Chapter 4 Triangle Congruence. By: Maya Richards 5 th 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).
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Chapter 4Triangle Congruence
By: Maya Richards
5th Period Geometry
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.
Reflection across the y-axis.
Dilation of 2x.
H
Classify each triangle by its angle measures.
30°
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.
Classify each triangle by its side lengths.
A
18
15
D
B
5
C
15
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.
B
Triangle Sum Theorem
The sum of the angle measures of a triangle is 180 degrees.
A
C
2
1
3
4
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.
L
N
M
R
T
S
C
Corresponding Sides
Corresponding Angles
A
E
B
D
F
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
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
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.
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
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
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
CPCTC = (“Corresponding Parts of Congruent Triangles are Congruent”)
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
E
F
(x+30)°
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°.