Download Chapter 4 Study Guide Answer Key

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Chapter 4 Study Guide
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Section 4.1 - Classify Triangles and Find Measures of Their Angles
A triangle is a polygon with three sides .
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triangle has no congruent sides.
An isosceles triangle has at least
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congruent sides.
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triangle has one right angle.
An obtuse triangle has one
An equiangular
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triangle has three congruent sides.
An acut~ triangle has three
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triangle has three
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When the sides of a polygon are extended, other angles are fonned. Yhe original angles are the
angles that form linear pairs with the interior angles are the
Triangle Sum Theorem:
Exterior Angle Theorem:
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angles. The
angles.
The sum of the measures of the interior angles of a triangle is ,
0
The measure of an exterior angle ofa triangle is equal to the sum of the measures of~
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Corollary
to the Triangle Sum Tlieorem:
The acute angles of a right triangle are
1.,[\ 2/\ 3~
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Classify the triangle by its sides and by its angles.
60
4.
FindmLABD and mLBDC.
.Find mLCAB and mLCBA .
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n~\\ts;~\~ne, eqU\Ii'\OS eq~\o.\f.m\
3x.~2X+30
X ~ 30
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Section 4.2 - Identify Congruent
In two congruent
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Figures
figures, all the parts of one figure are congruent to the corresponding
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parts of the other figure. In congruent
polygons, this means that the corresponding sides and the corresponding angles are congruent.
Third Angles Theorem:
congruent.
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are
In the diagram,
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"" .6.DEF.
B
1. Find the value ofx.
2. Find the value of)'.
D (5y-
3)0
~-
A~C
E
F
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Section 4.3 - Use Transformations
to Show Congruence
A rigid motion is a transformation that preserves
motion is also called an isometry.
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,and
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. A rigid
are rigid
motions.
Describe the transformation(s)
figure B.
you can use to move figure A onto
Tellwhether a rigid motion can move figure A onto figure
B. If so, describe the transformationls) that can be used. If
not, explain why the figures are not congruent.
3.
~
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Describe the rigid motion(s) that can be used to move
figure A onto figure B.
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Section 4.4 - Use the Side Lengths to Prove Triangles are Congruent
Side-Side-Side
(SSS) Congruence
of one triangle are congruent to three
Postulate:
of a second triangle, then the two triangles are
State the congruence statement for the triangles that can be proven congruent by SSS.
1.
2.
3.
:\1 (
A~B
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\
C
,\ f
G
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Section 4.5 - Use Sides and Angles to Prove Congruence
In a ,igh! "iang!', th, ,id" adja"n! tn th, ,igh! ang!' "" call,d th, ~
The side opposite the right angle is called the ¥eVUUfthe
Side-Angle-Side
(SAS) Congruence
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.
are congruent to two
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Hypote
to the
Postulate:
use-Leg (HL) Congruence
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and a
right triangle.
es
If two •.,
and the included
Theorem:
If the
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.
and the included ~
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of one triangle
of a second triangle, then the two triangles are
lea
and a
ofa second right triangle, then the ~are
of a ri~ht triangle are congruent
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.
Decide whether enough information is given to prove that the triangles
are congruent using the SAS Congruence Postulate.
1. £iPQI; £iRQS
p
2.
fj.NKJ,MKAI
J
R
Z
wlY
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p& -=
tvO
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LfQT~LlC-GS
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sPf::>
QT ~ &S
c.v' c:!
S ~~
OJ'<-1"Ct~.sc.A.l'-t..
Section 4.6 - Use Two More Methods to Prove Congruences
A
JJ~ow~
Angle-Side-Angle
proof uses arrows to show the flow of a logical argument.
(ASA) Congruence
Postulate:
are congruent to two ~
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Angle-ingle-Side
If two
and the includ~d
.
(AAS) Congruence
triangle are congruent to two
then the two triangles are
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Theorem:
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If two
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angw>
and the included
.5\ fl
of a second triangle, then the two triangles are
. 1"
and a non-in~luded
and the corresponding non-included
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Can the triangles be proven congruent with the information given in the
diagram? If so. state the postulate Gr theorem you would use.
1.
of one triangle
of one
of a second triangle,
Section 4.7 - Use Congruent Triangles to Prove Corresponding
Parts Congruent
Tell which triangles you can show are congruent in order to prove the
statement. What postulate or theorem would you use?
1. JK:::: LK
2. LRPQ:::: LTRS
K
p
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J
T
M
~ PR.Q ~ L} tTS
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Section 4.8 - Use Theorems about Isosceles and Equilateral Triangles
When an isosceles triangle has exactly two congruent sides, these two sides are the legs. The angle formed by the legs is the vertex
angle. The third side is the base of the isosceles triangle. The two angles adjacent to the base are called base angles.
Base Angles Theorem:
If two sides of a triangle are congruent, then the
Converse of Base Angles Theorem:
to the Base Angles Theorem:
Corollary
to the Converse of Base Angles Theorem:
1. Find WY:
opposite them are congruent.
If two angles ofa triangle are congruent, t en the ~
Corollary
Use the information
given values.
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If a triangle is
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opposite them are congruent.
' then it is e~~;~n:~~~~.-4. 1
Ifa triangle is equiangular, then it is~.
w
in the diagram to find the
x
0
2. Find mLWXY.
3. Find the values of x and y in the diagram at the right.
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2. ~
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-=-
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f7...2
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fA-=-
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Section 4.9 - Create an Image Congruent to a Given Triangle
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is an operation that moves or changes a geometric figure in some way to produce a new figure.
The new figure is called the image.
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moves every point of a figure the same distance in the same direction.
Coordinate notation: (x, y)
A~
(x + a, y + b)
uses a line of reflection to create a mirror image of the original figure.
Coordinate notation: (x, y)
m\I.ltlN\
A
--+
--+
(x, -y)
turns a figure about a fixed point, called the center of rotation.
Coordinate notation: (x, y)
--+
(-x, y)
A congruence transformation changes the position of the figure without changing its
Figure
Sketch
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sh~
or
ABeD has the vertices A(-5, 5), B(-1, 2), C(-3, OJ. and 0(-5.0).
ABeD and its image after the given transformation.
2. Translation: (x, .v) --+ (.x + 6, Y - 2)
Reflection in the x-axis
3.
Name all the angles that have Vas a vertex.
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Solve for x.
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12-0 .: 31 X
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State if the two triangles are congruent. If yes, state how you know.
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State what additional information is needed in order to know that the triangles ar congruent for the
reason given.
10. SAS
11. SSS
12. ASA
f)
---
Write a proof.
5.
4. GIVEN:AB==DB,BC-lAD
PROVE: /').ABC
==
GIVEN:
mLJKL
= mLMLK=
90°
JL==AfK
/:).DBC
PROVE:/:).JKL
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==
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GIVEN: LPQS=:::. LRQS
5.
GIVEN: L OltfN
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PROVE: 8.POS
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PROVE: 8.A1.IN~ !:::.NUI
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GIVEN: AB =:::.EB,FB~ eB
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GIVEN: RS
=:::.
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ST, PU=:::.PQ
-
PROVE: BG=:::.BD
LUPT=:::.LQPR
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PROVE: 8.PTU
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