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Transcript of Geometry
If the legs of the trapezoids are congruent then it is an isosceles
The mid segment of a trapezoid is the segment that connects the mid points of the legs of the trapezoid Base A B D C Base Leg Leg THEOREMS If a trapezoid is isosceles, then each pair of base angles is congruent.
If a trapezoid has one pair of congruent base angles then it is an isosceles trapezoid.
A trapezoid is isosceles if and only if its diagonals are congruent.
The mid segment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases. Quaderlatiral with exactly to pairs of consecutive congruent sides
the opposite sides of kite are not congruent or parallel THEOREMS If a quadrilateral is a kite then its diagonals are perpendicular
If a quadrilateral is a kite then exactly one pair of opposite angles is congruent Is a parallelogram with all four sides congruent THEOREMS If a parallelogram is a rhombus then its diagonals are perpendicular
if a parallelogram is a rhombus then each diagonal bisects a pair of opposite angles
If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus
if one diagonal of a parallelogram bisects a pair of opposite angles then the parallelogram is a rhombus
if one pair of consecutive sides of a parallelogram are congruent the parallelogram is a rhombus Is a parallelogram with four congruent sides and four right angles
All squares are equilateral THEOREM If a quadrilateral is both a rectangle and rhombus then it is a square Is a parellogram with four right angles Properties all four angles are right angles
opposite sides are parallel and congruent
opposite angles are congruent
consecutive angles are supplementary
diagonals bisect each other THEOREMS If a parallelogram is a rectangle then its diagonals are congruent
if the diagonals of a parallelogram are congruent then the parallelogram is a rectangle A diagonal of a polygon is a segment that connects any two non consecutive vertices
The sum of the angle measures of a polygon is the sum of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex THEOREMS The sum of the interior angle measures of an N/sided convex polygon is (N-2)x180 Is a quadrilateral with both pairs of opposite sides parallel THEOREMS If a quadrilateral is a parallelogram then its opposite sides are congruent
If a quadrilateral is a parallelogram then its opposite angles are congruent
if a quadrilateral is a parallelogram then its consecutive angles are supplementary
If a parallelogram has one right angle then it has four right angles
if a quadrilateral is a parallelogram then its diagonals bisect each other
if a quadrilateral is a parallelogram then each diagonal separates the parallelogram into two congruent triangles Tests for Parallelograms If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram
if both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a parallelogram
if the diagonals of the quadrilaterals bisect each other then the quadrilateral is a parallelogram
If one pair of opposite sides of a quadrilateral is both parallel and congruent then the quadrilateral is a parallelogram Relationships in Triangles Bisectors of Triangles Medians and altitudes of triangles Inequalities in One Triangle Indirect Proof The Triangle Inequality Inequalities in Two Triangles Hinge theorem: If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
Converse of the Hinge Theorem: If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second triangle, then the included angle measure of the first triangle is greater than the included angle measure in the second triangle. Theorem The sum of the lengths of any two sides of a triangle must be greater than the length of the third side Indirect reasoning- assuming that a conclusion is false and then showing that this assumption leads to a contradiction
Indirect Proof/Proof by contradiction: when you temporarily assume that what your trying to prove that the assumption is false and the original conclusion true Steps to Writing an Indirect Proof Step One: Identify the conclusion that you are asked to prove. Make the assumption that this conclusion is false by assuming that the opposite is true.
Step Two: Use logical reason to show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, corollary.
Step Three: Point out that since the assumption leads to a contradiction, the original conclusion, what you are asked to prove, must be true. Definition of inequality: for any real numbers A and B, A>B if and only if there is a positive number C such that A=B+C
Comparison Property: A<B, A=B, A>B
Transitive Property: If A<B<C, then A<C; If A>B>C, then A>C
Addition Property: If A>B, then A+C>B+C; If A<B then A+C<B+C
Subtraction Property: If A>B, then A-C>B-C; If A<B then A-C<B-C THEOREMS Exterior angle inequality theorem: The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles
Angle Side Relationships in triangles: If one side of a triangle is longer then another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side; If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle The median of a triangle is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side
A centroid is a point of concurrency of the medians of a triangle.
The altitude of a triangle is a segment of a vertex to the line containing the opposite side and perpendicular to the line containing that side
An altitude can lie in the interior, exterior, or on the side of the triangle.
An orthocenter is a point of concurrency of the altitudes of a triangle. THEOREM Centroid Theorem: the Medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side. THEOREMS Perpendicular Bisector Theorem- If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Converse of the Perpendicular Bisector Theorem- If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment
Circumcenter Theorem- The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.
Angle Bisectors- If a point is on a bisector of an angle, then it is equidistant from the sides of the angle
Converse of an Angle Bisector- 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.
Incenter-the angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle CHAPTER 6 CHAPTER 5 Congruent Triangles Classifying Triangles Angles of Triangles Congruent Triangles Proving Triangles Congruent/
SSS, SAS, ASA, AAS Isosceles and Equalatiral Triangles Congruence Transmissions Acute triangle- has three acute angles
Equiangular triangle- has three congruent angles
Obtuse Triangle- has one obtuse angle and two acute angles
Right Triangle- has one right angle and two acute angles Classification of triangles by angles Classification of triangles by sides Equilateral triangle- has three congruent sides
Isosceles triangle- has at least two congruent sides
Scalene triangle- has no congruent sides Auxiliary line- an extra line or segment drawn in a figure to help analyze geometric relationships
Exterior angles- are formed by one side of a triangle and an extension of an adjacent side
Remote interior angles- the angles of a triangle that are not adjacent to a given exterior angle
Flow proof- A proof that organizes statements in logical order, starting with the given statements. Each statement is written in a box with the reason verifying the statement written below the box. Arrows are used to indicate the order of the statements.
Corollary- theorem with a proof that follows as a direct result of another theorem Theorems Triangle angle-sum: the sum of the measures of the angles of a triangle is 180
Exterior angle theorem- the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles Triangle Angle-Sum Corollaries The acute angles of a right triangles are complimentary
There could be at most one right or obtuse angle in a triangle Congruent- if two geometric figures have the same shape or size
Congruent Polygons- when all the parts of one polygon are congruent to the corresponding parts or matching part of the other polygon Theorem Third angles theorem: if two angles of one triangles are congruent to two angles of a second triangles, then the third angles of the triangles are congruent. Postulates SSS- if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
SAS- If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent
ASA- 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
AAS- if two angles and the non included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent Included angle- the angle formed by two adjacent sides of a polygon
Included side: A side located between two consecutive angles of a polygon. Legs of an Isosceles Triangle- two congruent sides are
Vertex angle- is the angle with sides that are the legs
Base angles- the two angles formed by the base and the congruent sides Theorem Isosceles triangle theorem- if two sides of a triangle are congruent, then the angles opposite those sides are congruent
Converse of isosceles triangle theorem- if two angles of a triangle are congruent, then the sides opposite those angles are congruent Corollaries: Equilateral Triangle A triangle is equilateral if and only if it is equiangular
Each angle of an equilateral measures 60 Transformation- is an operation that maps an original geometric figure
Congruence transformation- is one in which the position of the image may differ from that of the original, but the two figures remain congruent.
Coordinate proofs- the use of figures in a coordinate plane and algebra to prove geometric concepts. The Three Main Types of Congruence Transformations Reflection/Flip- is a transformation over a line called the line of reflection
Translation/Slide- is transformation that moves all points of the original figure the same distance in the same direction.
Rotation/Turn- is a transformation around a fixed point called the center of a rotation, through a pacific angle, and a pacific direction. This triangle is an obtuse triangle since it has one obtuse angle EXAMPLE: EXAMPLE: Flow Proof EXAMPLE: SSS Congruence EXAMPLE: Proof Isosceles Triangle Theorem Reflection Translation Rotation EXAMPLE: correct answer: I EXAMPLE: What is ZX? The Answer is: ZX=6 CHAPTER 4