Ways to prove Ways to prove Triangles Congruent Triangles Congruent (SSS), (SAS), (ASA)(SSS), (SAS), (ASA)

Ways to prove Ways to prove Triangles Congruent Triangles Congruent (SSS), (SAS), (ASA)(SSS), (SAS), (ASA)

4-2 to 4-44-2 to 4-4

EXAMPLE 4 Use the Third Angles Theorem

Find m BDC.

So, m ACD = m BDC = 105° by the definition of congruent angles.

ANSWER

SOLUTION

A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, m ACD = 180° – 45° – 30° = 105° .

EXAMPLE 5 Prove that triangles are congruent

Plan for Proof

AC AC.

a. Use the Reflexive Property to show that

b. Use the Third Angles Theorem to show that

B D

Write a proof.

GIVEN AD CB, DC AB

ACD CAB, CAD ACB

PROVE ACD CAB

EXAMPLE 5 Prove that triangles are congruent

Plan in Action

1. Given

2. Reflexive Property of Congruence

STATEMENTS REASONS

3. Given

4. Third Angles Theorem

1. AD CB, DC BA

2. a. AC AC.

3. ACD CAB,CAD ACB

4. b. B D

5. ACD CAB Definition of5.

GUIDED PRACTICE for Examples 4 and 5

SOLUTION

4. DCN.In the diagram, what is m

CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75°

GUIDED PRACTICE for Examples 4 and 5

SOLUTION

(Proved from above sum)

By the definition of congruence, whatadditional information is needed toknow that

5.

NDC NSR.

CN NR, CDN NSR, DCN NRS

Given :

NDC NSR.Proved :

GUIDED PRACTICE for Examples 4 and 5

STATEMENT REASON

Given

Given

CDN NSR

DCN NRS

Therefore DC RS, DN SN as angles are congruent their sides are congruent.

EXAMPLE 1 Identify congruent parts

Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.

SOLUTION

The diagram indicates that JKL TSR.

Corresponding sides JK TS, KL SR, LJ RT

Corresponding angles J T, ∠K S, L R

EXAMPLE 2 Use properties of congruent figures

In the diagram, DEFG SPQR.

Find the value of x.a.

b. Find the value of y.

SOLUTION

You know that FG QR.a.

FG = QR

12 = 2x – 4

16 = 2x

8 = x

EXAMPLE 2 Use properties of congruent figures

b. You know that ∠F Q.

m F = m Q

68o = (6y + x)o

68 = 6y + 8

10 = y

EXAMPLE 3 Show that figures are congruent

PAINTING

If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain.

SOLUTION

From the diagram, A C and D B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC .

Then, 1 4 and 2 3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent.

EXAMPLE 3 Show that figures are congruent

The diagram shows AJ CK , KD JB , and DA BC . By the Reflexive Property, JK KJ . All corresponding parts are congruent, so AJKD CKJB.

GUIDED PRACTICE for Examples 1, 2, and 3

1. Identify all pairs of congruent corresponding parts.

SOLUTION

Corresponding sides: AB CD, BG DE, GH FE, HA FC

Corresponding angles: A C, B D, G E, H F.

In the diagram at the right, ABGH CDEF.

GUIDED PRACTICE for Examples 1, 2, and 3

In the diagram at the right, ABGH CDEF.

SOLUTION

2. Find the value of x and find m H.

(b) You know that H Fm H m F =105°

(a) You know that H F (4x+ 5)° = 105° 4x = 100 x = 25

GUIDED PRACTICE for Examples 1, 2, and 3

SOLUTION

In the diagram at the right, ABGH CDEF.

3. Show that PTS RTQ.

In the given diagram

PS QR, PT TR, ST TQ and

Similarly all angles are to each other, therefore all of the corresponding points of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem.

EXAMPLE 1 Use the SSS Congruence Postulate

Write a proof.

GIVEN KL NL, KM NM

Proof It is given that KL NL and KM NM

By the Reflexive Property, LM LN.

So, by the SSS Congruence Postulate, KLM NLM

PROVE KLM NLM

Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.

GUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

Yes. The statement is true.

1. DFG HJK

Side DG HK, Side DF JH,and Side FG JK.

So by the SSS Congruence postulate, DFG HJK.

GUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

2. ACB CAD

BC ADGIVEN :

PROVE : ACB CAD

PROOF: It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.

GUIDED PRACTICE for Example 1

Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

GUIDED PRACTICE for Example 1

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

QT TR , PQ SR, PT TSGIVEN :

PROVE : QPT RST

PROOF: It is given that QT TR, PQ SR, PT TS. So bySSS congruence postulate, QPT RST. Yes the statement is true

QPT RST 3.

EXAMPLE 1 Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE

STATEMENTS REASONS

BC DA, BC AD

ABC CDA

1. Given1. BC DAS

Given2. 2. BC AD

3. BCA DAC 3. Alternate Interior Angles Theorem

A

4. 4. AC CA Reflexive Property of Congruence

S

EXAMPLE 1 Use the SAS Congruence Postulate

STATEMENTS REASONS

5. ABC CDA SAS Congruence Postulate

5.

EXAMPLE 2 Use SAS and properties of shapes

In the diagram, QS and RP pass through the center M of the circle. What can you conclude about MRS and MPQ?

SOLUTION

Because they are vertical angles, PMQ RMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MS are all equal.

MRS and MPQ are congruent by the SAS Congruence Postulate.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

In the diagram, ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT SU and .SU VU

1. Prove that SVR UVR

STATEMENTS REASONS

1. SV VU 1. Given

3. 3. RV VR Reflexive Property of Congruence

2. 2. SVR RVU Definition of line

4. 4. SVR UVR SAS Congruence Postulate

GUIDED PRACTICE for Examples 1 and 2

2. Prove that BSR DUT

STATEMENTS REASONS

1. 1. GivenBS DU

2. 2. RBS TDU Definition of line

3. 3. RS UT Given

4. 4. BSR DUT SAS Congruence Postulate

EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem

Write a proof.

SOLUTION

Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram.

GIVEN WY XZ, WZ ZY, XY ZY

PROVE WYZ XZY

STATEMENTS REASONS

EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem

1. WY XZ 1. GivenH

2. 2. WZ ZY, XY ZY Given

4. 4. Definition of a right triangle

WYZ and XZY are right triangles.

L ZY YZ5. 5. Reflexive Property of Congruence

6. WYZ XZY 6. HL Congruence Theorem

3. 3. Definition of linesZ and Y are right angles

EXAMPLE 4 Choose a postulate or theorem

Sign Making

You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You know that RP QS and PQ PS . What postulate or theorem can you use to conclude that PQR PSR?

EXAMPLE 4 Choose a postulate or theorem

SOLUTION

RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent.

You are given that PQ PS . By the Reflexive Property, RP RP . By the definition of perpendicular lines, both

You can use the SAS Congruence Postulate to conclude that .PQR PSR

ANSWER

GUIDED PRACTICE for Examples 3 and 4

Use the diagram at the right.

3. Redraw ACB and DBC side by side with corresponding parts in the same position.

GUIDED PRACTICE for Examples 3 and 4

4.

Use the diagram at the right.

Use the information in the diagram to prove that ACB DBC

STATEMENTS REASONS

1. AB DC 1. GivenH

2. 2. AC BC, DB BC Given

4. 4. Definition of a right triangle

ACB and DBC are right triangles.

3. 3. Definition of linesC B

GUIDED PRACTICE for Examples 3 and 4

STATEMENTS REASONS

L BC CB5. 5. Reflexive Property of Congruence

6. ACB DBC 6. HL Congruence Theorem