Lesson 5.1 Congruence and Triangles
Jan 18, 2016
Lesson 5.1
Congruence and Triangles
Lesson 5.1 Objectives
Identify congruent figures and their corresponding parts.Prove two triangles are congruent.Apply the properties of congruence to triangles.
Congruent Triangles
When two triangles are congruent, then Corresponding angles are congruent. Corresponding sides are congruent.
Corresponding, remember, means that objects are in the same location. So you must verify that when the triangles
are drawn in the same way, what pieces match up?
Naming Congruent Parts
Be sure to pay attention to the proper notation when naming parts. ABC DEF
This is called a congruence statement.
A
B
C
D
E
F A D B E C F
and
AB DEBC EFAC DF
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Prove Triangles are Congruent
In order to prove that two triangles are congruent, we must Show that ALL corresponding angles
are congruent, and Show that ALL corresponding sides
are congruent.
We must show all 6 are congruent!
Side-Side-Side Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Side-Angle-Side Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Angle-Side-Angle Congruence
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Angle-Angle-Side Congruence
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.
Hypotenuse-Leg Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.
Abbreviate using HL
Tuesday’s Schedule
Collect signed syllabiCorrect lesson 5.1 day 1 assignmentReview lesson 5.1 assignmentLesson 5.1 day 2Lesson 5.1 day 2 assignment
Lesson 5.1 Day 2
Which postulate or theorem to use??
Which postulates/theorems can be used to prove triangle congruence?
SSS (side-side-side)SAS (side-angle-side)ASA (angle-side-angle)AAS (angle-angle-side)HL (hypotenuse-leg) HL can only be used in right
triangles!!
Decide whether or not the congruence statement is true. Explain your reasoning!
Reflexive Property of Congruence
The statement is true because ofSSS Congruence
The statement is not true because the vertices are
out of order.Because the segment is shared between two triangles, and yet it is the same segment
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem.
Reflexive Property of Congruence
SAS CongruenceNo
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem.
Reflexive Property of Congruence
Yes they are congruent!HL
Reflexive Property of Congruence
Not congruent
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem.
Reflexive Property of Congruence
Yes they are congruent!
ASA
Reflexive Property of Congruence
Yes they are congruent!
AAS
Wednesday
Collect signed syllabiCorrect/review 5.1 day 2Notes over lesson 5.2Assignment 5.2
Lesson 5.2
Proving Triangles are Congruent
Review
What does congruent mean?
Draw two triangles that appear to be congruent.
Label your drawings to make the two triangles congruent.
Complete the proof
If 2(x+12)=90, then x=33
1. 2(x+12)=90 1. Given
2. 2x+24=90 2. Distributive Property3. 2x=66 3. SPOE
4. X=33 4. DPOE
Complete the proof
Given
Given
Reflexive POC
SSS Congruence
Complete the proofGiven:
Prove:
,AD BC AD BC
DAB BCD
1. 1.Given
2. 2. Given
3. 3. AIA
4. 4. Reflexive
5. 4. SAS
AD BC
AD BCADB CBD
DAB BCD
BD BD
Construct a proof
1.
2.
3.
4.
1. Definition of midpoint2. Definition of midpoint3. Vertical angles
4. SAS
AB BE
DB BC
ABD EBC
ABD EBC
Surveying
MNP MKL Given
• Segment NM Segment KM– Definition of a midpoint
LMK PMN– Vertical Angles Theorem
KLM NPM– ASA Congruence
• Segment LK Segment PN– Corresponding Parts of Congruent Triangles
Lesson 5.3
Similar Triangles
Ratio
If a and b are two quantities measured in the same units, then theratio of a to b is a/b. It can also be written as a:b.
A ratio is a fraction, so the denominator cannot be zero.
Ratios should always be written in simplified form. 5/10 1/2
Proportional
If two ratios are equal after they are simplified, then they are said to be proportional.
6
1012
20
3
5
3
5
These two ratios are proportional.
Similarity of Trianlges
Two Triangles are similar when the following two conditions exist Corresponding angles are congruent Correspondng sides are proportional
Means that all side fit the same ratio.
The symbol for similarity is ~
ABC ~ FGH This is called a similarity statement.
Scale Factor
Since all the ratios should be equivalent to each other, they form what is called the scale factor.
We represent scale factor with the letter k.
This is most easily found by find the ratio of one pair of corresponding side lengths.
Be sure you know the polygons are similar.
k = 20/5
k = 4205
Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Theorem 8.2:Side-Side-Side Similarity
If the corresponding sides of two triangles are proportional, then the triangles are similar. Your job is to verify that all corresponding
sides fit the same exact ratio!
10 10
6
5 5
3
Theorem 8.3:Side-Angle-Side Similarity
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
Your task is to verify that two sides fit the same exact ratio and the angles between those two sides are congruent!
10
6
5
3
Using Theorems…which one do I use?
These theorems share the abbreviations with those from proving triangles congruent.
SSS SAS
So you now must be more specific SSS Congruence SSS Similarity SAS Congruence SAS Similarity
You chose based on what are you trying to show? Congruence Similarity