KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing?

Answer: A and D

Honors Geometry

Unit 4 Lesson 5Proving Triangles Congruent

Objectives

• I can identify corresponding parts of congruent triangles

• I can use the definition of congruent triangles

• I can discover and apply theorems about triangles

Recall

• Congruent polygons– All sides are congruent (same length)– All angles are congruent (same measure)

• Congruent triangles– 3 congruent sides– 3 congruent angles

– Total of 6 congruent parts

Corresponding Parts

• When two (or more) polygons are congruent, it is necessary to identify their corresponding parts– Use dash marks on sides and arcs on angles!

• Corresponding parts – between congruent polygons, corresponding parts have the same measure and are found in the same position

– Corresponding parts = matching parts– CPCF (Corresponding parts of congruent figures)

Locate Corresponding Parts

Identify Corresponding PartsIdentify all of the congruent corresponding parts. Mark them on the diagram using the appropriate symbols.

Sides:

Angles:

Congruence Statements

• Write a valid congruence statement by listing corresponding parts in the same order

– Corresponding parts will appear in the same position

– Also, you can read a congruence statement and determine which parts are corresponding

Example – Congruence Statement• Same example:

• Corresponding parts:

• Congruence Statement:

Example - Congruence Statement

• Statement:

• Notice: Since angles A and H were congruent, they appear first (same as B & J, C & K)

• Notice: Segments AB and HJ are congruent, and those two letters appear first…

• There are multiple correct answers!

A. AB. BC. CD. D

A. ΔLMN ΔRTS

B. ΔLMN ΔSTR

C. ΔLMN ΔRST

D. ΔLMN ΔTRS

Write a congruence statement for the triangles.

A. AB. BC. CD. D

A. L R, N T, M S

B. L R, M S, N T

C. L T, M R, N S

D. L R, N S, M T

Name the corresponding congruent angles for the congruent triangles.

List all the congruent parts and write a congruence statement.

Algebra

O P CPCF

mO = mP Definition of congruence

6y – 14 = 40 Substitution

In the diagram, ΔITP ΔNGO. Find the values of x and y.

6y = 54 Add 14 to each side.

y = 9 Divide each side by 6.

AlgebraIn the diagram, ΔITP ΔNGO. Find the values of x and y.

CPCFNG = IT Definition of congruence

x – 2y = 7.5 Substitution

x – 2(9) = 7.5 y = 9

x – 18 = 7.5 Simplify.

x = 25.5 Add 18 to each side.

A. AB. BC. CD. D

Algebra

A. x = 4.5, y = 2.75

B. x = 2.75, y = 4.5

C. x = 1.8, y = 19

D. x = 4.5, y = 5.5

In the diagram, ΔFHJ ΔHFG. Find the values of

x and y.

Congruent Triangles

• For the rest of the lesson, we will focus on proving that triangles are congruent

• Recall that congruent triangles share 6 corresponding, congruent parts

• There are 4 shortcuts – ways to show that all six parts are congruent by only using 3– You only have to do half the work!

The Four Congruence Postulates

• You must follow the specific order named by the postulate

• A – a pair of congruent angles• S – a pair of congruent sides

• Included angle – an angle found between two congruent sides

• Included side – a side found between two congruent angles

SSS Congruence

SAS Congruence

ASA Congruence

AAS Congruence

Notice…

• There is a combination of letters that we did NOT use – because it will NOT prove that all 6 parts are congruent

• One angle and the next two sides… or two sides and the next angle

Hints…

• To determine WHICH of the 4 postulates is illustrated in a particular example– Read all given information– Use the information to find congruent parts– Find any other congruent parts • Vertical pairs, shared side, etc

– Mark congruent parts on the diagram– Determine in which order the parts appear• Is a side between two marked angles?• Is an angle between two marked sides?

A. AB. BC. CD. D

A. SSS

B. ASA

C. SAS

D. not possible

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, choose not possible.

A. AB. BC. CD. D

A. SSS

B. ASA

C. SAS

D. not possible

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, choose not possible.

A. AB. BC. CD. D

A. SSA

B. ASA

C. SSS

D. not possible

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, choose not possible.

EXTENDED RESPONSE Triangle DVW has vertices D(–5, –1), V(–1, –2), and W(–7, –4). Triangle LPM has vertices L(1, –5), P(2, –1), and M(4, –7).a. Graph both triangles on the same coordinate

plane.b. Use the distance formula to make a conjecture as to

whether the triangles are congruent. Explain your reasoning.

Use SSS congruence…

Find the side lengths of the two triangles

Triangle DVW Triangle LPM

CONCLUSION: by SSS Congruence LPMDVW

1. A2. B3. C

Use SSS Congruence

A. yes

B. no

C. cannot be determined

Determine whether ΔABC ΔDEF for A(–5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1).

Video Clip: Proof

• http://ed.ted.com/lessons/scott-kennedy-how-to-prove-a-mathematical-theory

• 4:39• Open browser first

Use SAS to Prove Triangles are Congruent

Draw a picture

3. Vertical Angles Theorem

3. FGE HGI

2. Midpoint Theorem2.

Prove: ΔFEG ΔHIG

4. SAS4. ΔFEG ΔHIG

Given:EI HF; G is the midpoint of both EI and HF.

1. Given1. EI HF; G is the midpoint ofEI; G is the midpoint of HF.

ReasonsStatements

A. AB. BC. CD. DA. Reflexive B. Symmetric

C. Transitive D. Substitution

3. SSS3. ΔABG ΔCGB

2. _________2. ? Property

1.

ReasonsStatements

1. Given

Recap