2 angles that add up to 90 degrees, don’t have to be adjacent
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supplementary
2 angles that add up to 180 degrees, don’t have to be adjacent
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Linear pair
adjacent supplementary angles
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vertical angles
when 2 lines intersect, the non-adjacent angles are vertical angles
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vertical angles theorem
all vertical angles are congruent
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transversal
a line that intersects two coplanar lines at two different points
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corresponding angles
lie on the same side of the transversal and same sides of the intersecting lines
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same-side interior angles
lie on the same side of the transversal and between the intersected lines
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alternate exterior angles
lie on opposite sides of the transversal and outside the intersected lines
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alternate interior angles
are nonadjacent angles that lie on opposite sides of the transversal between the intersected lines
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parallel lines
lie in the same plane and never intersect →- indicates parallel lines same direction. ll symbolizes parallel too.
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same-side interior angles postulate
if two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary
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alternate interior angles theorem
if two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure
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corresponding angles theorem
if two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure
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alternate exterior angles are…
congruent
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transitive property
if a=b and b=c then a=c
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subtraction property
an equal value subtracted or removed from two equal items will result in a new equal amount
if a=b then a-c=b-c
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substitution
if a=b then b can replace a in any case
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converse of the same-side interior angles postulate
if two lines are cut by a transversal so that pair of same-side interior angles are supplementary, then the lines are parallel
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converse of the alternate interior angles theorem
if two lines are cut by a transversal so that any pair of alternate interior angles are congruent, they are parallel
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converse of the corresponding angle theorem
if two lines are cut by a transversal so that any pair of corresponding angles are congruent, then the lines are parallel
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the parallel postulate
through a point “P” not on the “L” there is exactly one line parallel to “L”
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Perpendicular bisector theorem
if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment
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Perpendicular line slope relationship
opposite reciprocal
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Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment
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distance formula
d=√(x₂-x₁)² + (y₂-y₁)²
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point-slope formula
y - y₁ = m ( x - x₁ )
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midpoint formula
(x₁ + x₂ / 2) , (y₁ + y₂ / 2)
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slope formula
rise/ run y₂-y₁ / x₂-x₁
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slope intercept form
y = mx + b
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parallel line slope relationship
same slope
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CPCFC
corresponding parts of congruent figures are congruent
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CPCTC
corresponding parts of triangles are congruent (biconditional)
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Bioconditional
p if and only if q - statement can be written in that form
Two triangles are congruent if and only if corresponding pairs of angles are congruent
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Contrapositive
“if p then q” “if not q, then not p”
contrapositive of true statement is true
ex. if corresponding pairs of sides or corresponding pairs of angles are not congruent, then the triangles are not congruent
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ASA
angle, included side, angle
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ASA Triangle Congruence Theorem
if two angles and the included side of one triangle are congruent to. two angles and the included side of another triangle, then the triangles are congruent
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reflexive property
a=a
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midpoint
of a line segment is the point that divides the segment into two segments that have the same length
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right angles
all right angles are congruent; 90 degrees
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bisector
goes through an angle (angle bisector), segment (segment bisector), line to form two equal parts, angles, etc
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congruent supplements theorem
if two angles are supplements of the same angle (or congruent angles) then the two angles are congruent
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congruent complements theorem
if two angles are complements of the same angle (or congruent angles), then the two angles are congruent
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regular polygon
all angle measures are the same, all side lengths are the same (equiangular and equilateral)
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SAS
side, included angle, side
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SAS Triangle Congruence Theorem
if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the the triangles are congruent
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perpendicular bisector
perpendicular bisector of a line segment is a line perpendicular to the segment at the segments midpoint. Form right angles
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SSS
side, side, side
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SSS Triangle Congruence Theorem
if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
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two triangle congruence that don’t work
AAA & SSA
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AAS
angle, angle, side
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AAS Triangle Congruence Theorem
if two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.
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HL
hypotenuse, leg
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HL Triangle Congruence Theorem
if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent
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pythagorean theorem
a² + b² = c² (a bottom, b side, c hypotenuse)
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Triangle Sum Theorem
the sum of the angle measures of a triangle is 180 degrees
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Polygon Angle Sum Theorem
the sum of the measures of the interior angles of a convex polygon with “n” sides is n-2(180)
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concave
in, dip in shape “M”
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convex
out, up ;“O”;”G"
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exterior angle (polygon)
an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles
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remote interior angle
an interior angle that is not adjacent to the exterior angle
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exterior angle theorem
the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles
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isosceles triangle (know parts)
a triangle with at least two congruent sides
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isosceles triangle theorem
if two sides of a triangle are congruent, then the two angles opposite the sides are congruent
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Converse of the Isosceles Triangle Theorem
if two angles of a triangle are congruent, then the two sides opposite the angles are congruent
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Equilateral Triangle
a triangle with 3 congruent sides
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Equiangular Triangle
a triangle with 3 congruent angles
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Equilateral triangle theorem
if a triangle is equilateral, then it is equiangular
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Converse of the equilateral triangle congruence theorem
if a triangle is equiangular, then it is equilateral
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Triangle Inequality Theorem
the sum of any two side lengths of a triangle is greater than the third-side length \*\***tip - add two smallest sides****
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Side-Angle Relationships in Triangles
if two sides of a triangle are not congruent, then the larger angle is opposite the longer side (if AC>BC, then
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Angle-Side Relationship in Triangles
if two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (if BC)
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A circle that contains all the vertices of a polygon is circumscribed about the polygon
circumcircle? circumcenter?
circle is called circumcircle
the center of the circle is called the circumcenter
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Point of Concurrency
Three or more lines are concurrent if they intersect at the same point. The point of intersection is called the point of concurrency.
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Circumcenter Theorem
the perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of the triangle PA=PB=PC
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Acute Triangles (circumcenter location)
circumcenter is inside the triangle
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Right Triangle (circumcenter location)
circumcenter is on the triangle
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Obtuse Triangle (circumcenter location)
circumcenter is outside the triangle
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the point from a point to a line is the…
length of the perpendicular segment from the point to the line
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Angle Bisector Theorem
if a point is on the bisector of an angle, then it is equidistant from the sides of the angle
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Converse of the Angle Bisector Theorem
if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle
AC = BC so…
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When is a circle inscribed in a polygon? What’s an inscribed circle called?
if each side of the polygon is tangent (on) to the circle. The inscribed circle is called the incircle
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The center of a circle inscribed in a triangle is called?
The *incenter* of the triangle
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Incenter Theorem
the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle
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Median
of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint
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Centroid of a Triangle
the intersection (point of concurrency) of the three medians of a triangle
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Where is the centroid regarding the triangle? What is it for a triangle?
inside the triangle; center of gravity of a triangle
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Centroid Theorem
the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side
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Altitude of a Triangle
perpendicular segment from a vertex to the line containing the opposite side; every triangle has 3 altitudes; not a bisector, doesn’t necessarily go to midpoint
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Location of Altitude
can be inside, outside, or on the triangle
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Orthocenter & Location?
the intersection (or point of concurrency) of the lines that contain the altitudes; can be inside, outside, or on the triangle
Studied by 15 people
