4.9

What Is Left To Prove?

Pg. 30

Parts of Congruent Triangles

4.10 – What Is Left To Prove? Parts of Congruent Triangles

Now that you know how to prove triangles are congruent, how can we prove more about their individual parts?

4.56 – CONGRUENT TRIANGLESWhen you have proven that two triangles are congruent, what can you say about their corresponding parts?

a. Examine the two triangles at right and the proof below. What is the given? What are you trying to prove?

Given: ABCD is a kiteProve: D B

b. Complete the missing reasons for #4 and #5 above.

Given: ABCD is a kiteProve: D B

reflexive prop.SSS

c. Notice there is no reason given for Statement #6. Why do you know those angles will be congruent based on this proof?

All corresponding parts are =

reflexive prop.SSS

reflexive prop.SSS

c. This reason is called "Corresponding Parts of Congruent Triangles Are Congruent." It can be shortened to CPCTC. Or you can write an arrow diagram to show the meaning by stating: ≅ ∆ ≅ parts.Complete the reason for the proof above.

CPCTC

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Cows Poop Cause They Can

4.57 – DIAGONALS OF A RECTANGLEUse the proof below to show that the diagonals of a rectangle are congruent.

Given: ABCD is a rectangleProve: AC = BD

Given: ABCD is a rectangleProve: AC = BD

2. AD BC 2. opp. side rect. =3. D C 3. def. rectangle

4. DC DC 4. reflexive prop.

5. ADC BCD 5. SAS

1. is a rectangleABCD 1. given

6. AC BD 6. CPCTC

4.58 – DIAGONALS OF A RHOMBUSWhat can congruent triangles tell us about the diagonals of and angles of a rhombus? Prove that the diagonals of a rhombus bisect the angles.

Given: ABCD is a rhombusProve: ABD CBD

Given: ABCD is a rhombusProve: ABD CBD

2. AD DC CB BA 2. def. rhombus3. DB DB 3. reflexive prop.

4. ABD CBD 4. SSS

1. is a rhombusABCD 1. given

5. ABD CBD 5. CPCTC

4.59 – DIAGONALS OF A RHOMBUSProve that if one pair of opposite sides are congruent and parallel, the shape is a parallelogram.

2. //AB CD 2. given3. ABD CDB 3. alt. interior s4. DB DB 4. reflexive prop.

5. ABD CDB 5. SAS

1. AB CD 1. given

6. AD BC 6. CPCTC7. is a ABCD 7. both. opp. sides =

4.60 – PROOF BY CONTRADICTIONSometimes you cannot prove something directly and need to prove it by disproving other ideas. Come up with a way to disprove the following claims.

a. The product of an odd number and an even number is always odd.

5 2 = 10

3 -2 = -6

b. A number minus another number will always be smaller.

5 – 2 = 3

3 – (-2) = 5

c. A quadrilateral with perpendicular diagonals is a kite.

rhombus

d. All quadrilaterals with two pairs of congruent sides is a parallelogram.

kite

e. Interior angles of a pentagon are always 108°

Non-regular pentagon