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Rzqust [24]
3 years ago
7

How are the surface area and volume of a rectangular prism related to each other?

Mathematics
1 answer:
MatroZZZ [7]3 years ago
7 0
The surface area of a rectangular prism is equal to the volume of a rectangular prism on their cooresponding sides.
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Decide if the following statement is valid or invalid. If two sides of a triangle are congruent then the triangle is isosceles.
Naya [18.7K]

Answer:

Step-by-step explanation:

Properties of an Isosceles Triangle

(Most of this can be found in Chapter 1 of B&B.)

Definition: A triangle is isosceles if two if its sides are equal.

We want to prove the following properties of isosceles triangles.

Theorem: Let ABC be an isosceles triangle with AB = AC.  Let M denote the midpoint of BC (i.e., M is the point on BC for which MB = MC).  Then

a)      Triangle ABM is congruent to triangle ACM.

b)      Angle ABC = Angle ACB (base angles are equal)

c)      Angle AMB = Angle AMC = right angle.

d)      Angle BAM = angle CAM

Corollary: Consequently, from these facts and the definitions:

Ray AM is the angle bisector of angle BAC.

Line AM is the altitude of triangle ABC through A.

Line AM is the perpendicular bisector of B

Segment AM is the median of triangle ABC through A.

Proof #1 of Theorem (after B&B)

Let the angle bisector of BAC intersect segment BC at point D.  

Since ray AD is the angle bisector, angle BAD = angle CAD.  

The segment AD = AD = itself.

Also, AB = AC since the triangle is isosceles.

Thus, triangle BAD is congruent to CAD by SAS (side-angle-side).

This means that triangle BAD = triangle CAD, and corresponding sides and angles are equal, namely:

DB = DC,

angle ABD = angle ACD,

angle ADB = angle ADC.

(Proof of a).  Since DB = DC, this means D = M by definition of the midpoint.  Thus triangle ABM = triangle ACM.

(Proof of b) Since angle ABD = angle ABC (same angle) and also angle ACD = angle ACB, this implies angle ABC = angle ACB.

(Proof of c) From congruence of triangles, angle AMB = angle AMC.  But by addition of angles, angle AMB + angle AMC = straight angle = 180 degrees.  Thus 2 angle AMB = straight angle and angle AMB = right angle.

(Proof of d) Since D = M, the congruence angle BAM = angle CAM follows from the definition of D.  (These are also corresponding angles in congruent triangles ABM and ACM.)

QED*

*Note:  There is one point of this proof that needs a more careful “protractor axiom”.  When we constructed the angle bisector of BAC, we assumed that this ray intersects segment BC.  This can’t be quite deduced from the B&B form of the axioms.  One of the axioms needs a little strengthening.

The other statements are immediate consequence of these relations and the definitions of angle bisector, altitude, perpendicular bisector, and median.  (Look them up!)

Definition:  We will call the special line AM the line of symmetry of the isosceles triangle.  Thus we can construct AM as the line through A and the midpoint, or the angle bisector, or altitude or perpendicular bisector of BC. Shortly we will give a general definition of line of symmetry that applies to many kinds of figure.

Proof #2 (This is a slick use of SAS, not presented Monday.  We may discuss in class Wednesday.)

The hypothesis of the theorem is that AB = AC.  Also, AC = AB (!) and angle BAC = angle CAB (same angle).  Thus triangle BAC is congruent to triangle BAC by SAS.

The corresponding angles and sides are equal, so the base angle ABC = angle ACB.

Let M be the midpoint of BC.  By definition of midpoint, MB = MC. Also the equality of base angles gives angle ABM = angle ABC = angle ACB = angle ACM.  Since we already are given BA = CA, this means that triangle ABM = triangle ACM by SAS.

From these congruent triangles then we conclude as before:

Angle BAM = angle CAM (so ray AM is the bisector of angle BAC)

Angle AMB = angle AMC = right angle (so line MA is the perpendicular bisector of  BC and also the altitude of ABC through A)

QED

Faulty Proof #3.  Can you find the hole in this proof?)

In triangle ABC, AB = AC.  Let M be the midpoint and MA be the perpendicular bisector of BC.

Then angle BMA = angle CMA = right angle, since MA is perpendicular bisector.  

MB = MC by definition of midpoint. (M is midpoint since MA is perpendicular bisector.)

AM = AM (self).

So triangle AMB = triangle AMC by SAS.

Then the other equal angles ABC = ACB and angle BAM = angle CAM follow from corresponding parts of congruent triangles.  And the rest is as before.

QED??

8 0
2 years ago
Adam and his sister Beatrice combined their allowance of $7 each so they could buy a movie for $12 they bought $1 containers of
Fittoniya [83]
14 - 12 = 2 dollars left over to buy containers. They each got one container each.
8 0
3 years ago
Read 2 more answers
Which shows the correct substitution of the values a, b, and c from the equation –2 = –x + x2 – 4 into the quadratic formula?
Serhud [2]
X=[-(-1)+or-sqrt ((-1)^2-4x1x-2)]/(2x1)

7 0
3 years ago
Read 2 more answers
Samantha has a pig enclosure that is 4 feet by 6 feet in her backyard. She wants to double the area of the enclosure by increasi
luda_lava [24]

Answer:

Step-by-step explanation:

The answer is 48 feet because 4x6=24 and 24x2=48

4 0
3 years ago
Figure ABCD is a parallelogram. What is the value of x? A B O 6 ET O 7 8 5x + 3 38 09 D С​
kondor19780726 [428]

Answer:

Given: It is given that  and .

To Prove: ABCD is a parallelogram.

Statements                                                                            

1.  and            

Reason: Given                                      

2.  are alternate interior angles      

Reason: Def of alternate interior angles

3.                                                

Reason: Alternate interior angles are congruent

4.                                                              

Reason: Reflexive property

5.                                                  

Reason: SAS congruency theorem

6.                                                        

Reason: Corresponding Parts of Congruent triangles are Congruent (CPCTC)

7. ABCD is a parallelogram                                  

Reason: Parallelogram Side Theorem

Step-by-step explanation:

please mark me brainliest

6 0
2 years ago
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