Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:
-
Upload
hester-rhoda-poole -
Category
Documents
-
view
230 -
download
0
Transcript of Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:
![Page 1: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/1.jpg)
Congruent Triangles
Geometry Chapter 4
![Page 2: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/2.jpg)
4.1 Triangles and Angles
Classification by Sides:
![Page 3: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/3.jpg)
Triangles and Angles
Classification by Angles
![Page 4: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/4.jpg)
Parts of Triangles
leg leg
base
leg
leg
hypotenuse
Vertex angle
Base angle Base angle
ExteriorangleInterior
angle
![Page 5: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/5.jpg)
Theorems Involving Triangles
The sum of the measures of the angles of a triangle = 180°
The measure of the exterior angle of a triangle = the sum of the two remote interior angles. 12
3
A
B
C
![Page 6: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/6.jpg)
Corollaries to Triangle Theorems
The acute angles of a right triangle are complementary.
Each angle of an equiangular triangle has a measure of 60°.
In a triangle, there can be at most one right angle or one obtuse angle.
¬
![Page 7: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/7.jpg)
Examples
Sides of lengths 2mm, 3mm and 5mm.
Sides of lengths 3m, 3m, 3m.
Sides of lengths 8m, 8m, 5m.
![Page 8: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/8.jpg)
Examples
Angles of measures 90, 25, 65.
Angles of measures 60, 60, 60.
Angles of measures 80, 70, 30.
Angles of measures 140, 30, 10.
![Page 9: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/9.jpg)
Examples
A triangle has angles that measure x, 7x, and x. Find x.
![Page 10: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/10.jpg)
Examples
A right triangle has angle measures of x and (2x-21). Find x.
![Page 11: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/11.jpg)
Examples
Find the measure of the exterior angle shown.
![Page 12: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/12.jpg)
4.2 Congruence and Triangles
Congruent – same size, same shape
Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent
A C
B
D
E
F
If ΔABC ΔDEF,thenA D AB DEB E BC EFC F AC DF
![Page 13: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/13.jpg)
Theorems about Congruent Figures
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
If R M and S N, then T O
R
S
T M
N
O
![Page 14: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/14.jpg)
Examples
110°
87°
10m
72°
L M
NO
If LMNO EFGH, find x and y.
E F
GH
(7y + 9)°
(2x +3)m
![Page 15: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/15.jpg)
Examples
![Page 16: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/16.jpg)
4.3-4.3 Proving Triangles Congruent
SSS – Side Side Side – If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
If AB DE BC EF AC DF, then
ABC DEF
![Page 17: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/17.jpg)
SAS
SAS – Side Angle Side – 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.
If AB DE BC EF B E, then ABC DEF
![Page 18: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/18.jpg)
ASA
ASA – Angle Side Angle – 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.
If A D C F AC DF,then ABC DEF
![Page 19: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/19.jpg)
AAS
AAS – Angle Angle Side – If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
If A D C F AB DE,then ABC DEF
![Page 20: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/20.jpg)
HL
HL – Hypotenuse Leg – If the hypotenuse and leg of one RIGHT triangle are congruent to the hypotenuse and leg of another RIGHT triangle then the triangles are congruent.
C
A
B F
D
E
If ABC,DEF Right s,AB DE, AC DF, thenABC DEF.
![Page 21: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/21.jpg)
4.5 Using Congruent Triangles
Definition of Congruent Triangles (rewritten)
Corresponding Parts of Congruent
Triangles are CongruentCPCTC is used often in proofs involving
congruent triangles.
![Page 22: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/22.jpg)
1.
M
TS
R
AA is the midpoint of MT.A is the midpoint of SR.
MS ll TR
A is the midpoint of MT.A is the midpoint of SR.
1. Given
![Page 23: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/23.jpg)
UR ll ST R and T are right angles
UR ll ST R and T are right angles
1. 1. Given
U
R S
T
![Page 24: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/24.jpg)
4.6 Isosceles, Equilateral and Right Triangles
If two sides of a triangle are congruent, then the angles opposite are congruent. (Base angles of an isosceles triangle are congruent.
Converse – If two angles of a triangle are congruent, then the sides opposite are congruent.
A
B
C
If BA BC, then A C.
If A C, then BA BC.
![Page 25: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/25.jpg)
More Corollaries
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular then it is equilateral.
A
B
C
![Page 26: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/26.jpg)
Examples
Find x and y.
35 x
y
![Page 27: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/27.jpg)
Examples
Find the unknown measures.
50 ?
?
![Page 28: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/28.jpg)
Examples
Find x.
33 in(x-11) in
![Page 29: Congruent Triangles Geometry Chapter 4. 4.1 Triangles and Angles Classification by Sides:](https://reader031.fdocuments.net/reader031/viewer/2022031914/56649cee5503460f949bc4da/html5/thumbnails/29.jpg)
Examples
Find x and y.
y
40
x