Geometry, circles with tangents

Question

i am grading a test for my son, and confused on a problem, in geometery the question is a picture of a circle middle point p two tangent lines point x with 100, they want the answer for x, not other coordinates listed

Answer 1

Let the third time be the CHARM:

 

Let Y be the point on the circle where segment PY and segment XY intersect

 

Let Z be the point on the circle where segment PZ and segment XZ intersect

 

measure of angle PYX = 90° and measure of angle PZX = 90° because:

RADIUS OF A CIRCLE IS PERPENDICULAR TO A LINE DRAWN TANGENT

TO THE CIRCLE AT THE POINT OF TANGENCY

(in your drawing points Y and Z are the points of tangency)

 

Given:  arc YZ = 100° from the drawing

 

measure of angle YPZ = 100° because a CENTRAL ANGLE IS EQUAL

IN MEASURE TO THE ARC IT CUTS OUT

 

Looking at quadrilateral XYPZ, the sum of its interior angles is 360°

 

angle Y = 90°,  angle P = 100°,  angle Z = 90°;    360 -280 = 80

 

Therefor the measure of angle X = 80°

 

I worked this problem using Polygons and the sum of their interior angles.

It can also be worked using congruent triangles.

 

Let me know if you want to see the triangular method.