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