Chapter 4 Triangle Congruence

1
Chapter 4Triangle Congruence

• By Maya Richards
• 5th Period Geometry

2
Section 4-1 Congruence and Transformations

• Transformations
• Translations slides
• Reflections flips
• Rotations turns
• Dilations gets bigger or smaller (only one that
  changes size)

3

• Rotation of 180 degrees around the point (-0.5,
  -0.5)

• Translation 6 units right and 2 units up.

Reflection across the y-axis.
Dilation of 2x.
4
Section 4-2 Classifying Triangles
5
Example 1
H

• Classify each triangle by its angle measures.

30

1. Triangle EHG

Angle EHG is a right angle, so triangle EHG is a
right triangle.
120
E
30
60
60
G
F
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.
6
Example 2
A
Classify each triangle by its side lengths.
18
15
D
B
5
C
15

1. Triangle ABC

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.
7
Section 4-3 Angle Relationships in Triangles
B
A
C

• Triangle Sum Theorem
• The sum of the angle measures of a triangle is
  180 degrees.
• angle A angle B angle C 180

8

• 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.

2
1
3
4
9

• 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
T
S
10
Example 1
11
Section 4-4 Congruent Triangles
C

• 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

A
E
B
D
F
12
Section 4-5 Triangle Congruence SSS and SAS
A

• Side-Side-Side Congruence (SSS)
• If three sides of one triangle are congruent to
  three sides of another triangle, then the
  triangles are congruent.

B
C
E
F
D
13

• 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.

B
A
C
F
D
E
14
Example 1
Q

• Use SSS to explain why triangle PQR is congruent
  to triangle PSR.

P
R
S
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.
15
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.

B
C
A
D
F
E
16

• 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.

B
A
F
C
D
E
17

• 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.

B
A
F
C
D
E
18
Example 1
19
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.

20
Example 1
21
Section 4-8 Introduction to Coordinate Proof

• Coordinate proof a style of proof that uses
  coordinate geometry and algebra

22
Example 1
23
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.
• Converse of Isosceles Theorem
• If two angles of a triangle are congruent , then
  the sides opposite the angles are congruent.

A
angle B is congruent to angle C
B
C
D
DE is congruent to DF
F
E
24
Example 1
(x30)
R
Find the angle measure.
T
2x
S
B. Measure of angle S
M of angle S is congruent to M of angle R.
Isosceles Triangle Theorem
2x (x 30)
Substitute the given values.
x 30
Subtract x from both sides.
Thus M of angle S 2x 2(30) 60.