UW bisects ∠VUX. Complete the proof that △UWX≅△UWV.
U
V
W
X
Statement Reason
1
UW
bisects ∠VUX Given
2 ∠X≅∠V Given
3 ∠VUW≅∠WUX
Additive Property of Angle Measure
Additive Property of Length
All right angles are congruent
Angles forming a linear pair sum to 180°
Definition of angle bisector
Definition of equilateral triangle
Definition of midpoint
Vertical Angle Theorem
4
UW
≅
UW
Reflexive Property of Congruence
5 △UWX≅△UWV AAS

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12