Notes
Section 7.3
Name ______________________________
Proofs Using Triangle Congruence Theorems.
Now that you know several triangle congruence theorems, it’s time to put them to use!
Here’s a list of the theorems, definitions, and properties that you may find useful:
DEFINITIONS
Use
Name
Name?
No
Midpoint
No
Isosceles Triangle
No
Parallelogram
No
Equilateral Triangle
No
Kite
No
Square
No
Trapezoid
No
Circle
Yes
Angle bisector
No
Supplementary angles
No
Complementary angles
Yes
Reflection
No
Median
No
Altitude
PROPERTIES
Use
Name
Name?
Yes
Transitive property of
congruence/equality
Yes
Reflexive Property of
congruence/equality
What it says
A point divides a segment into two equal/congruent parts if and only
if it is the midpoint of the segment.
A triangle is isosceles if and only if it has at least two sides of equal
length (or that are congruent).
A quadrilateral is a parallelogram if and only if both pairs of opposite
sides are parallel.
A triangle is equilateral if and only if it has three equal/congruent
sides.
A quadrilateral is a kite if and only if it has two distinct pairs of
consecutive sides of the same length.
A quadrilateral is a square if and only if it has four equal sides and
four right angles.
A quadrilateral is a trapezoid if and only if it has at least one pair of
parallel sides.
A circle is the set of all points in a plane at a certain distance (its
radius) from a certain point (its center).
All radii of a circle are equal/congruent.
If a ray in the interior of an angle has its endpoint on the vertex of the
angle and it divides the angle into two congruent/equal angles, then it
is an angle bisector.
Two angles are supplementary if and only if their measures add to
180°.
Two angles are complementary if and only if their measures add to
90°
For a point P not on a line m, the reflection image of P over line m is
the point Q if and only if m is the perpendicular bisector of PQ .
The median of a triangle is the line segment drawn from the vertex of
a triangle to the midpoint of the opposite side. (The median of a
triangle bisects the opposite side.)
The altitude of a triangle is the line segment drawn from the vertex of
a triangle perpendicular to the opposite side.
What it says
If a @ b and b @ c then a @ c
If a = b and b = c then a = c.
a@a
a=a
Yes
Yes
Symmetric Property of
If a @ b , then b @ a
congruence/equality
If a = b, then b = a
Angle Addition Property If ABC and CBD are adjacent, then
mABC + mCBD = mABD
THEOREMS
Use
Name
Name?
No
Vertical Angle
Theorem
Yes
SSS Theorem
Yes
SAS Theorem
Yes
ASA Theorem
Yes
AAS or SAA Theorem
Yes
No
CPCFC
Alternate Interior
Angle Theorem
Corresponding Angle
Theorem
Alternate Exterior
Angle Theorem
Converse to the
Alternate Interior
Angle Theorem
Converse to the
Corresponding Angle
Theorem
Converse to the
Alternate Exterior
Angle Theorem
Polygon sum theorem
No
No
No
No
No
No
No
No
No
Yes
No
Angle Congruence
Theorem
Segment congruence
Theorem
ABCD Theorem
Figure Reflection
Theorem
Isosceles Triangle Base
Angles Theorem
What it says
Vertical angles are congruent/have equal measures
If in two triangles, three sides of one are congruent to three sides of
the other, then the triangles are congruent.
If, in two triangles, two sides and the included angle of one are
congruent to two sides and the included angle of the other, then the
triangles are congruent.
If, in two triangles, two angles and the included side of one are
congruent to two angles and the included side of the other, then the
two triangles are congruent.
If, in two triangles, two angles and the non-included side of one are
congruent respectively to two angles and the corresponding nonincluded side of the other, then the triangles are congruent.
Corresponding Parts of Congruent Figures are Congruent
|| lines  alternate interior angles are congruent/equal
|| lines  corresponding angles are congruent/equal
|| lines  alternate exterior angles are congruent/equal
Alternate interior angles are congruent/equal  || lines
Corresponding angles are congruent/equal  || lines
Alternate exterior angles are congruent/equal  || lines
The sum of the measures of the angles in a triangle is 180
The sum of the measures of the angles in a quadrilateral is 360.
If the measures are =, then the angles are congruent. You can write:
If mABC = mCDE, then ABC @ CDE,
If the lengths are =, then the segments are congruent. You can write:
If AB = CD, then AB @ CD
Isometries preserve Angle Measure, Betweenness, Collinearity and
Distance.
If you reflect every vertex of a figure and connect them in order, the
reflected figure is congruent to the original.
The base angles of an isosceles triangle are congruent.
(If D , then D .)
1.
R
A
Given:
Prove:
Statements
I
N
RI bisects ARN
RIA  RIN
RIA  RIN
Reasons
2.
A
R
I
T
Given:
Prove:
Statements
3.
N
I is the midpoint of TA
T and A are right angles
ITR  IAN
Reasons
A
S
T
M
H
Given:
A  H
T is the midpoint of AH
Prove:
MA @ SH
Statements
Reasons
4.
G
M
O
E
Given:
GO ^ EM
Prove:
Statements
GEO  GMO
Reasons
5. The perimeter of COG is 26. Find the value of x.
Is CON  CGN? Please explain why or why not.
C
3x
2x + 3
O
x
N
G
2x - 1
6.
F
M
S
H
Given:
FHS  MAT
HS = y and AT = 2x + 5
F = 90 and M = 3x + y
a. Find the value of x and y.
b. Find AT.
T
A