Congruent Triangles

• In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure.

• This means there will be corresponding sides that are congruent.

• There will also be corresponding angles that are congruent.

•A coordinate proof involves placing geometric figures in a coordinate plane.

What are the corresponding sides? What are the

corresponding angles?

Congruent Triangles

• Which triangles are congruent by the SSS Postulate?

Not congruent by SSS

Congruent Triangles

Are these congruent by SAS?

How about:

Are these congruent by HL?

Not right triangles!

Congruent Triangles

Are these triangles congruent by ASA?

Yes!

Are these triangles congruent by ASA?

No

Congruent Triangles

Are these triangles congruent by AAS?

How about now?

NoYes

Congruent Triangles

There are a couple of methods for organizing your thoughts when proving triangle congruency.

The first is to use a two-column proof.

The second is to use a flow proof.

Congruent Triangles

Given: Prove:

Congruent TrianglesA flow proof uses arrows to show the flow of a logical argument.

• Just like a flow chart does!

Congruent TrianglesSo remember:

SSS, SAS, ASA, AAS Postulates and the HL Theorem will help everyone to be congruent.

But be careful during a test! Make sure you don’t need to call AAA to bail out your SSA.

Congruent Triangles

Problems

• Pg 239 #15-16

• Pg 245 # 17-18

• Pg 247 #9-10

• Pg 254 #15-16

Triangle Relationships

– Inequality

The longest side and largest angle are opposite each other.

The shortest side and smallest angle are opposite each other.

Triangle Relationships

SOLUTION

Draw a diagram and label the side lengths. The peak angle is opposite the longest side so, by Theorem 5.10, the peak angle is the largest angle.

Triangle Relationships

– Inequality

Is it possible to construct a triangle with the given lengths? 3, 5, 9

Not Possible

5+9 > 3

3+9 > 5

5+3 > 9 Does not work!

Is it possible to construct a triangle with the given lengths? 6, 8, 10

6+8 > 10

8+10 > 6

6+10 > 8 It is Possible!

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Triangle Relationships

What can we say about angle 1?

Think about it:

The angles of a triangle have to sum to 180.

The angles that form a line must sum to 180.

+ =

Thus

soand

Triangle Relationships

– Hinge Theorem

Triangle Relationships

– Hinge Theorem

60

35

CD, BC, BD, AB, AD

Triangle Relationships

Problems

• Pg 287 #1-10, 15-28

• Pg 294 #1-13

Triangle Relationships

– Perpendicular bisectors

Does this triangle have a perpendicular bisector?

How do you know?

Yes, segment BDTheorem 5.3 proves D is on the perpendicular bisector and BD makes a right angle with AC at its midpoint.

Triangle Relationships

Where is the point of concurrency in this triangle?

What special type of point of concurrency is this?

What is special about the red lines?

Point G

It is a circumcenter.

They are congruent.**Note: The circumcenter can be outside of the triangle if you have an obtuse triangle!

Triangle Relationships

Problems

• Math I

• Pg 266 #1-18

• Pg 268 #1-9

Triangle Relationships

• Angle Bisectors

What is the angle bisector?

How do you know?

FH

The Angle Bisectors Theorem

Triangle RelationshipsA soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost R or the left goalpost L?

SOLUTIONThe congruent angles tell you that the goalie is on the bisector of LBR. By the Angle Bisector Theorem, the goalie is equidistant from BR and BL .

So, the goalie must move the same distance to block either shot.

Triangle Relationships

• Medians

What are the medians?

Where is the centroid?

What is the length of DG?

BG, CE, AF

At D

DG = 6

Triangle Relationships

• Altitudes

Fun Fact: The orthocenter likes to travel!

Triangle Relationships

Problems

• Math I

• Pg 274 #1-12, 14-17

• Pg 280-281 #1-20

• Pg 282 #1-5