In convex quadrilateral ABCD, AB = CD, <ABC = 77 degrees and <BCD = 150 degrees.
Let P be the intersection of the perpendicular bisectors of line segment BC and line segment AD.
Find <BPC.
To find the measure of angle <BPC, we need to understand the properties of a convex quadrilateral and perpendicular bisectors.
A convex quadrilateral is a four-sided polygon in which all the interior angles are less than 180 degrees.
The perpendicular bisector of a line segment is a line that divides the line segment into two equal parts and is perpendicular to the line segment at its midpoint.
To find angle <BPC, we'll use the fact that the intersection of perpendicular bisectors of two sides of a quadrilateral is the circumcenter of the quadrilateral.
The circumcenter is the center of the circle through the vertices of the quadrilateral. It is equidistant from all the vertices.
Now let's solve the problem step by step:
Step 1: Given that AB = CD, this tells us that segment AB is congruent to segment CD.
Step 2: Given that <ABC = 77 degrees and <BCD = 150 degrees, we can observe that angles ABC and BCD are supplementary angles. (They add up to 180 degrees.)
Step 3: Because the perpendicular bisectors of BC and AD intersect at point P, point P is the circumcenter of quadrilateral ABCD.
Step 4: The circumcenter is equidistant from all the vertices of the quadrilateral.
Step 5: Since AB = CD, point P is equidistant from A and D.
Step 6: As P is the circumcenter, PA = PD.
Step 7: This means triangle BPC is an isosceles triangle because BP and CP are radii of the circumcircle, so BP = CP.
Step 8: In an isosceles triangle, the base angles are congruent.
Step 9: Therefore, angle <BPC is congruent to angle <BAC. (Angle <BAC is one of the base angles of triangle BPC.)
Step 10: Since angle BAC is given as 77 degrees, angle <BPC is also 77 degrees.
Thus, we have found that angle <BPC is 77 degrees.