BY: KELLAN HIRSCHLER AND KATHERINE ROSENCRANCE

Section 3.7 Angle-Side Theorems

Theorem 20: If two sides of a triangle are congruent, the angles opposite the sides are congruent. Symbolic form: If , then (If sides, then angles)

Given: E

A T

EA ETConclusion: A E

Theorem 21: If two angles of a triangle are congruent, the sides opposite the angles are congruent.Symbolic form: If , then (If angles, then sides)

Given: P

I E

I E

Conclusion: PI PE

Prove: PIK PGK

P

I GK

S R

1. IK KG 1. GivenI 2. G 2. Given

3. PI PG 3. If angles, then sides4. PIK PGK 4. SAS (1,2,3)

Theorem: If two sides of a triangle are not congruent, then the angles opposite them are not congruent and the larger angle is opposite the longer side.

Symbolic Form: If , then .

Longer

Larger

Shor

ter

Smaller

Theorem: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Symbolic Form: If , then . Sh

orte

r Longer

SmallerLarger

2 ways to prove a triangle is isosceles:If , then is isosceles.

If , then is isosceles.

Equilateral triangle <=> Equiangular triangle

<=>

H

O R S E

Given: OHR EHS

OH HE

Prove: HR HS

S R1. OHR EHS 1. Given2. OH HE 2. Given

3. O H 3. If sides, then angles

4. OHR EHS 4. ASA (1,2,3)

5. HR HS 5. CPCTC

REVIEWO

W L

Given: OW OL

Conclusion: W L

R

O W

Given: O W Conclusion: RORW

K A T I

E

Given: KTAI

KEI is isos. with and KEIE

S R

Prove: KET IEA

1. AI KT 1. Given

KEI is isos. with

IE andKE

;

3. K I3. If the triangle is isos., then the base angles are congruent.

2. IE KE

2. If the triangle is isos., then the legs are congruent

4. KET

IEA

4. SAS (1,2,3)

A

K

M

S

E TR

S R

1. RK ET ;

MRMT ;

A is the mp. of MR;

S is the mp. of MT

1. Given

2. R T 2. If sides, then angles.

3.

RAAM ;TSSM

3. Mp. divides segments into 2 congruent segments.

RAK 5. TSE5. SAS ( 4, 2, 1)

4. RA TS4. Same as 3.

6. AK SE 6. CPCTC

Given: RK ET

MT MR

A is the mp of MR

S is the mp of MT

Prove: AK SE

Works Cited

Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry: For Enjoyment and Challenge. Evanston, Illinois: McDougal Littell, 1991. Print.