Geometry Regents with Accurate
Solutions
Collinear Points - ANS-points that lie on the same line
complementary angles - ANS-have a sum of 90 degrees
supplementary angles - ANS-have a sum of 180 degrees
exterior angle theorem - ANS-in a triangle; two interior angles equal the opposite
exterior angle (x+y=z)
centroid - ANS-in a triangle; intersection of medians, (2x+x=3x)
orthocenter - ANS-in a triangle; intersection of altitudes
incenter - ANS-in a triangle; intersection of angle bisectors, center of inscribed circle
circumcenter - ANS-in a triangle; intersection of perpendicular bisectors, center of
circumscribed circle
acute triangle - ANS-all angles less than 90 degrees, orthocenter, incenter,
circumcenter, centroid all inside triangle
obtuse triangle - ANS-one angle greater than 90 degrees, orthocenter, circumcenter
located outside, incenter, centroid located inside triangle
equilateral triangle - ANS-all angles equal (120 degrees) ortho-, in-, circumcenter, and
centroid all inside and interest at same location
right triangle - ANS-one angle is 90 degrees, ortho- and circumcenter on triangle,
incenter and centroid inside triangle
distance - ANS-square root of (X2-X1)^2+(Y2-Y1)^2
sum of interior angles - ANS-180(n-2)
sum of exterior angles - ANS-360 degrees
single interior angle - ANS-(180(n-2))/2
, single exterior angle - ANS-360/n
equation of a line - ANS-y=mx+b OR Y-Y1=m(X-X1)
midpoint - ANS-(X1+X2)/2 AND (Y1+Y2)/2
slope - ANS-(Y1-Y2) / (X1-X2)
negative slope reciprocals - ANS-found in perpendicular lines
same slopes - ANS-found in parallel lines
equation of median - ANS-1) find midpoint 2) find slope from vertex to opposite midpoint
3) write equation using slope and midpoint
equation of altitude - ANS-1) fnd slope of opposite side 2) take negative reciprocal or
slope 3) write eqaution using vertec and the negative reciprocal slope
equation of opposite bisector - ANS-1) find midpoint 2) find slope of opposite side and
take negative reciprocal 3) write equation using negative reciprocal and midpoint
proving right triangle - ANS-how: using distance 3X and pythatgoreom theroem to show
Pyth Thm works
proving isosceles triangle - ANS-how: distance 3X and Pythagorean theroem to show at
least 2 sides are congruent and Pyth Thm works
proving parallelogram - ANS-how: distance 4X to show both pairs of opposite sides are
congruent
proving rhombus - ANS-how: distance 4X to show all sides are congruent
proving rectangle - ANS-how: distance 6X to show both pairs of opposite sides are
congruent and diagonals are congruent
proving square - ANS-how: distance 6X to show all sides are congruent and diagonals
are congrent
proving trapzeoid - ANS-how: slope 4X to show only one pair of parallel lines
proving isosceles trapzeoid - ANS-how: slope 4X distance 2X to show one pair of
parallel sides and the nonparallel sides are congruent
logic: p=true, q=true - ANS-~p=F, ~q=F, pVq=T, p^q=T, if p then q = T, if p and if q = T
logic: p=true, q=false - ANS-~p=F, ~q=T, pVq=T, p^q=F, if p then q = F, if p and if q = F
Solutions
Collinear Points - ANS-points that lie on the same line
complementary angles - ANS-have a sum of 90 degrees
supplementary angles - ANS-have a sum of 180 degrees
exterior angle theorem - ANS-in a triangle; two interior angles equal the opposite
exterior angle (x+y=z)
centroid - ANS-in a triangle; intersection of medians, (2x+x=3x)
orthocenter - ANS-in a triangle; intersection of altitudes
incenter - ANS-in a triangle; intersection of angle bisectors, center of inscribed circle
circumcenter - ANS-in a triangle; intersection of perpendicular bisectors, center of
circumscribed circle
acute triangle - ANS-all angles less than 90 degrees, orthocenter, incenter,
circumcenter, centroid all inside triangle
obtuse triangle - ANS-one angle greater than 90 degrees, orthocenter, circumcenter
located outside, incenter, centroid located inside triangle
equilateral triangle - ANS-all angles equal (120 degrees) ortho-, in-, circumcenter, and
centroid all inside and interest at same location
right triangle - ANS-one angle is 90 degrees, ortho- and circumcenter on triangle,
incenter and centroid inside triangle
distance - ANS-square root of (X2-X1)^2+(Y2-Y1)^2
sum of interior angles - ANS-180(n-2)
sum of exterior angles - ANS-360 degrees
single interior angle - ANS-(180(n-2))/2
, single exterior angle - ANS-360/n
equation of a line - ANS-y=mx+b OR Y-Y1=m(X-X1)
midpoint - ANS-(X1+X2)/2 AND (Y1+Y2)/2
slope - ANS-(Y1-Y2) / (X1-X2)
negative slope reciprocals - ANS-found in perpendicular lines
same slopes - ANS-found in parallel lines
equation of median - ANS-1) find midpoint 2) find slope from vertex to opposite midpoint
3) write equation using slope and midpoint
equation of altitude - ANS-1) fnd slope of opposite side 2) take negative reciprocal or
slope 3) write eqaution using vertec and the negative reciprocal slope
equation of opposite bisector - ANS-1) find midpoint 2) find slope of opposite side and
take negative reciprocal 3) write equation using negative reciprocal and midpoint
proving right triangle - ANS-how: using distance 3X and pythatgoreom theroem to show
Pyth Thm works
proving isosceles triangle - ANS-how: distance 3X and Pythagorean theroem to show at
least 2 sides are congruent and Pyth Thm works
proving parallelogram - ANS-how: distance 4X to show both pairs of opposite sides are
congruent
proving rhombus - ANS-how: distance 4X to show all sides are congruent
proving rectangle - ANS-how: distance 6X to show both pairs of opposite sides are
congruent and diagonals are congruent
proving square - ANS-how: distance 6X to show all sides are congruent and diagonals
are congrent
proving trapzeoid - ANS-how: slope 4X to show only one pair of parallel lines
proving isosceles trapzeoid - ANS-how: slope 4X distance 2X to show one pair of
parallel sides and the nonparallel sides are congruent
logic: p=true, q=true - ANS-~p=F, ~q=F, pVq=T, p^q=T, if p then q = T, if p and if q = T
logic: p=true, q=false - ANS-~p=F, ~q=T, pVq=T, p^q=F, if p then q = F, if p and if q = F
