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Classifying Triangles

Proving Congruence

Coordinate Proof

Congruence in Right Triangles

Isosceles Triangles

Classifying Triangles for $100

Classify the following triangle by sides and angles. Give all possible names:

Answer

Acute, equiangular, equilateral, isosceles

Back

Classifying Triangles for $200

Define: Isosceles Triangle

Answer

Isosceles Triangle – A three sided polygon where two or more sides are congruent

Back

Classifying Triangles for $300

Classify the following triangle by sides and angles. Give all possible names:

Answer

Isosceles, Right

Back

Classifying Triangles for $400

Classify the following triangle by sides and angles. Give all possible names:

Answer

Back

Scalene

Classifying Triangles for $500Given that the two triangles below

are congruent, then triangle ABC is congruent to _____. Also, identify the congruent, corresponding parts.

A

B

C

D

E

F

Triangle ABC is congruent to Triangle EDF.AB = EDBC = DFAC = EF<A = <E<B = <D<C = <F

Answer

Back

Proving Congruence for $100

List all the ways to prove congruence in right triangles:

Answer

HA – Hypotenuse- Angle

HL – Hypotenuse - Leg

LL – Leg - Leg

LA – Leg - Angle

Back

Proving Congruence for $200

List all the ways to prove congruence in triangles:

Answer

ASA – Angle – Side – Angle

SAS – Side – Angle – Side

AAS – Angle – Angle – Side

SSS – Side – Side - Side

Back

Proving Congruence for $300

Given triangle ABC is congruent to triangle PQR, m<B = 3x+4, and m<Q = 8x-6, find m<B and m<Q

Answer

m<B = m<Q => CPCTC3x+4 = 8x – 610 = 5x2 = xm<B = 3x+ 4 = 3*2+4 = 10 degreesm<Q = 8x-6 = 8*2-6 = 10 degrees

Back

Proving Congruence for $400

Given: RS = UT; RT = US

Prove: Triangle RST = Triangle UTS

Answer

Back

Statements ReasonsRS = UT Given

RT = US Given

ST = ST Reflexive Property of Congruence

Triangle RST is congruent to triangle UTS

SSS

Proving Congruence for $500

Can you prove that triangle FDG is congruent to triangle FDE from the given information? If so, how?

Answer

Back

Yes, ASA or AAS

Congruence in Right Triangles for $100

Is it possible to prove that two of the triangles in the figure below are congruent? If so, name the right angle congruence theorem that allows you to do so.

Answer

Back

Yes, Hypotenuse – Leg Congruence

Congruence in Right Triangles for $200

Given that AD is perpendicular to BC, name the right angle congruence theorem that allows you to IMMEDIATELY conclude that triangle ABD is congruent to triangle ACD

Answer

Back

Hypotenuse – Angle Congruence

Congruence in Right Triangles for $300

Name the right angle congruence theorem that allows you to conclude that triangle ABD is congruent to triangle CBD

Answer

Leg- Leg Congruence

Back

Congruence in Right Triangles for $400

Is there enough information to prove that triangles ABC and ADC are congruent? If so, name the right angle congruence theorem that allows you to do so.

Answer

Yes, Hypotenuse – Leg Congruence

Back

Congruence in Right Triangles for $500

What additional information will allow you to prove the triangles congruent by the HL Theorem?

Answer

AC is congruent to DC or

BC is congruent to EC

Back

Isosceles Triangles for $100

If a triangle is isosceles, then the ___________ are congruent

Answer

Back

If a triangle is isosceles, then the base angles are congruent

Isosceles Triangles for $200

The angle formed by the congruent sides of an isosceles triangle is called the ____________

Answer

Back

The angle formed by the congruent sides of an isosceles triangle is called the vertex angle

Isosceles Triangles for $300

Name the congruent angles in the triangle below. Justify your answer:

A

B

C

Answer

Back

<A <C by the Isosceles Triangle Theorem

Isosceles Triangles for $400

Given ABC is an equilateral triangle, BD is the angle bisector of <ABC, Prove that triangle ABD is a right triangle

A

B

CD

Answer

Back

Statements ReasonsAB = BC = AC

BD is the angle bisector of <ABC

Given

<ABD = <DBC Definition of Angle bisector

<ABD = 60 degrees Definition of a equilateral triangle

<ABD+<DBC = 60 Angle Sum Theorem

<ABD + <ABD = 60 Substitution

<ABD = 30 Simplify

<BAD = 60 Definition of a equilateral triangle

<BAD + <ADB + <ABD = 180 degrees Triangle Sum Theorem

60 + 30 + <ADB = 180 Substitution

ADB = 90 Simplify

Triangle ABD is a right triangle Definition of right triangles

Isosceles Triangles for $500

Given ABC is an isosceles right triangle, and BD is the angle bisector of <ABC, Prove that triangle ABD is isosceles

A

B

CD

Answer

Back

Statements ReasonsAB = BCABC is a right triangle

BD is the angle bisector of <ABC

Given

<ABD = <DBC Definition of Angle bisector

<ABD = 90 degrees Definition of a right, isosceles triangle

<ABD+<DBC = 90 Angle Sum Theorem

<ABD + <ABD = 90 Substitution

<ABD = 45 Simplify

<BAC = <BCD Isosceles Triangle Theorem

<BAC + <BCA + <ABC = 180 degrees Triangle Sum Theorem

<BAC + <BAC + 90 = 180 Substitution

<BAC = 45 Simplify

<BAC = <ABD Substitution

AD = BD Isosceles Triangle Theorem

Triangle ABD is isosceles Definition of Isosceles Triangles

Coordinate Proof for $100

Draw the following triangle on a coordinate plane. Label the coordinates of the vertices:

An equilateral triangle where the length of the base is 2a and the height is b

Answer

Back

A (0,0) B (2a,0)

C (a,b)

Coordinate Proof for $200

Draw the following triangle on a coordinate plane. Label the coordinates of the vertices:

A Scalene Triangle

Answer

Back

A (0,0) B (a,0)

C (b,c)

Coordinate Proof for $300

Draw the following triangle on a coordinate plane. Label the coordinates of the vertices:

An isosceles triangle with base a and height c

Answer

Back

A (0,0) B (a,0)

C (a/2,c)

Coordinate Proof for $400

Write a coordinate proof to prove that if a line segment joins the midpoints of two sides of a triangle, then its length is equal to one-half the length of the third side.

Answer

BackA (0,0) B (a,0)

C (b,c)

S (b/2,c/2) T ((a+b)/2,c/2)

ST = √(((a+b)/2) – (b/2))^2 + (c/2 – c/2)^2)

ST = √((a/2)^2 + 0)

ST = a/2

AB = √(((a-0)/2) – (b/2))^2 + (0 -0)^2)

AB = √((a)^2 + 0)

AB = a

Thus, ST = ½ AB

Coordinate Proof for $500

Use coordinate proof to prove that a triangle with base a and height b such that the vertex aligns vertically with the midpoint of the base is isosceles

Answer

BackA (0,0) B (a,0)

C (a/2,b)

CA = √((a/2 – 0)^2 + (b – 0)^2)CA = √((a/2)^2 + b^2)AB = √((a – a/2)^2 + (b -0)^2)AB = √((a/2)^2 + b^2)Thus CA = AB so Triangle ABC is Isosceles