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