Congruent Triangles

Today’s Learning Goals We will determine three different ways to

know if two triangles are identical. We will begin to see identical triangles in

different shapes.

Definitions Congruent means that two shapes or objects are

exactly the same. The symbol means congruent.

Consider the two pairs of line segments below. Which pair appears to be congruent line segments?

One way to figure out if two objects are congruent is to lay them on top of each other. If they line up perfectly, then they are congruent.

Nice… .CDAB a) b)

6 in

6 in

6 cm14 cmA

B

C DP

QR

S

Congruent Objects Consider the following two pairs of stars.

Which pair of stars appears to be congruent?

a) b)

Great…the stars above b) appear to be congruent. The line segments and stars were easy to see which were congruent. Often times, we must reason about congruency using the side lengths and angles of the shapes.

Triangle Review Previously, we discussed the triangle inequality. In

particular, we found out when a triangle could be made given only the three side lengths.

For example, could you make a triangle with the side lengths 5, 8, and 13? Nice…you cannot make a triangle with these three side lengths because 5 + 8 = 13. So the three side lengths form a straight line…not a triangle!

13

8 5

Triangle Review Could you make a triangle with the side

lengths 25, 13, and 7? Good…you cannot make a triangle with these three side lengths because 13 + 7 < 25. So the three side lengths will never meet to form a triangle!

13

25

7

Triangle Review Could you make a triangle with the side

lengths 8, 12, and 15? Great…you can make a triangle with these three side lengths because 8 + 12 > 15.

How many triangles can you make with these three given side lengths?

15

12 8

Yes…you can only make one triangle with these three side lengths because when these sides come together, the triangle is rigid.

Congruent Triangles Consider all of the triangles below with side lengths of

8, 12, and 15. Are these triangles different?

15

12 815

12

8 12

815

8 12

15No…these triangles are all the same. They are just rotated or flipped in different ways.

So, if all three sides of two triangles are the same, then the two triangles are congruent. This property is known as SSS (side-side-side).

Congruent Triangles Consider the following two pairs of triangles.

Which pair has two congruent triangles? Explain how you know.

6 8

136

9

14 4

9

11

4

9

11A

B C E

D

FP

Q

R

S

T

U

Nice…PQR STU by SSS. SSS is one way to know if two triangles are congruent.

Congruent Triangles Suppose we were given two angle

measurements and the included side length like in the picture below. How many triangles could you make with this information?

72 3413 cm

Yes…only one because the side lengths would be extended until they met in only one place.

If two triangles have two angles and an included side that are the same, then the two triangles are congruent. This property is known as ASA (angle-side-angle).

Congruent Triangles Consider the following two pairs of triangles.

Which pair has two congruent triangles?

Excellent…PQR STU by ASA. So, SSS and ASA are two ways to know if two

triangles are congruent. There is one other way to know if two triangles are congruent.

68 2913 m

68 2913.2 ma) b)

24 4918 ft

18 ft

24 49A

B

C D

E

F P

Q

R S

T

U

Congruent Triangles Suppose we were given two side lengths and an

included angle like in the picture below. How many triangles could you make given this information?

8 km

21 km

123

Great…only one because there is only one side length that would work to connect the existing sides. If two triangles have two side lengths and an

included angle that are the same, then the two triangles are congruent. This property is known as SAS (side-angle-side).

Congruent Triangles Consider the following two pairs of triangles.

Which pair has two congruent triangles?

a) b)

53 5327

27

16

mm 16

mm

21 mm

21 mm

L

M N S

T

UW

X

Y

D

E

F

8 ft

8 ft12.5 ft

12.5 ft

Excellent…LMN TSU and WXY DEF by SAS. So, SSS, ASA, and SAS are three different ways

to know if two triangles are congruent.

Partner Work You have 20 minutes to work on the following

problems with your partner.

For those that finish earlyAre the following congruent? Explain why or why

not.

26

43 7843

78

26

Big Idea from Today’s Lesson SSS, SAS, and ASA are three ways to

know that two triangles are congruent.

Homework Complete Homework Worksheet