Download Fine work. If you do not look at these as right triangles

Congruent Right Triangles
In this section, you will extend your knowledge of congruence into the realm of right triangles.
You will learn shortcuts for proving congruence in right triangles. Then you will use these
congruence theorems to discover other new and useful facts.
Objectives
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Experiment with congruent triangles.
Explore the right triangle congruence shortcuts theorems — HL, LL, HA, LA.
Discover the relationship between the right triangle congruence theorems and the
congruence theorems for nonright triangles.
Prove the Perpendicular Bisector Theorem.
Prove the Angle Bisector Theorem and its Converse.
Congruent Right Triangles
To begin, recall what you know about a pair of congruent triangles: they have the same size and
same shape, and all of their corresponding side lengths and angle measures are equal.
What if the two congruent triangles are right triangles?
You can experiment with some here.
Notice that congruent right triangles have the same properties that were listed on the previous
slide — same size and shape, three pairs of congruent sides, and three pairs of congruent angles.
Also, recall the congruence postulates and theorems that “work” for all triangles.
Can these be simplified for right triangles? After all, two right triangles automatically have one
pair of congruent angles already — the right angles.
You will try to build a right triangle below by requesting the lengths of the hypotenuse and one
leg.
You will be given two sets of these pieces so that you can build two right triangles. Each set of
pieces includes a right angle, hypotenuse, and leg.
Are they congruent? Is it possible to build two triangles that are not congruent?
You may have seen that your triangles were always congruent.
If two right triangles have congruent hypotenuses and one pair of congruent legs, the triangles
are congruent. This is the HL Congruence Theorem. (“HL” stands for “hypotenuse-leg”.)
HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse
and corresponding leg of another right triangle, then the triangles are congruent.
Given:
A and
Conclusion:
D are right angles.
ABC
DEF
Reason: HL
Try the experiment again, this time requesting the lengths of two legs of a right triangle.
Again, you will be given two sets of the legs you requested, along with a right angle to go with
each set of legs.
Build two triangles and see if congruence is guaranteed.
You may have noticed that if both pairs of legs of right triangles are congruent, the triangles
themselves are guaranteed to be congruent.
You have discovered the LL Congruence Theorem. (“LL” stands for “leg-leg”.)
LL Theorem
If the legs of one right triangle are congruent to the corresponding legs of another right triangle,
then the triangles are congruent.
Given:
I and
Conclusion:
L are right angles.
GHI
JKL
Reason: LL
So why does HL guarantee congruence?
Recall that when you know two side lengths in a right triangle, you can calculate the third with
the Pythagorean Theorem. So, if two pairs of sides are congruent, the third pair must also be
congruent.
By the same argument you could conclude that LL also guarantees congruence. Since LL means
that two pairs of legs are congruent, you know by the Pythagorean Theorem that the hypotenuses
are also congruent.
Thus, LL is also a special case of SSS.
You can verify this for yourself in the example below.
Thus, HL is just a special case of SSS congruence!
Are there more shortcut congruence theorems for right triangles?
Experiment with “hypotenuse-angle”. Request a hypotenuse length and an acute angle measure.
Then use those pieces to build two right triangles.
Are they guaranteed to be congruent?
Indeed, HA, or “hypotenuse-angle”, works.
In other words, if two right triangles have congruent hypotenuses and one pair of congruent
angles (in addition to the congruent right angles), the triangles are guaranteed to be congruent.
HA Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the
hypotenuse and corresponding acute angle of another right triangle, then the triangles are
congruent.
Given:
M and
O
Conclusion:
P are right angles.
R
MNO
Reason: HA
PQR
Now experiment with “leg-angle.”
Note that there are two possible legs to consider. One leg is included between the right angle and
the other given angle, and one leg is not.
First, experiment with the leg that is included between the two given angles. Are two triangles
built with these pieces always congruent?
What about the other “leg-angle” case, in which the leg is not included between the given
angles? Does this guarantee congruent triangles?
Experiment below.
You may have noticed that both cases of “leg-angle” guarantee congruence.
This is the fourth and final right triangle congruence theorem, the LA Congruence Theorem.
LA Theorem: If one leg and an acute angle of one right triangle are congruent to one leg and an
acute angle of another right triangle, then the two right triangles are congruent.
How do HA and LA relate to the congruence theorems for nonright triangles?
When you consider that the right angles are automatically congruent, you can see that they are
simplified versions of either ASA or AAS.
You have now discovered four congruence theorems for use with right triangles. They are
summarized in the table below.
Notice that each is really just a special case of an earlier congruence theorem.
In addition, each has only two letters. Can you see why, with right triangles, two is enough?
Well done. The hypotenuses are congruent, but that is not enough to ensure congruence. HA or HL
would work, but just H is not good enough.
True or False? Based on the information marked in the diagram,
ABC
DEF.
True
Very nice. You have two pairs of sides and their included angles congruent (SAS).
But this is also a right triangle, and the congruent sides are both legs, so you could also use LL.
Based only on the information marked in the diagram, which two congruence theorems could be
given as reasons why
ABC
DEF?
A. HL
B. LLC. HAD. LAE. ASAF. SAS
Fine work. If you do not look at these as right triangles, you have the AAS case, because the
congruent side is not included between the congruent angles.
When you do consider them as right triangles, you have hypotenuse-angle ("HA") congruence.
Based only on the information marked in the diagram, which two congruence theorems could be
given as reasons why
GHI
JKL?
A. HLB. LL
C. HAD. LAE. AASF. ASA
Excellent. As "non-right" triangles, this is the ASA case, because the congruent sides are
included between the congruent angles.
As right triangles, this is the leg-angle ("LA") case.
Based only on the information marked in the diagram, which two congruence theorems could be
given as reasons why
MNO
PQR?
D. LAE. AASF. ASA
A. HLB. LLC. HA
It's time to put the right triangle congruence theorems you've learned to work for you.
The next several pages present four useful new theorems about points, lines, and angles. A key to
each proof is congruent right triangles — even though the theorems themselves are not about
triangles at all!
Before getting to the proofs, be sure you recall that the distance from a point to a line is always
measured perpendicularly.
This is illustrated here.
The first theorem describes how any point on a perpendicular bisector is related to the segment it
bisects.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints
of the segment.
Experiment to confirm this for yourself. Once you have seen that it seems to be true, click
through the proof.
The converse of the Perpendicular Bisector Theorem is also true.
Perpendicular Bisector Converse: If a point is equidistant from the endpoints of a segment, then
it lies on the perpendicular bisector of that segment.
Experiment to convince yourself of this, then click through the sequence of buttons to see a proof.
Again, you will see that right triangle congruence is a key to the proof.
Just as a segment can be bisected by a line, so can an angle. The Angle Bisector Theorem relates
any point on the angle bisector to the sides of the angle.
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Experiment to confirm this theorem. Then click through the sequence of buttons to see a proof.
The converse of the Angle Bisector Theorem is true as well.
Angle Bisector Converse: If a point is in the interior of an angle and equidistant from the two
sides of the angle, then it lies on the bisector of the angle.
Experiment to confirm this for yourself. Then click through the sequence of buttons to see a proof.
Notice the use, once again, of a right triangle congruence theorem.
Nice job! The distance from a point to a line is always measured by the perpendicular segment. (This is
also always the shortest distance from the point to the line.)
What is the distance from point C to
?
A. B. 10.8
10.5
C. 14.6 D. can't be determined
Well done. Point S is equidistant from the sides of
So, if m
SED = 30 , then m
What is m
RED. Therefore, S is on the angle bisector.
RED = 60 .
RED?
A. 15 B. 17
D. 34
C. 30
E. 60
F. can't be determined
Outstanding! There is no guarantee that
because you do not know if
bisects
True or false? Based only on the given information, it is guaranteed that
Given:
.
.
ABC
False
Excellent. In this example, you know that point C is on the perpendicular bisector of
you can conclude that C is equidistant from A and B, or, in other words,
.
. Therefore,
True or false? Based only on the given information, it is guaranteed that
Given:
.
ABC
True
You have seen a great deal about congruence in right triangles in this section.
Four congruence theorems, given again in the table below, were presented and then used in
proofs.
You are now ready to extend this knowledge into the topic of similar right triangles.