Download Section 4-6: Triangle Congruence

Chapter 4
Triangle
Congruence
By: Maya Richards
5th Period Geometry
Section 4-1: Congruence and
Transformations

Transformations:




Translations – slides
Reflections – flips
Rotations – turns
Dilations – gets bigger or smaller (only one that changes
size)
Rotation of 180 degrees around
the point (-0.5, -0.5)
Translation 6 units right and 2
units up.
Dilation of 2x.
Reflection across the y-axis.
Section 4-2: Classifying Triangles
Example 1
Classify each triangle by its angle measures.
30°
A. Triangle EHG
Angle EHG is a right angle, so
triangle EHG is a right triangle.
30°
120° 60°
60°
B. Triangle EFH
Angle EFH and angle HFG form a linear pair, so they are supplementary.
Therefore measure of angle EFH + measure of angle HFG = 180°.
By substitution, measure of angle EFH + 60° = 180°.
So measure of angle EFH = 120°.
Triangle EFH is an obtuse triangle by definition.
Example 2
Classify each triangle by its side
lengths.
18
15
A. Triangle ABC
15
5
From the figure, AB is congruent to AC.
So AC = 15, and triangle ABC is equilateral.
B. Triangle ABD
By the Segment Addition Postulate, BD = BC + CD = 15 + 5 = 20.
Since no sides are congruent, triangle ABD is scalene.
Section 4-3: Angle Relationships in
Triangles
Triangle Sum Theorem

The sum of the angle measures of a
triangle is 180 degrees.
angle A + angle B + angle C = 180°


Angle 4 is an exterior angle.
Its remote interior angles are angle 1
and angle 2.
Exterior Angle Theorem
The measure of an exterior angle of a triangle is
equal to the sum of the measures of its remote
interior angles.
•Measure of angle 4 = measure of angle 1 +
measure of angle 2.
Third Angles Theorem

If two angles of one triangle are congruent
to two angles of another triangle, then the
third pair of angles are congruent.
Angle N is congruent to angle T
L
N
M
R
S
T
Example 1
Section 4-4: Congruent Triangles
Corresponding Sides



AB is congruent to DE
BC is congruent to EF
AC is congruent to DF
Corresponding Angles



A is congruent to D
B is congruent to E
C is congruent to F
Section 4-5: Triangle Congruence:
SSS and SAS
Side-Side-Side Congruence (SSS)
If three sides of one triangle are
congruent to three sides of another
triangle, then the triangles are
congruent.
Side-Angle-Side Congruence (SAS)
If two sides and the included angle of
one triangle are congruent to two sides
and the included angle of another
triangle, then the triangles are
congruent.
Example 1
Use SSS to explain why triangle
PQR is congruent to triangle PSR.
It is given that PQ is congruent to PS and that QR is congruent to SR.
By the Reflexive Property of Congruence, PR is congruent to PR.
Therefore triangle PQR is congruent to triangle PSR by SSS.
Section 4-6: Triangle Congruence:
ASA, AAS, and HL
Angle-Side-Angle Congruence
(ASA)
If two angles and the included side of
one triangle are congruent to two angles
and the included side of another triangle,
then the triangles are congruent.
Angle-Angle-Side Congruence
(AAS)
If two angles and a nonincluded side of
one triangle are congruent to the
corresponding angles and nonincluded
side of another triangle, then the
triangles are congruent.
Hypotenuse-Leg Congruence
(Hy-Leg)
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and a leg of another triangle, then the
triangles are congruent.
Example 1
Section 4-7: Triangle Congruence:
CPCTC
CPCTC = (“Corresponding Parts of Congruent
Triangles are Congruent”)
 Can be used after you have proven that two
triangles are congruent.
Example 1
Section 4-8: Introduction to
Coordinate Proof

Coordinate proof – a style of proof that uses
coordinate geometry and algebra
Example 1
Section 4-9 Isosceles and Equilateral
Triangles
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite the sides
are congruent.
angle B is congruent to angle C
Converse of Isosceles Theorem
If two angles of a triangle are congruent , then the sides opposite the angles
are congruent.
DE is congruent to DF
Example 1
(x+30)°
Find the angle measure.
2x°
B.
Measure of angle S
M of angle S is congruent to M of angle R.
2x° = (x + 30)°
x = 30
Thus M of angle S = 2x° = 2(30) = 60°.
Isosceles Triangle Theorem
Substitute the given values.
Subtract x from both sides.