Unit 5 ReviewState Standards

2: Write geometric proofs.

5: Prove triangles are congruent.

12: Find and use measures of sides and angles in triangles.

13: Prove angle relationships.

16: Perform basic constructions.

17: Prove theorems by using coordinate geometry.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

Classifying Triangles by sides

Scalene: no congruent sides Isosceles: at least two congruent sides Equilateral: all three sides congruent

by angles Acute: 3 acute angles

Equiangular: all angles congruent (60o) Right: 1 right angle Obtuse: 1 obtuse angle

Theorems on Triangles Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180o.

Exterior Angles Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Congruent Triangles definition

all corresponding sides are congruent all corresponding angles are congruent

postulates and theorems to prove congruent triangles

SSS SAS ASA AAS HL

Using Congruent Triangles

C.P.C.T.C. is used to indicate that the corresponding parts of already proven congruent triangles are congruent.

Isosceles and Equilateral Triangles The Base Angles Theorem, its converse, and its corollaries are used with Isosceles Triangles.

The angles opposite the congruent sides of an isosceles triangle are congruent If two angles of a triangle are congruent, then the sides opposite the angles are congruent. A triangle is equilateral if and only if it is also equiangular.

Coordinate Proofs Coordinate proofs place figures on a coordinate plane and then use the distance formula, midpoint formula, slope formula, postulates, theorems, and definitions to prove assumptions.

2 2

2 1 2 1distance x x y y

1 2 1 2,2

midp i2

o nt x x y y

2 2 2pythagorean theorem c a b

Marker Boards Out!

Classify the triangle by its angles.

acute

Classify the triangle by its angles.

right

Classify the triangle by its sides

equilateral

also isosceles

Classify the triangle by its sides

scalene

Classify the triangle by its angles and its sides.

right, scalene

Identify the hypotenuse of right triangle DBDC.

BC

Identify the base of isosceles triangle DABC.

AC

Identify the legs of right triangle DABD.

and AD BD

Find the measure of the numbered angle.

48x

42 90x

Find the measure of the numbered angle.

34x

68 102x

2 58 180m 2 122m

Is 1 4?

1 122 22 180m 1 36m

2 3m m

4 122 20 180m 4 38m

1 4

In the diagram, . Complete the statement.TJM PHS

_____P T

In the diagram, . Complete the statement.TJM PHS

_____JT HP

In the diagram, . Complete the statement.TJM PHS

_____m J 59

Name the included angle between the pair of sides given.

MTR

Name the included angle between the pair of sides given.

Q

Name the included side between the pair of angles given.

WY

and XWY XYW

Name the included side between the pair of angles given.

HU

and CUH CHU

Which postulate or theorem can be used to prove the two triangles congruent?

SSS

SAS

Which postulate or theorem can be used to prove the two triangles congruent?

,JHK JLK

AAS

Which postulate or theorem can be used to prove the two triangles congruent?

,JUI LUI

SAS

Which postulate or theorem can be used to prove the two triangles congruent?

,DKA TKS

AAS

Which postulate or theorem can be used to prove the two triangles congruent?

,TNS UHS

not possible

Which postulate or theorem can be used to prove the two triangles congruent?

,MAE TAE

SAS

Which postulate or theorem can be used to prove the two triangles congruent?

,CAR RDC

State the third congruence that must be given to prove that using the indicated postulate or theorem.DEF MNF

,

SAS

DE MN D M

DF MF

D

E

F

M

N

State the third congruence that must be given to prove that using the indicated postulate or theorem.DEN MEN

,

ASA

D M DNE MNE

DN MN

D

E

M

N

Statements Reasons

,WU YV XU ZVWX YZ

Given

Z WXUZYV W

Corresponding s Post.

WXU YZW ASA

If , which angles are congruent?RI IT

1 3

If , which angles are congruent?TN IT

11 8

If 1 6, which segments are congruent?

RG HG

Find the missing coordinates without using any new variables.

,0a

0,b

Find the missing coordinates without using any new variables.

,c b

,a d

Solve for and .x y

72x

36y

Solve for and .x y

45x

135y

Solve for .x

54 180

2 54 180

63

63x x

2 63x

31.5x

Statements Reasons

,WU YV XU ZVWX YZ

Given

Z WXUZYV W

Corresponding s Post.

WXU YZW ASA

Statements Reasons

is the midpoint of is the midpoint of

B AEB CD

Given

,AB EB CB DB Def. Midpoint

ABD EBC Vertical Angles Theorem

ABD EBC SAS

Given: is the midpoint of is the midpoint of

Prove:

B AEB CD

ABD EBC

48 0 0 48,

2 2

24,24

(48, 48)

Use the given information and diagram to find the coordinates of .H OBH BDH

0 24 8 0,

2 2

12,4

Use the given information and diagram to find the coordinates of .H OBC OAC