use-the-diagrams-on-pages-153-and-154-to-prove-the-following-theorems-theorem-4-5

Question

Use the diagrams on pages 153 and 154 to prove the following theorems.
Theorem $4-5$

William Scherer · Accepted Answer

problem number 14. They want us to prove that a B equals a C. Given that line l is the perpendicular by sector of B C and A is on line ele. All right, so first, we&#x27;re going to say l is the perpendicular or by sector of a C. Okay, that&#x27;s given. And also I think we should mention, um, line A X is the same as line l Okay, we can see that from the diagram. And it was given that point A is on the perpendicular by sector. Okay, so we know that be X is congrats. Went to X c. Okay. And that&#x27;s the definition of bisect. We said a X is perpendicular by sector. Okay. And we know that angle. A xsi and a X B are right angles. Okay? And that&#x27;s the definition off perpendicular by sector. All right. And because they&#x27;re both right angles, angle a xsi is congratulate to angle a x B. And that&#x27;s because all right angles Arkan growing. And we also know that a X is congratulate to itself. All right, that&#x27;s the reflexive property. Okay, so if I could just draw this right angle in here, we have a side angle side side angle sites. Let&#x27;s make the letters in order. I&#x27;m gonna say triangle be X a is congruent to see x a All right triangle be x A is congratulate to triangle c x a. Okay, so from there we can say that a B equals a c. All right. A B is the 1st and 3rd letters, and a C is the 1st and 3rd letters, so they correspond to each other. So that means a B equals a C. And the reason for that is because could corresponding parts of congruent triangles are congruent, so a B equals a C. Uh, I gotta make three marks. There you go.

Amrita Bhasin · Answer

we know that a bee&#x27;s a seat. Be exit. Maxine Axe is a X. These were all given. Therefore we know by the sss er side side side rule of concurrency Abie access congruent triangles a C X, We also know angle a xB an angle Xia equivalent because corresponding angles of congruent triangles are equal. Therefore, we know that angle X b plus angle XY is 180 degrees because they&#x27;re complimentary angles. And so we know that if we&#x27;re adding up complimentary angles, 90 plus 90 is gonna end up being 180. Therefore, this proves that a lies on the perpendicular by sector of BC.

Charles Carter · Answer

you are an angle by sector. It&#x27;s saying that if there&#x27;s a point p on ankle by sector than it&#x27;s perpendicular distance from the to raise in the original angle, that these two distances will be congruent. Okay, so we&#x27;re asked to prove this theorem using this picture. Um, it&#x27;s a couple of things work, Um, better mentioned in the theory. Okay, so we have that BZ Rabies E by 60 angle. So you know that these two angles are congrats. You know, the P is on BZ, so it&#x27;s just good late. Well, these things so Oh, just for sake of argument, it&#x27;s just a way of strong. Can I use your points? X y and P? Okay, we know that because the shortest distance to those lines they have to be perpendicular. But it&#x27;s also part of the given. And so we need to prove that p X is congruent p y. So I&#x27;m gonna go about it by saying first that BP, it&#x27;s King Grant to be peeing. That&#x27;s because of the reflexive property. I&#x27;m going about this to kind of prove that the two triangles Air King Guerette We know that the small ankle and that 90 degree angle are also congruent. And we know that this is a 90 degree, uh, triangle can so kind of this several angle. Why? BP is king grin, Teoh angle ex BP. Guess it&#x27;s by definition of the ankle by sector, which means that the sign of those two angles will also be congruent in a sign dealing with the opposite side. That means the opposite side Y p and X p So do the sign. I think a why BP is equal to the length of P. Why are y p and about about BP and the sign of and go ex BP this *** at two p except for BP. So then because BP is the same for both and the sun is the same for both. We can say that BP times the sign the middle ex BP is equal to BP. It comes a sign of angle. Why BP, which is equal to the length of P X, which is equal to the length of P. Y. Okay. And so there is what we were needing to prove. Okay, hopefully that does it. Thank you very much.

Amrita Bhasin · Answer

We know that sign of a mount of son of B equals to co sign April&#x27;s be over two times Sign of a minus B over to Therefore we know that A is gonna be angled PB acts and B is gonna be angle PB Why therefore the assumption that April&#x27;s be over too cannot equal pi over too.

Amrita Bhasin · Answer

okay. We know that because of the common side, que es must be congruent. Tick you ask because we know congruent triangles or convert by the S A s s s rule. In this case, that means side, side side with their front of the triangle Thi s cures converted to triangle. Archy asked. Therefore, we know opposite angles of Congress. Of course, finding sides of congruent triangles are equal. Therefore, we know the quadrilateral has grown opposite sides and concurrent opposite angles. Therefore, angle cute is angle asked an angle t is angle are therefore we know the given quadrilateral Haas to be a parallelogram.

Adam Dehollander · Answer

in this firm were told that a D is congruent to BC in Austin that a D is a love b c. Now we see that if a d and B c a parallel, then we have these lines D, c and B C that intercepting two parallel lines and by the alternating angles the M. We know that the measure of angle and the measure of angle see are the same. And similarly, the measure angle B and D are offset the same. And now we can use this to see that if we drew a diagnosis between B and D, then we can figure out that this diagnose would be the same triangle a D B as it is for Triangle See CP. And therefore, since they have the same angle and the same side, that means well, we have the same angle say angle A and an angle. See, we have the same side d B But then we also have the same side. 80 is same, spc And so that means this side length must also be equal to each other. And now by the M 54 Since we have two years of congruent sides here a d b c and also a d d c. But the M 54 this is a parallelogram.

Problem

Use the diagrams on pages 153 and 154 to prove th…

Problem 14 Hard Difficulty

Use the diagrams on pages 153 and 154 to prove the following theorems.
Theorem $4-5$

Answer

$\overline{A X} \cong \overline{A X}$ - reflexive property
$\overline{X B} \cong \overline{X C}-$ property of bisector
$\angle A X B \cong \angle A X C$ -both right angles- property of perpendicular line
By theorem SAS, $A B=A C$

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$\overline{A X} \cong \overline{A X}$ - reflexive property$\overline{X B} \cong \overline{X C}-$ property of bisector$\angle A X B \cong \angle A X C$ -both right angles- property of perpendicular lineBy theorem SAS, $A B=A C$

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Christine G.

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Maria G.

Samuel H.

(A few) Circle Theorems

Angles and Cirlce Theorems with chords and tangents

Ray C. Jurgensen,Richard G. Brown,John W. JurgensenGeometry

Christine G.

Heather Z.

Maria G.

Samuel H.

$\overline{A X} \cong \overline{A X}$ - reflexive property
$\overline{X B} \cong \overline{X C}-$ property of bisector
$\angle A X B \cong \angle A X C$ -both right angles- property of perpendicular line
By theorem SAS, $A B=A C$

Ray C. Jurgensen,Richard G. Brown,John W. Jurgensen

Geometry