Geometry 1 Unit 4Congruent Triangles

Casa Grande Union High SchoolFall 2008

Geometry 1Unit 4

4.1 Triangles and Angles

Scalene- No sides congruent

Equilateral- all 3 sides are congruent

Isosceles- at least 2 sides are congruent

Acute Triangle- All anglesare less than 90°

Obtuse Triangle- 1 angle greater than 90 ° Right Triangle- 1 angle

measuring 90°

Classifying Triangles Example 1

◦ Name each triangle by its sides and angles

A. B.

C.

Vertex (plural vertices)◦ The points joining the sides of a triangle

Adjacent sides◦ Sides sharing a common vertex

Side AB is adjacent to side BC

A

BC

Interior angle◦ Angle on the inside of a triangle

Exterior angle◦ Angle outside the triangle that is formed by extending

one side

A

B

C

Interior angleExterior angle

Triangle Sum Theorem◦ The sum of the three interior angles of a triangle

is 180º

Exterior Angle Theorem◦ The measure of the exterior angle of a triangle is

equal to the sum of the two nonadjacent interior angles. Example: m∠1=m∠A+ m∠B

B

AC

1

Example 2

Find the measure of each angle.

2x + 10

x x + 2

Example 3

Given that ∠ A is 50º and ∠B is 34º, what is the measure of ∠BCD?

What is the measure of ∠ACB? DA

B

C

Right triangle vocabulary

Legs◦ Sides that form the right angle

Hypotenuse◦ Side opposite the right angle

Legs

Hypotenuse

Corollary to the Triangle Sum Theorem◦ The acute angles of a right triangle are

complementary. m∠A+ m∠B = 90°

Example 4

A. Given the following triangle, what is the length of the hypotenuse?

B. What are the length of the legs?

C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement?

13

12

5

Legs◦ The two congruent sides of an isosceles triangle.

Base◦ The noncongruent side of an isosceles triangle.

Base Angles◦ The two angles that contain the base of an isosceles

triangle.

Vertex Angle◦ The noncongruent angle in an isosceles triangle.

Legs

Base Angles

Vertex Angle

Base

Example 5

A. Given the following isosceles triangle, what is the measurement of segment AC?

B. What is the measurement of angle A?

A

B C75º

15

7

Example 6

Find the missing measures

80°

53°

Example 7

Given: ∆ABC with mC = 90°Prove: mA + mB = 90°

Statement Reason

1. mC = 90°

2. mA + mB + mC = 180°

3. mA + mB + 90° = 180°

4. mA + mB = 90°

Geometry 1Unit 4

4.2 Congruence and Triangles

Congruent Figures Congruent Figures

◦ Figures are congruent if corresponding sides and angles have equal measures.

Corresponding Angles of Congruent Figures◦ When two figures are congruent, the angles that

are in corresponding positions are congruent.

Corresponding Sides of Congruent Figures◦ When two figures are congruent, the sides that

are in corresponding positions are congruent.

Congruent Figures For the triangles below, ∆ABC ≅ ∆PQR

◦ The notation shows congruence and correspondence.

◦ When writing congruence statements, be sure to list corresponding angles in the same order.

A

B

C

P

Q

R

Corresponding Angles

Corresponding Sides

A ≅ P AB ≅ PQ

B ≅ Q BC ≅ QR

C ≅ R CA ≅ RP

Complete the congruence statement for the two given triangles:

DEFWhat side corresponds with DE?

What angle corresponds with E?

D

E

F

S

V

T

Example 1

Example 2

In the diagram, ABCD ≅ KJHLa. Find the value of x.b. Find the value of y.

A B

D C

9 cm

6 cm86°

91°

113°

J

H

KL

(5y – 12)°

(4x – 3) cm

Third Angles Theorem◦ If 2 angles of 1 triangle are congruent to 2 angles

of another triangle, then the third angles are also congruent.

Example 3

Given ABC PQR, find the values of x and y.

A

B CP

QR

85°

50°

(6y – 4)°

(10x + 5)°

Example 4 Decide whether the triangles are congruent.

Justify your answer.

F

HE

G

J

58°

58°

Example 5

Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN.

Prove: ∆MNO ≅ ∆QPO

P

M

O

Q

N Statements Reasons

1.

2. Alt. Interior Angles Theorem

3. Vertical Angles Theorem

4.O is the midpoint of MQ and PN

5. Def of Midpoint

6. ∆MNO ≅ ∆QPO

Geometry 1Unit 4

4.3 Proving Triangles are Congruent: SSS and SAS

Warm-Up

Complete the following statementBIG

B

I

G

R

A

T

Definitions Included Angle

◦ An angle that is between two given sides.

Included Side ◦ A side that is between two given angles.

Example 1 Use the diagram.

Name the included angle between the pair of given sides.

KP

J L

PLandKP

LKandPK

KLandJK

Triangle Congruence Shortcut SSS

◦ If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Triangle Congruence Shortcuts SAS

◦ If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Example 2 Complete the congruence statement. Name the congruence shortcut used.

ST

U

V

WSTW

Example 3 Determine if the following are congruent. Name the congruence shortcut used.

J

H

I

L

MN

HIJ LMN

Example 4 Complete the congruence statement. Name the congruence shortcut used.

B

OX

C

A

R

XBO

Example 5 Complete the congruence statement. Name the congruence shortcut used.

SPQ

S

P

Q

T

Constructing Congruent Triangles Construct segment DE as a segment

congruent to AB Open your compass to the length of AC.

Place the point of your compass on point D and strike an arc.

Open the compass to the width of BC. Place the point of your compass on E and strike an arc. Label the point where the arcs intersect as F.

A

C

B

Example 6Given: AB ≅ PB, MB ⊥ APProve: ∆MBA ≅ ∆MBP

AB P

M

Statements Reasons

1. MB ⊥ AP

2. Perpendicular lines form right angles

3. Right angles are congruent

4. AB ≅ PB

5. MB ≅ MB

6.

Example 7

Use SSS to show that ∆NPM ≅∆DFEN(-5, 1)P(-1, 6)M(-1, 1)D(6, 1)F(2, 6)E(2, 1)

Geometry 1Unit 4

4.4 Proving Triangles are Congruent: ASA and AAS

Triangle Congruence Shortcuts ASA

◦ If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Triangle Congruence Shortcuts AAS

◦ If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

Example 1 Complete the congruence statement. Name the congruence shortcut used.

Q U

ADQUA

Example 2 Complete the congruence statement. Name the congruence shortcut used.

RMQ

M R

Q

N

P

Example 3 Determine if the following are congruent. Name the congruence shortcut used.

A

B

C

F

E

DABC FED

Example 4

Given: B ≅C, D ≅F; M is the midpoint of DF.Prove: ∆BDM ≅∆CFM

B

D M

C

F

Statements Reasons

1.

2.

3. Def of Midpoint

4.

Geometry 1Unit 4

4.5 Using Congruent Triangles

Warm-up State which postulate or theorem you can

use to prove that the triangles are congruent.

Then, write the congruence statement.

C

G

H

S

Example 1

Given: NO is parallel to MP, MN is parallel to PO

Prove MN = OP(Prove Δ MNO Δ OPM)

Mark the given information first

Then, mark the deduced information

PO

N M

Statements Reasons1.NO||MP, MN|| PO 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

Example 2

Given: HJ || KL, JK || HLProve: LHJ ≅ JKL

H J

L K

Example 3

Given: MS || TR, MS ≅ TR

Prove: A is the midpoint of MT.

S

M

A

T

R

Statements Reasons

1.

2.

3.

4.

5.

6.

Geometry 1Unit 4

4.6 Isosceles, Equilateral, and Right Triangles

Warm-Up 1

Find the measure of each angle.

90°

90°

30°

60°a

b

Warm-Up 2

Find the measure of each angle.

110

15090

Base Angles Theorem◦ If two sides of a triangle are congruent, then the

angles opposite them are also congruent. Converse of the Base Angles theorem

◦ If two angles of a triangle are congruent, then the sides opposite them are also congruent

Example 1

35°

x

Example 2

15°

b

a

Example 3

Find each missing measure

63°

10 cmm n

p

Equilateral Triangles If a triangle is equilateral, then it is

equiangular.

If a triangle is equiangular, then it is equilateral.

Hypotenuse-Leg (HL) If the hypotenuse and the leg of a right

triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.

Example 4 Find the value of x

12 in

2x in

Example 5 Find the value of x and y.

yx

Example 6 Find the value of x and y.

75°

x°

y°

x°

Geometry 1Unit 4

4.7 Triangles and Coordinate Proof

Warm-up

What is the midpoint formula?

What is the distance formula?

What are some postulates and theorems you have learned about triangles this chapter?

Vocabulary Coordinate Proof

◦ A proof involving placing geometric figures on a coordinate plane.

◦ Uses the midpoint formula, distance formula, postulates and theorems to prove statements about the figure

Placing Figures in a Coordinate Plane Complete the activity on p. 243 individually. Compare your results to those of your

partners. What did you learn?

Example 1 A right triangle has legs of 3 units and 4

units. Place the triangle on a coordinate grid.

Label the vertices, then find the length of the hypotenuse.

3

4

Example 2 In the diagram, ΔABO ≅ ΔCBO. Find the coordinates of point B.

C(10,0)

A(0,10)

O(0,0)

B

Example 3 Write a plan to prove that OU bisects TOV.

V(3,5)

U(0,5)

T(-3,5)

O(0,0)

Example 4 Find the coordinates of P.

P

N(h,0)

M(0,k)

Constructions review Duplicate the given triangle. Write the steps that you used to construct

the new triangle