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Distance Formula
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Length of a segment
having endpoints
(x1 , y1 ) and ( x2 , y2 ) |
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2 angles which have a sum of 180° |
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2 angles having a sum of 90° |
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adjacent supplementary angles
[image] |
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Triangle Angle Sum Theorem |
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The angles of a triangle have a sum of 180° |
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1. has endpoints which are the midpoints of two sides
2. is parallel to the third side of the triangle
3. has a length half of the third side
2(length of midsegment) = length of third side |
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The measure of the exterior angle of a triangle is equal to the sum of the two remote interior angles.
[image] |
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intersect to form right angles
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equal slopes
never intersect
coplanar |
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Any value is equal to itself.
Any figure is congruent to it.
3=3
AB≅AB
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The segment from the vertex of a triangle, perpendicular to the opposite side.
Also called the height.
The intersection of the three altitudes of a triangle is called the ORTHOCENTER. |
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The intersection of the three altitudes of a triangle. |
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The intersection of the three angle bisectors of a triangle.
To inscribe a circle in a triangle, you must first construct at least two angle bisectors then construct a perpendicular from the incenter to a side. |
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The intersection of the three perpendicular bisectors of a triangle.
To circumscribe a circle around a triangle, you must first construct at least two perpendicular bisectors of the sides, then use the distance from the circumcenter to a vertex as the radius of your circle. |
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If two sides of a triangle are known, how do you find the possible range of values for the third side?
Let two sides be x and y.
x>y |
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The missing third side is between the difference and sum of the two given sides.
The third side is between (x-y) and (x+y)
The third side is greater than (x-y) and less than (x+y)
(x-y) < missing side < (x+y) |
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How can you determine if three segments lengths can form a triangle?
Let the three sides be x,y, and z.
The smallest values are x and y. |
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The sum of the two smallest side lengths must be more than the third side length.
x + y > z
(Cannot be equal or less) |
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If a=b then b=a
If m<ABC= 40°, then 40° = m<ABC. |
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Midpoint of a segment having endpoints
(x1, y1)(x2, y2) |
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Slope "m" of the line containing two given points
(x1, y1 ) and (x2, Y2) |
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Parallel lines are coplanar and do not intersect.
Parallel lines have same slope "m" in the equations
y=mx+b
and y-y1 = m(x-x1) |
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The slopes of perpendicular lines are opposite reciprocals.
If one is 2/3, the other is -3/2
The product of the slopes of perpendicular lines = -1
2 • -3 = -1
3 2 |
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̴q → ̴p
The contrapositive has the same truth value as the original conditional. |
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p→ q, q→ r, p→ r
If p, then q. If q, then r.
∴ If p, then r. |
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p→q
p
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∴q
It does not work backwards.
p→q
q
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no conclusion possible |
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a segment, ray , line or plane that intersects a segment at its midpoint at a right angle |
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an angle with degree measure less than 90° |
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an angle with degree measure between 90° and 180° |
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an angle with measure 180° |
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two rays with the same endpoint which are collinear
<----•-----------------•------------------•---->
A B C
ray BA and ray BC are opposite rays. |
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alternate interior angles |
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if two parallel lines are cut by a transversal, the alternate interior angles are congruent.
[image]
m∠5=m∠6 and m∠7= m∠8 |
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same-side interior angles |
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If two parallel lines are cut by a transversal, the same-side interior angles are supplementry and have a sum of 180°.
[image]
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If two parallel lines are cut by a transversal the corresponding angles are congruent (equal). [image] |
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[image]
List the sides of the triangle in increasing order. |
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[image]
List the angles in decreasing order. |
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[image]
What is the possible range of values for s?
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same-side exterior angles |
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If two parallel lines are cut by a transversal, same-side exterior angles are supplementary.
[image]
a+b=180 |
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alternate exterior angles |
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[image]
If two parallel lines are cut by a transversal, alternate interior angles are congruent.
<1=<8 and <2=<7 |
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The intersection of the three medians of a triangle.
[image] |
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construct an angle bisector |
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http://www.hstutorials.net/math/geometry/constructions/bisect_acute_angle.htm[image] |
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Construct the Perpendicular Bisector |
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http://www.hstutorials.net/math/geometry/constructions/perp_bisector.htm[image] |
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1. used to find the length of a segment on a number line
2. if the endpoints are A and B
the length of AB = l A-B l
3. Absolute value will produce a positive answer. |
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share a common side and a common vertex, but no common interior points.
[image] |
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construct a perpendicular from a point not on a line |
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http://www.hstutorials.net/math/geometry/constructions/perpendicular_not_on_line.htm
[image] |
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The median is the segment from the vertex of a triangle to the midpoint of the opposite side.[image] |
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What is the formula to find the sum of all the interior angles of a convex polygon with "n" sides? |
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180(n-2)
or
180(#sides - 2) |
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SSS congruency of triangles |
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ASA congruency of triangles |
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SAS congruency of triangles |
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What is the sum of the exterior angles of a convex polygon? |
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What is the formula to find the measure of one exterior angle of a regular convex polygon, if you know how many sides it has? |
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What is the formula to determine how many sides a regular polygon has, if you know the measure of an exterior angle? |
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360/ext. angle
or
360÷(measure of one exterior angle) |
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[image]A segment connecting two non-consecutive vertices of a polygon. |
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A quadrilateral with 2 pairs of opposite parallel sides |
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Opposite sides of a parallelogram are__?__ |
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Opposite angles of a parallelogram are__?__ |
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2 consecutive angles of a parallelogram are _?_ |
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Supplementary
add up to equal 180 |
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The diagonals of a parallelogram __?__ |
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The diagonals of a rectangle are__?__ |
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Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of one is less than the included angle of the other, then the third side of the first triangle is shorter than the third side of the second triangle. |
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The diagonals of a rhombus are __?__ |
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perpendicular
and
bisect opposite angles
[image] |
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A parallelogram with 4 congruent sides.
The diagonals of a rhombus are perpendicular.
The diagonals of a rhombus bisect the opposite angles.
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midsegment of a trapezoid |
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Definition
[image]
MN is the midsegment.
M and N are the midpoints of the legs.
MN is parallel to AD and BC.
MN =½(AD + BC) |
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When a rhombus is given one diagonal, what type of triangles are formed? |
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2 congruent isosceles triangles |
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When both diagonals are drawn in a rhombus, the four triangles formed are __?__ |
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congruent right triangles |
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formula to find arc length in terms of [image], if you know the degree measure of the central angle m° and radius r of the intercepted arc.
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equation of a circle with center (h,k) and radius r |
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how to find chord lengths in a circle |
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