Isosceles, Equilateral, and Right Triangles

Chapter 4.6

Isosceles Triangle Theorem

Isosceles The 2 Base s are • Base angles are the angles opposite the equal

sides.

Isosceles Triangle Theorem

A C

B

If AB BC, then A C

Isosceles Triangle Theorem

A C

B

If A C then AB BC

Sample ProblemSolve for the variables• mA = 32°• mB = (4y)° • mC = (6x +2)°

A C

B

6x + 2 = 32

6x = 30

x = 5

32 + 32 + 4y = 180

4y + 64 = 180

4y = 116

y = 29

Find the Measure of a Missing Angle

120o

180o – 120o = 60o

30o 30o

30o

180o – 30o = 150o

75o

75o

1. A

2. B

3. C

4. D

A. 25

B. 35

C. 50

D. 130

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. Which statement correctly names two congruent angles?

A.

B.

C.

D.

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

B. Which statement correctly names two congruent segments?

A.

B.

C.

D.

Equilateral Triangle Theorem

Equilateral Equiangular

Each angle = 60o !!!

603

180

Answer: 105

Use Properties of Equilateral Triangles

Subtraction

Linear pair Thm.

Substitution

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. x = 15

B. x = 30

C. x = 60

D. x = 90

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 30

B. 60

C. 90

D. 120

Don’t be an ASS!!!• Angle Side Side does not work!!!

– (Neither does ASS backward!)

• It can not distinguish between the two different triangles shown below.

However, if the angle is a right angle, then they are no longer called sides. They are called…

Hypotenuse-Leg Theorem

• If the hypotenuse and one leg of a right triangle are congruent to the corresponding parts in another right triangle, then the triangles are congruent.

ABC XYZ Why?

HL TheoremB

CA

X Z

Y

Prove XMZ YMZ

X Y

Z

M

Step Reason

YZXZ Given

XYZM GivenmZMX = mZMY = 90o Def of lines

ZMZM Reflexive

HL Thm

ZMX ZMY

Corresponding Parts Corresponding Parts of Congruent Triangles of Congruent Triangles are Congruentare Congruent

Given Given ΔΔABC ABC ΔΔXYZXYZ You can state that:You can state that:

A A XX B B YY C C ZZ

AB AB XY XY BCBC YZYZ CACA ZXZX

CA

CBAD

23

Suppose you know that ABD CDB by SAS Thm. Which additional pairs of sides and angles can be found congruent using Corr. Parts of s are ?

Complete the following two-column proof.

Proof:

4.

ReasonsStatements

1. Given

2. Isosceles Δ Theorem

1.

2.

3. 3. Given

4. Def. of midpoint

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

Proof:

4.

ReasonsStatements

4. Def. of midpoint

5. ______

6. 6. ?

5. ΔABC ΔADC ?

Complete the following two-column proof.

SAS Thm.Corr. Parts of s are

Homework

Ch 4-6

• pg 248

1 – 10, 14 – 27, 32, 33, 37 – 39, & 48

2,

22121 yyxx

Reminder!

Midpoint Formula:

Video C