 4 = Angle 1+ Angle 2

Let us do some questions based on these theorems.

Ques. - In the figure  if QT is perpendicular PR , Angle TQR = 400  and Angle SPR =300, find x and y .
perpendicular
Solution : In triangle QTR
Angle TQR +Angle QRT +Angle QTR =1800
400 + y + 900 =1800
  y =1800-1300
     = 500
Angle QSP = Angle SPR +Angle SRP

Reason:Exterior angle = sum of interior opposite angles
  x = 300 +y
  x = 300 +500
  x =800

Ques. - In the figure below XY II MN , Angle YXZ  =350  and angle ZMN =530, find angle MZN.

Solution:

We know that XY is parallel to MN.

Angle MNZ = Angle ZXY  ( alternate interior angles)
                      = 350
Now in triangle MZN

Angle ZMN +Angle MNZ +Angle MZN =1800

530 +350 +Angle MZN =1800

Angle MZN = 1800-880

Angle =920

Ques - In the figure given below  If PQ and RS intersect at T, such that angle PRT =400, angle RPT=950 and angle TSQ=750, find SQT.


Solution:

In triangle PRT

400+950+Angle RTP =1800

 Angle RTP =1800-1350

Angle RTP =450

Angle STQ =Angle RTP ( vertically Opposite angle )

                    =450

Again in triangle TQS

Angle STQ + Angle SQT + Angle TSQ = 1800 ( Angle sum property)

450+ Angle SQT +750 =1800

Angle SQT =1800-1200

Angle SQT = 600

Ques. In the figure if PQ is perpendicular to PS, PQ II SR Angle SQR =280  and Angle QRT =650, then find the values of x and y .

Solution:

Since PQ II SR

Angle QRT = Angle PQR (alternate interior angles )

650 = x +280

 X= 650-280

    = 370

In triangle PQS

Angle PSQ +Angle PQS + QPS = 1800 ( angle sum property)

 Y +x +900 =1800

 Y+370+900=1800

Y =1800-1270

   = 530

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