Geo postulatestheorems list 2011Visual Clue Complementary Two angles whose measures
have a sum of 90Transversal A line that intersects two lines in the same Pairs of angles formed by two lines and a transversal that make an F pattern Angle bisector A ray that begins two angles of equal measure Segment bisector A ray, line or segment that divides a segment Perpendicular A line that bisects a segment and is Altitude A segment from a vertex of a triangle Definitions, Postulates and Theorems Page 2 of 11 Definitions Visual Clue Geometric mean The value of x in proportion
numbers (x is the geometric mean between a Sine, sin For an acute angle of a right triangle, the ratio e angle to the measure Cosine, cos For an acute angle of a right triangle the ratio of the side adjacent to the angle to the measure Tangent, tan For an acute angle ngle, the ratio the angle to the measure of the side adjacent (opp/adj)
Algebra Postulates Name Visual Clue If the same number is added to equal numbers, then the sums are equal If the same number is subtracted from equal numbers, then the differences are equal Multiplication Prop.Of equality If equal numbers are multiplied by the same number, then the products are equal If equal numbers are divided by the same number, then the quotients are equal A number is equal to itself Symmetric Property of Equality If a = b then b = a
Substitution Prop.Of If values are equal, then one value may be substituted for the other.
Transitive Property of If a = b and b = c then a = c
Distributive Property a(b + c) = ab + ac
Congruence Postulates Name Visual Clue Reflexive Property of Congruence AA
Symmetric Property of
Transitive Property of Congruence B then CA
Definitions, Postulates and Theorems Page 5 of 11 Visual Clue Similarity If two angles of one triangle are equal in measure triangles are similar Similarity Theorem then the triangles are similar.
Similarity Theorem are congruent, then the triangles are similar.Theorem ngle are congruent to two are equal in measure to the corresponding sides and angle of another triangle, then the triangles are ngle are equal in measure the triangles are congruent 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.
Theorem The sum of the measure of the angles of a triangle Corollary The acute angles of a right triangle are complementary.Exterior angle theorem An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote Theorem If a line parallel to a side of a triangle intersects the Theorem Definitions, Postulates and Theorems Page 7 of 11 Triangle Postulates And Theorems Visual Clue Theorem The centriod of a triangle is located 2/3 of the distance from each vertex to the midpoint of the Midsegment Theorem A midsegment of a triangle is parallel to a side of half the length of that Theorem
are not congruent, then Theorem Theorem The sum of any two side lengths of a triangle is Theorem and the third sides are not the larger included angle.Theorem and the third sides are not from the longer third side.Pythagorean Theorem If the sum of the squares of the lengths of two
the triangle is a right Pythagorean Theorem Theorem
length times the square root of 2.
triangle, the length of the hypotenuse is 2 times the length of the shorter leg, shorter leg times the square root of 3.
Law of Sines For any triangle ABC cCbBaAsinsinsin
For any triangle, ABC with Definitions, Postulates and Theorems Page 9 of 11 Polygon Postulates And Theorems Visual Clue Theorem If a parallelogram is a rhombus then its Theorem If a parallelogram is a rhombus, then each Theorem If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.Theorem
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
Theorem If one pair of consecutive sides of a Theorem If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.Theorem If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
Theorem If a quadrilateral is Theorem If a quadrilateral is Theorem
If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.Theorem
If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.
Theorem A trapezoid is isosceles if and only if its Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.
Definitions, Postulates and Theorems Page 11 of 11 Circle Postulates And Theorems Name Visual Clue Theorem If a tangent and a secant (or chord) intersect on a ency, then the measure of the angle formed is half the measure of its Theorem If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of the intercepted arcs.
Theorem If a tangent and a secant, two tangents or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measure of its intercepted arc.Theorem If two chords intersect in the interior of a circle, then the products of the lengths of the segments of Theorem If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.Theorem If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared
.The equal of a circle with center (h, k) and radius r
Chapter 5 Congruence Postulates &theorems - ’s
SSS Postulate If three sides of one triangle are congruent, respectively, to three ... I have to use the definition or the SSS Congruence postulate. (riverview.wednet.edu)
Geometry Online Sample Lesson - Oak Meadow: Homeschooling ...
SSS POSTULATE Is it always necessary to show that all of the corresponding parts of two triangles ... By definition of congruent segments, all corresponding segments are (hanlonmath.com)
Practice A Triangle Congruence: Sss And Sas
Triangle Congruence: SSS and SAS For each definition, tell what angle measures or side lengths of the quadrilateral ... Side-Side-Side (SSS) Congruence Postulate (oakmeadow.com)
Geometry sample3 of 29Side-Side-Side Congruence If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.Abbreviation: SSS The tail of an orca whale can be viewed as two triangles that share a common side.Write a two-column proof to prove that
and Prove:Proof: Determine whether
for R(2,5), Z(1,1), T(5,2), L(-3,0), K(-7,1), and J(-4,4).Explain.Use the Distance Formula to show that the corresponding sides are congruent.5 of 29Congruent Triangles using Two Sides and the Included AngleDraw a triangle and label its vertices A, B, and C.1.Select a point K on line m.
Use a compass to construct
on m such that 2.Construct an angle congruent to
as a side of the angle and point K as the vertex.Construct such that 3.Draw
to complete 4.Cut out
and place it over
compare to ?5.You can also construct congruent triangles given two sides and the included angle.6 of 29X is the midpoint of X is the midpoint of Flow Proof:prove that they are congruent, write not possible.Each pair of corresponding sides is congruent.The triangles are congruent by the SSS Postulate.a.The triangles have three pairs of corresponding angles congruent.This does not match the SSS Postulate or the SAS Postulate.It is not possible to prove the triangles congruent.b.Draw a triangle and label the vertices.
Name two sides and the included angle.1.FIND THE ERROR Carmelita and Jonathan are trying to determine
is congruent to 2.8 of 298.PRECISION FLIGHT The United States Navy Flight Demonstration Squadron, the Blue Angels, fly in a formation that can be viewed as two triangles with a common side.
Write a two-column proof to prove that
if T is the midpoint of
and Determine whether
given the coordinates of the vertices.Explain.9.J(-3,2), K(-7,4), L(-1,9), F(2,3), G(4,7), H(9,1)10.J(-1,1), K(-2,-2), L(-5,-1), F(2,-1), G(3,-2), H(2,5)11.J(-1,-1), K(0,6), L(2,3), F(3,1), G(5,3), H(8,1)12.J(3,9), K(4,6), L(1,5), F(1,7), G(2,4), H(-1,3)Write a flow proof.13.Given: Prove: 14.14 of 29The Bank of China Tower in Hong Kong has triangular trusses for structural support.
These trusses form congruent triangles.In this lesson, we will explore two additional methods of proving triangles congruent.between them, the included side.Do these measures form a unique triangle?Congruent Triangles Using Two Angles and Included SideDraw a triangle and label its vertices A, B, and C.1.Draw any line m and select a point l.Construct
such that 2.Construct an angle congruent to
at L using
as a side of the angle.3.Construct an angle congruent to
at K using
as a side of the angle.
Label the point where the newsides of the angles meet J.4.16 of 29Copy , , and
on another piece of patty paper and cut them out.2.Assemble them to form a triangle in which the side is not the included side of the angles.3.AnalyzePlace the original
over the assembled figure.How do the two triangles compare?1.Make a conjecture about two triangles with two angles and the nonincluded side of one triangle congruent to two angles and the nonincluded side of the other triangle.2.In the activity, you see that two triangles that have two pairs of corresponding angles congruent and congruent nonincluded sides are congruent.Theorem 4.5Angle-Angle-Side Congruence If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.Abbreviation: AAS17 of 29Prove:Proof:Prove:Flow Proof:20 of 29Given: Prove: 7.PARACHUTES Suppose
each measure 7 feet,
each measure 5.5 feet, and
Justify your answer.Write a flow proof.8.Given: Prove: 9.21 of 29Given: Prove: 10.Given: Prove: 11.Given: Prove: 12.Given: Prove: 13.Given: Z is the midpoint of Prove: Write a paragraph proof.14.23 of 29Given: Prove:
For Exercises 21 and 22, use the following information.Beth is planning a garden.She wants the triangular sections,
and , to be congruent.F is the midpoint of , and DG = 16 measure 4 feet and the measure of
is 29.Determine whether
Justify your answer.21.Suppose F is the midpoint of , and .
Justify your answer.25 of 29Answer the question that was posed at the beginning of the lesson.How are congruent triangles used in construction?Include the following in your answer:explain how to determine whether the triangles are congruent, andwhy it is important that triangles used for structural support are congruent.In
are angle bisectors and
What is the measure of ?26A.52B.76C.128D.31.ALGEBRA For a positive integer x, 1 percent of x percent of 10,000 a flow proof.(Lesson 4-4)32.Given: Prove: 33.Given: Prove: 34.26 of 29Verify that each of the following preserves congruence and name the congruence transformation.(Lesson 4-3)Write each statement in if-then form.(Lesson 2-3)36.Happy people rarely correct their faults.37.A champion is afraid of losing.
Classify each triangle according to its sides.
(To review classification by sides, see Lesson 4-1.)22.214.171.124.28 of 29Draw
so that XY = centimeters.Use a protractor to draw a ray from Y that is perpendicular to .Open your compass to a width of 10 centimeters.Place the point at X and draw a long arc to intersect the ray.Label the intersection Z and draw
to complete Does the model yield a unique triangle?4.Can you use the lengths of the hypotenuse and a leg to show right triangles are congruent?5.Make a conjecture about the case of SSA that exists for right triangles.The two activities provide evidence for four ways to prove right triangles congruent.Write a paragraph proof of each theorem.6.Theorem 4.67.Theorem 4.78..
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